p value statistics hypothesis testing

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the P -value is large, say more than \(\alpha\), then it is "likely."

If the P -value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
• Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to \(\alpha\). If the P -value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than \(\alpha\), do not reject the null hypothesis.

Example S.3.2.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than \(\alpha\) = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to \(\alpha\) = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than \(\alpha\) = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.

p-value Calculator

What is p-value, how do i calculate p-value from test statistic, how to interpret p-value, how to use the p-value calculator to find p-value from test statistic, how do i find p-value from z-score, how do i find p-value from t, p-value from chi-square score (χ² score), p-value from f-score.

Welcome to our p-value calculator! You will never again have to wonder how to find the p-value, as here you can determine the one-sided and two-sided p-values from test statistics, following all the most popular distributions: normal, t-Student, chi-squared, and Snedecor's F.

P-values appear all over science, yet many people find the concept a bit intimidating. Don't worry – in this article, we will explain not only what the p-value is but also how to interpret p-values correctly . Have you ever been curious about how to calculate the p-value by hand? We provide you with all the necessary formulae as well!

🙋 If you want to revise some basics from statistics, our normal distribution calculator is an excellent place to start.

Formally, the p-value is the probability that the test statistic will produce values at least as extreme as the value it produced for your sample . It is crucial to remember that this probability is calculated under the assumption that the null hypothesis H 0 is true !

More intuitively, p-value answers the question:

Assuming that I live in a world where the null hypothesis holds, how probable is it that, for another sample, the test I'm performing will generate a value at least as extreme as the one I observed for the sample I already have?

It is the alternative hypothesis that determines what "extreme" actually means , so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event, calculated under the assumption that H 0 is true:

Left-tailed test: p-value = Pr(S ≤ x | H 0 )

Right-tailed test: p-value = Pr(S ≥ x | H 0 )

Two-tailed test:

p-value = 2 × min{Pr(S ≤ x | H 0 ), Pr(S ≥ x | H 0 )}

(By min{a,b} , we denote the smaller number out of a and b .)

If the distribution of the test statistic under H 0 is symmetric about 0 , then: p-value = 2 × Pr(S ≥ |x| | H 0 )

or, equivalently: p-value = 2 × Pr(S ≤ -|x| | H 0 )

As a picture is worth a thousand words, let us illustrate these definitions. Here, we use the fact that the probability can be neatly depicted as the area under the density curve for a given distribution. We give two sets of pictures: one for a symmetric distribution and the other for a skewed (non-symmetric) distribution.

• Symmetric case: normal distribution:

• Non-symmetric case: chi-squared distribution:

In the last picture (two-tailed p-value for skewed distribution), the area of the left-hand side is equal to the area of the right-hand side.

To determine the p-value, you need to know the distribution of your test statistic under the assumption that the null hypothesis is true . Then, with the help of the cumulative distribution function ( cdf ) of this distribution, we can express the probability of the test statistics being at least as extreme as its value x for the sample:

Left-tailed test:

p-value = cdf(x) .

Right-tailed test:

p-value = 1 - cdf(x) .

p-value = 2 × min{cdf(x) , 1 - cdf(x)} .

If the distribution of the test statistic under H 0 is symmetric about 0 , then a two-sided p-value can be simplified to p-value = 2 × cdf(-|x|) , or, equivalently, as p-value = 2 - 2 × cdf(|x|) .

The probability distributions that are most widespread in hypothesis testing tend to have complicated cdf formulae, and finding the p-value by hand may not be possible. You'll likely need to resort to a computer or to a statistical table, where people have gathered approximate cdf values.

Well, you now know how to calculate the p-value, but… why do you need to calculate this number in the first place? In hypothesis testing, the p-value approach is an alternative to the critical value approach . Recall that the latter requires researchers to pre-set the significance level, α, which is the probability of rejecting the null hypothesis when it is true (so of type I error ). Once you have your p-value, you just need to compare it with any given α to quickly decide whether or not to reject the null hypothesis at that significance level, α. For details, check the next section, where we explain how to interpret p-values.

As we have mentioned above, the p-value is the answer to the following question:

What does that mean for you? Well, you've got two options:

• A high p-value means that your data is highly compatible with the null hypothesis; and
• A small p-value provides evidence against the null hypothesis , as it means that your result would be very improbable if the null hypothesis were true.

However, it may happen that the null hypothesis is true, but your sample is highly unusual! For example, imagine we studied the effect of a new drug and got a p-value of 0.03 . This means that in 3% of similar studies, random chance alone would still be able to produce the value of the test statistic that we obtained, or a value even more extreme, even if the drug had no effect at all!

The question "what is p-value" can also be answered as follows: p-value is the smallest level of significance at which the null hypothesis would be rejected. So, if you now want to make a decision on the null hypothesis at some significance level α , just compare your p-value with α :

• If p-value ≤ α , then you reject the null hypothesis and accept the alternative hypothesis; and
• If p-value ≥ α , then you don't have enough evidence to reject the null hypothesis.

Obviously, the fate of the null hypothesis depends on α . For instance, if the p-value was 0.03 , we would reject the null hypothesis at a significance level of 0.05 , but not at a level of 0.01 . That's why the significance level should be stated in advance and not adapted conveniently after the p-value has been established! A significance level of 0.05 is the most common value, but there's nothing magical about it. Here, you can see what too strong a faith in the 0.05 threshold can lead to. It's always best to report the p-value, and allow the reader to make their own conclusions.

Also, bear in mind that subject area expertise (and common reason) is crucial. Otherwise, mindlessly applying statistical principles, you can easily arrive at statistically significant, despite the conclusion being 100% untrue.

As our p-value calculator is here at your service, you no longer need to wonder how to find p-value from all those complicated test statistics! Here are the steps you need to follow:

Pick the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

Tell us the distribution of your test statistic under the null hypothesis: is it N(0,1), t-Student, chi-squared, or Snedecor's F? If you are unsure, check the sections below, as they are devoted to these distributions.

If needed, specify the degrees of freedom of the test statistic's distribution.

Enter the value of test statistic computed for your data sample.

Our calculator determines the p-value from the test statistic and provides the decision to be made about the null hypothesis. The standard significance level is 0.05 by default.

Go to the advanced mode if you need to increase the precision with which the calculations are performed or change the significance level .

In terms of the cumulative distribution function (cdf) of the standard normal distribution, which is traditionally denoted by Φ , the p-value is given by the following formulae:

Left-tailed z-test:

p-value = Φ(Z score )

Right-tailed z-test:

p-value = 1 - Φ(Z score )

Two-tailed z-test:

p-value = 2 × Φ(−|Z score |)

p-value = 2 - 2 × Φ(|Z score |)

🙋 To learn more about Z-tests, head to Omni's Z-test calculator .

We use the Z-score if the test statistic approximately follows the standard normal distribution N(0,1) . Thanks to the central limit theorem, you can count on the approximation if you have a large sample (say at least 50 data points) and treat your distribution as normal.

A Z-test most often refers to testing the population mean , or the difference between two population means, in particular between two proportions. You can also find Z-tests in maximum likelihood estimations.

The p-value from the t-score is given by the following formulae, in which cdf t,d stands for the cumulative distribution function of the t-Student distribution with d degrees of freedom:

Left-tailed t-test:

p-value = cdf t,d (t score )

Right-tailed t-test:

p-value = 1 - cdf t,d (t score )

Two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

p-value = 2 - 2 × cdf t,d (|t score |)

Use the t-score option if your test statistic follows the t-Student distribution . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails – the exact shape depends on the parameter called the degrees of freedom . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from the normal distribution N(0,1).

The most common t-tests are those for population means with an unknown population standard deviation, or for the difference between means of two populations , with either equal or unequal yet unknown population standard deviations. There's also a t-test for paired (dependent) samples .

🙋 To get more insights into t-statistics, we recommend using our t-test calculator .

Use the χ²-score option when performing a test in which the test statistic follows the χ²-distribution .

This distribution arises if, for example, you take the sum of squared variables, each following the normal distribution N(0,1). Remember to check the number of degrees of freedom of the χ²-distribution of your test statistic!

How to find the p-value from chi-square-score ? You can do it with the help of the following formulae, in which cdf χ²,d denotes the cumulative distribution function of the χ²-distribution with d degrees of freedom:

Left-tailed χ²-test:

p-value = cdf χ²,d (χ² score )

Right-tailed χ²-test:

p-value = 1 - cdf χ²,d (χ² score )

Remember that χ²-tests for goodness-of-fit and independence are right-tailed tests! (see below)

Two-tailed χ²-test:

p-value = 2 × min{cdf χ²,d (χ² score ), 1 - cdf χ²,d (χ² score )}

(By min{a,b} , we denote the smaller of the numbers a and b .)

The most popular tests which lead to a χ²-score are the following:

Testing whether the variance of normally distributed data has some pre-determined value. In this case, the test statistic has the χ²-distribution with n - 1 degrees of freedom, where n is the sample size. This can be a one-tailed or two-tailed test .

Goodness-of-fit test checks whether the empirical (sample) distribution agrees with some expected probability distribution. In this case, the test statistic follows the χ²-distribution with k - 1 degrees of freedom, where k is the number of classes into which the sample is divided. This is a right-tailed test .

Independence test is used to determine if there is a statistically significant relationship between two variables. In this case, its test statistic is based on the contingency table and follows the χ²-distribution with (r - 1)(c - 1) degrees of freedom, where r is the number of rows, and c is the number of columns in this contingency table. This also is a right-tailed test .

Finally, the F-score option should be used when you perform a test in which the test statistic follows the F-distribution , also known as the Fisher–Snedecor distribution. The exact shape of an F-distribution depends on two degrees of freedom .

To see where those degrees of freedom come from, consider the independent random variables X and Y , which both follow the χ²-distributions with d 1 and d 2 degrees of freedom, respectively. In that case, the ratio (X/d 1 )/(Y/d 2 ) follows the F-distribution, with (d 1 , d 2 ) -degrees of freedom. For this reason, the two parameters d 1 and d 2 are also called the numerator and denominator degrees of freedom .

The p-value from F-score is given by the following formulae, where we let cdf F,d1,d2 denote the cumulative distribution function of the F-distribution, with (d 1 , d 2 ) -degrees of freedom:

Left-tailed F-test:

p-value = cdf F,d1,d2 (F score )

Right-tailed F-test:

p-value = 1 - cdf F,d1,d2 (F score )

Two-tailed F-test:

p-value = 2 × min{cdf F,d1,d2 (F score ), 1 - cdf F,d1,d2 (F score )}

Below we list the most important tests that produce F-scores. All of them are right-tailed tests .

A test for the equality of variances in two normally distributed populations . Its test statistic follows the F-distribution with (n - 1, m - 1) -degrees of freedom, where n and m are the respective sample sizes.

ANOVA is used to test the equality of means in three or more groups that come from normally distributed populations with equal variances. We arrive at the F-distribution with (k - 1, n - k) -degrees of freedom, where k is the number of groups, and n is the total sample size (in all groups together).

A test for overall significance of regression analysis . The test statistic has an F-distribution with (k - 1, n - k) -degrees of freedom, where n is the sample size, and k is the number of variables (including the intercept).

With the presence of the linear relationship having been established in your data sample with the above test, you can calculate the coefficient of determination, R 2 , which indicates the strength of this relationship . You can do it by hand or use our coefficient of determination calculator .

A test to compare two nested regression models . The test statistic follows the F-distribution with (k 2 - k 1 , n - k 2 ) -degrees of freedom, where k 1 and k 2 are the numbers of variables in the smaller and bigger models, respectively, and n is the sample size.

You may notice that the F-test of an overall significance is a particular form of the F-test for comparing two nested models: it tests whether our model does significantly better than the model with no predictors (i.e., the intercept-only model).

Can p-value be negative?

No, the p-value cannot be negative. This is because probabilities cannot be negative, and the p-value is the probability of the test statistic satisfying certain conditions.

What does a high p-value mean?

A high p-value means that under the null hypothesis, there's a high probability that for another sample, the test statistic will generate a value at least as extreme as the one observed in the sample you already have. A high p-value doesn't allow you to reject the null hypothesis.

What does a low p-value mean?

A low p-value means that under the null hypothesis, there's little probability that for another sample, the test statistic will generate a value at least as extreme as the one observed for the sample you already have. A low p-value is evidence in favor of the alternative hypothesis – it allows you to reject the null hypothesis.

Books vs e-books

False positive, geometric distribution.

• Biology (100)
• Chemistry (100)
• Construction (144)
• Conversion (295)
• Ecology (30)
• Everyday life (262)
• Finance (570)
• Health (440)
• Physics (510)
• Sports (105)
• Statistics (182)
• Other (182)
• Discover Omni (40)

• Search Search Please fill out this field.

What Is P-Value?

Understanding p-value.

• P-Value in Hypothesis Testing

The Bottom Line

• Corporate Finance
• Financial Analysis

P-Value: What It Is, How to Calculate It, and Why It Matters

Yarilet Perez is an experienced multimedia journalist and fact-checker with a Master of Science in Journalism. She has worked in multiple cities covering breaking news, politics, education, and more. Her expertise is in personal finance and investing, and real estate.

In statistics, a p-value is a number that indicates how likely you are to obtain a value that is at least equal to or more than the actual observation if the null hypothesis is correct.

The p-value serves as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means stronger evidence in favor of the alternative hypothesis.

P-value is often used to promote credibility for studies or reports by government agencies. For example, the U.S. Census Bureau stipulates that any analysis with a p-value greater than 0.10 must be accompanied by a statement that the difference is not statistically different from zero. The Census Bureau also has standards in place stipulating which p-values are acceptable for various publications.

Key Takeaways

• A p-value is a statistical measurement used to validate a hypothesis against observed data.
• A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true.
• The lower the p-value, the greater the statistical significance of the observed difference.
• A p-value of 0.05 or lower is generally considered statistically significant.
• P-value can serve as an alternative to—or in addition to—preselected confidence levels for hypothesis testing.

Jessica Olah / Investopedia

P-values are usually found using p-value tables or spreadsheets/statistical software. These calculations are based on the assumed or known probability distribution of the specific statistic tested. P-values are calculated from the deviation between the observed value and a chosen reference value, given the probability distribution of the statistic, with a greater difference between the two values corresponding to a lower p-value.

Mathematically, the p-value is calculated using integral calculus from the area under the probability distribution curve for all values of statistics that are at least as far from the reference value as the observed value is, relative to the total area under the probability distribution curve.

The calculation for a p-value varies based on the type of test performed. The three test types describe the location on the probability distribution curve: lower-tailed test, upper-tailed test, or two-tailed test .

In a nutshell, the greater the difference between two observed values, the less likely it is that the difference is due to simple random chance, and this is reflected by a lower p-value.

The P-Value Approach to Hypothesis Testing

The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. The null hypothesis, also known as the conjecture, is the initial claim about a population (or data-generating process). The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture.

In practice, the significance level is stated in advance to determine how small the p-value must be to reject the null hypothesis. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests. P-values provide a solution to this problem.

Even a low p-value is not necessarily proof of statistical significance, since there is still a possibility that the observed data are the result of chance. Only repeated experiments or studies can confirm if a relationship is statistically significant.

For example, suppose a study comparing returns from two particular assets was undertaken by different researchers who used the same data but different significance levels. The researchers might come to opposite conclusions regarding whether the assets differ.

If one researcher used a confidence level of 90% and the other required a confidence level of 95% to reject the null hypothesis, and if the p-value of the observed difference between the two returns was 0.08 (corresponding to a confidence level of 92%), then the first researcher would find that the two assets have a difference that is statistically significant , while the second would find no statistically significant difference between the returns.

To avoid this problem, the researchers could report the p-value of the hypothesis test and allow readers to interpret the statistical significance themselves. This is called a p-value approach to hypothesis testing. Independent observers could note the p-value and decide for themselves whether that represents a statistically significant difference or not.

Example of P-Value

An investor claims that their investment portfolio’s performance is equivalent to that of the Standard & Poor’s (S&P) 500 Index . To determine this, the investor conducts a two-tailed test.

The null hypothesis states that the portfolio’s returns are equivalent to the S&P 500’s returns over a specified period, while the alternative hypothesis states that the portfolio’s returns and the S&P 500’s returns are not equivalent—if the investor conducted a one-tailed test , the alternative hypothesis would state that the portfolio’s returns are either less than or greater than the S&P 500’s returns.

The p-value hypothesis test does not necessarily make use of a preselected confidence level at which the investor should reset the null hypothesis that the returns are equivalent. Instead, it provides a measure of how much evidence there is to reject the null hypothesis. The smaller the p-value, the greater the evidence against the null hypothesis.

Thus, if the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the investor can confidently conclude that the portfolio’s returns and the S&P 500’s returns are not equivalent.

Although this does not provide an exact threshold as to when the investor should accept or reject the null hypothesis, it does have another very practical advantage. P-value hypothesis testing offers a direct way to compare the relative confidence that the investor can have when choosing among multiple different types of investments or portfolios relative to a benchmark such as the S&P 500.

For example, for two portfolios, A and B, whose performance differs from the S&P 500 with p-values of 0.10 and 0.01, respectively, the investor can be much more confident that portfolio B, with a lower p-value, will actually show consistently different results.

Is a 0.05 P-Value Significant?

A p-value less than 0.05 is typically considered to be statistically significant, in which case the null hypothesis should be rejected. A p-value greater than 0.05 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected.

What Does a P-Value of 0.001 Mean?

A p-value of 0.001 indicates that if the null hypothesis tested were indeed true, then there would be a one-in-1,000 chance of observing results at least as extreme. This leads the observer to reject the null hypothesis because either a highly rare data result has been observed or the null hypothesis is incorrect.

How Can You Use P-Value to Compare 2 Different Results of a Hypothesis Test?

If you have two different results, one with a p-value of 0.04 and one with a p-value of 0.06, the result with a p-value of 0.04 will be considered more statistically significant than the p-value of 0.06. Beyond this simplified example, you could compare a 0.04 p-value to a 0.001 p-value. Both are statistically significant, but the 0.001 example provides an even stronger case against the null hypothesis than the 0.04.

The p-value is used to measure the significance of observational data. When researchers identify an apparent relationship between two variables, there is always a possibility that this correlation might be a coincidence. A p-value calculation helps determine if the observed relationship could arise as a result of chance.

U.S. Census Bureau. “ Statistical Quality Standard E1: Analyzing Data .”

• Terms of Service
• Editorial Policy
• Privacy Policy
• Your Privacy Choices

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

9.5: The p value of a test

• Last updated
• Save as PDF
• Page ID 36128

• Danielle Navarro
• University of New South Wales

In one sense, our hypothesis test is complete; we’ve constructed a test statistic, figured out its sampling distribution if the null hypothesis is true, and then constructed the critical region for the test. Nevertheless, I’ve actually omitted the most important number of all: the p value . It is to this topic that we now turn. There are two somewhat different ways of interpreting a p value, one proposed by Sir Ronald Fisher and the other by Jerzy Neyman. Both versions are legitimate, though they reflect very different ways of thinking about hypothesis tests. Most introductory textbooks tend to give Fisher’s version only, but I think that’s a bit of a shame. To my mind, Neyman’s version is cleaner, and actually better reflects the logic of the null hypothesis test. You might disagree though, so I’ve included both. I’ll start with Neyman’s version…

softer view of decision making

One problem with the hypothesis testing procedure that I’ve described is that it makes no distinction at all between a result this “barely significant” and those that are “highly significant”. For instance, in my ESP study the data I obtained only just fell inside the critical region - so I did get a significant effect, but was a pretty near thing. In contrast, suppose that I’d run a study in which X=97 out of my N=100 participants got the answer right. This would obviously be significant too, but my a much larger margin; there’s really no ambiguity about this at all. The procedure that I described makes no distinction between the two. If I adopt the standard convention of allowing α=.05 as my acceptable Type I error rate, then both of these are significant results.

This is where the p value comes in handy. To understand how it works, let’s suppose that we ran lots of hypothesis tests on the same data set: but with a different value of α in each case. When we do that for my original ESP data, what we’d get is something like this

When we test ESP data (X=62 successes out of N=100 observations) using α levels of .03 and above, we’d always find ourselves rejecting the null hypothesis. For α levels of .02 and below, we always end up retaining the null hypothesis. Therefore, somewhere between .02 and .03 there must be a smallest value of α that would allow us to reject the null hypothesis for this data. This is the p value; as it turns out the ESP data has p=.021. In short:

p is defined to be the smallest Type I error rate (α) that you have to be willing to tolerate if you want to reject the null hypothesis.

If it turns out that p describes an error rate that you find intolerable, then you must retain the null. If you’re comfortable with an error rate equal to p, then it’s okay to reject the null hypothesis in favour of your preferred alternative.

In effect, p is a summary of all the possible hypothesis tests that you could have run, taken across all possible α values. And as a consequence it has the effect of “softening” our decision process. For those tests in which p≤α you would have rejected the null hypothesis, whereas for those tests in which p>α you would have retained the null. In my ESP study I obtained X=62, and as a consequence I’ve ended up with p=.021. So the error rate I have to tolerate is 2.1%. In contrast, suppose my experiment had yielded X=97. What happens to my p value now? This time it’s shrunk to p=1.36×10−25, which is a tiny, tiny 163 Type I error rate. For this second case I would be able to reject the null hypothesis with a lot more confidence, because I only have to be “willing” to tolerate a type I error rate of about 1 in 10 trillion trillion in order to justify my decision to reject.

probability of extreme data

The second definition of the p-value comes from Sir Ronald Fisher, and it’s actually this one that you tend to see in most introductory statistics textbooks. Notice how, when I constructed the critical region, it corresponded to the tails (i.e., extreme values) of the sampling distribution? That’s not a coincidence: almost all “good” tests have this characteristic (good in the sense of minimising our type II error rate, β). The reason for that is that a good critical region almost always corresponds to those values of the test statistic that are least likely to be observed if the null hypothesis is true. If this rule is true, then we can define the p-value as the probability that we would have observed a test statistic that is at least as extreme as the one we actually did get. In other words, if the data are extremely implausible according to the null hypothesis, then the null hypothesis is probably wrong.

common mistake

Okay, so you can see that there are two rather different but legitimate ways to interpret the p value, one based on Neyman’s approach to hypothesis testing and the other based on Fisher’s. Unfortunately, there is a third explanation that people sometimes give, especially when they’re first learning statistics, and it is absolutely and completely wrong . This mistaken approach is to refer to the p value as “the probability that the null hypothesis is true”. It’s an intuitively appealing way to think, but it’s wrong in two key respects: (1) null hypothesis testing is a frequentist tool, and the frequentist approach to probability does not allow you to assign probabilities to the null hypothesis… according to this view of probability, the null hypothesis is either true or it is not; it cannot have a “5% chance” of being true. (2) even within the Bayesian approach, which does let you assign probabilities to hypotheses, the p value would not correspond to the probability that the null is true; this interpretation is entirely inconsistent with the mathematics of how the p value is calculated. Put bluntly, despite the intuitive appeal of thinking this way, there is no justification for interpreting a p value this way. Never do it.

P-Value And Statistical Significance: What It Is & Why It Matters

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.

Hypothesis testing

When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.

The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.

The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.

The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).

What a p-value tells you

A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).

The level of statistical significance is often expressed as a p-value between 0 and 1.

The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.

Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.

Example: Test Statistic and p-Value

Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.

P-value interpretation

The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.

A p-value less than or equal to your significance level (typically ≤ 0.05) is statistically significant.

A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.

This suggests the effect under study likely represents a real relationship rather than just random chance.

For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05. 

It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).

Therefore, we reject the null hypothesis and accept the alternative hypothesis.

Example: Statistical Significance

Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.

What does a p-value of 0.001 mean?

A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.

Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.

Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.

A p-value more than the significance level (typically p > 0.05) is not statistically significant and indicates strong evidence for the null hypothesis.

This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.

Note : when the p-value is above your threshold of significance,  it does not mean that there is a 95% probability that the alternative hypothesis is true.

One-Tailed Test

Two-Tailed Test

How do you calculate the p-value ?

Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.

Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.

These tables help you understand how often you would expect to see your test statistic under the null hypothesis.

Understanding the Statistical Test:

Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.

For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.

Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.

This factor is particularly important to consider when comparing results across different analyses.

Example: Choosing a Statistical Test

If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.

How to report

A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).

Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).

Example: Reporting the results

In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD  = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).

The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:

“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.

The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)

• Do not use 0 before the decimal point for the statistical value p as it cannot equal 1. In other words, write p = .001 instead of p = 0.001.
• Please pay attention to issues of italics ( p is always italicized) and spacing (either side of the = sign).
• p = .000 (as outputted by some statistical packages such as SPSS) is impossible and should be written as p < .001.
• The opposite of significant is “nonsignificant,” not “insignificant.”

Why is the p -value not enough?

A lower p-value  is sometimes interpreted as meaning there is a stronger relationship between two variables.

However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).

To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .

When do you reject the null hypothesis?

In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.

Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.

The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.

What does p-value of 0.05 mean?

If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Are all p-values below 0.05 considered statistically significant?

No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.

A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.

Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.

How does sample size affect the interpretation of p-values?

Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.

With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.

Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.

Can a non-significant p-value indicate that there is no effect or difference in the data?

No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.

There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.

Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.

Can P values be exactly zero?

While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001

Further Information

• P-values and significance tests (Kahn Academy)
• Hypothesis testing and p-values (Kahn Academy)
• Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “ p “< 0.05”.
• Criticism of using the “ p “< 0.05”.
• Publication manual of the American Psychological Association
• Statistics for Psychology Book Download

Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply.  BMJ: British Medical Journal ,  309 (6958), 874.

Goodman, S. N., & Royall, R. (1988). Evidence and scientific research.  American Journal of Public Health ,  78 (12), 1568-1574.

Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In  Seminars in hematology  (Vol. 45, No. 3, pp. 135-140). WB Saunders.

Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value.  Epidemiology (Cambridge, Mass.) ,  9 (1), 7-8.

Statistics Made Easy

An Explanation of P-Values and Statistical Significance

In statistics, p-values are commonly used in hypothesis testing for t-tests, chi-square tests, regression analysis, ANOVAs, and a variety of other statistical methods.

Despite being so common, people often interpret p-values incorrectly, which can lead to errors when interpreting the findings from an analysis or a study. 

This post explains how to understand and interpret p-values in a clear, practical way.

Hypothesis Testing

To understand p-values, we first need to understand the concept of hypothesis testing .

A  hypothesis test  is a formal statistical test we use to reject or fail to reject some hypothesis. For example, we may hypothesize that a new drug, method, or procedure provides some benefit over a current drug, method, or procedure. 

To test this, we can conduct a hypothesis test where we use a null and alternative hypothesis:

Null hypothesis – There is no effect or difference between the new method and the old method.

Alternative hypothesis – There does exist some effect or difference between the new method and the old method.

A p-value indicates how believable the null hypothesis is, given the sample data. Specifically, assuming the null hypothesis is true, the p-value tells us the probability of obtaining an effect at least as large as the one we actually observed in the sample data. 

If the p-value of a hypothesis test is sufficiently low, we can reject the null hypothesis. Specifically, when we conduct a hypothesis test, we must choose a significance level at the outset. Common choices for significance levels are 0.01, 0.05, and 0.10.

If the p-values is  less than  our significance level, then we can reject the null hypothesis.

Otherwise, if the p-value is  equal to or greater than  our significance level, then we fail to reject the null hypothesis. 

How to Interpret a P-Value

The textbook definition of a p-value is:

A p-value is the probability of observing a sample statistic that is at least as extreme as your sample statistic, given that the null hypothesis is true.

For example, suppose a factory claims that they produce tires that have a mean weight of 200 pounds. An auditor hypothesizes that the true mean weight of tires produced at this factory is different from 200 pounds so he runs a hypothesis test and finds that the p-value of the test is 0.04. Here is how to interpret this p-value:

If the factory does indeed produce tires that have a mean weight of 200 pounds, then 4% of all audits will obtain the effect observed in the sample, or larger, because of random sample error. This tells us that obtaining the sample data that the auditor did would be pretty rare if indeed the factory produced tires that have a mean weight of 200 pounds. 

Depending on the significance level used in this hypothesis test, the auditor would likely reject the null hypothesis that the true mean weight of tires produced at this factory is indeed 200 pounds. The sample data that he obtained from the audit is not very consistent with the null hypothesis.

How Not  to Interpret a P-Value

The biggest misconception about p-values is that they are equivalent to the probability of making a mistake by rejecting a true null hypothesis (known as a Type I error).

There are two primary reasons that p-values can’t be the error rate:

1.  P-values are calculated based on the assumption that the null hypothesis is true and that the difference between the sample data and the null hypothesis is simple caused by random chance. Thus, p-values can’t tell you the probability that the null is true or false since it is 100% true based on the perspective of the calculations.

2. Although a low p-value indicates that your sample data are unlikely assuming the null is true, a p-value still can’t tell you which of the following cases is more likely:

• The null is false
• The null is true but you obtained an odd sample

In regards to the previous example, here is a correct and incorrect way to interpret the p-value:

• Correct Interpretation: Assuming the factory does produce tires with a mean weight of 200 pounds, you would obtain the observed difference that you  did  obtain in your sample or a more extreme difference in 4% of audits due to random sampling error.
• Incorrect Interpretation: If you reject the null hypothesis, there is a 4% chance that you are making a mistake.

Examples of Interpreting P-Values

The following examples illustrate correct ways to interpret p-values in the context of hypothesis testing.

A phone company claims that 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 85% responded yes. The p-value associated with this sample data turned out to be 0.018.

Correct interpretation of p-value:  Assuming that 90% of the customers actually are satisfied with their service, the researcher would obtain the observed difference that he  did  obtain in his sample or a more extreme difference in 1.8% of audits due to random sampling error.

A company invents a new battery for phones. The company claims that this new battery will work for at least 10 minutes longer than the old battery. To test this claim, a researcher takes a simple random sample of 80 new batteries and 80 old batteries. The new batteries run for an average of 120 minutes with a standard deviation of 12 minutes and the old batteries run for an average of 115 minutes with a standard deviation of 15 minutes. The p-value that results from the test for a difference in population means is 0.011.

Correct interpretation of p-value:  Assuming that the new battery works for the same amount of time or less than the old battery, the researcher would obtain the observed difference or a more extreme difference in 1.1% of studies due to random sampling error.

Hey there. My name is Zach Bobbitt. I have a Master of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Table of Contents

What is p-value , p value vs alpha level, p values and critical values, how is p-value calculated, p-value in hypothesis testing, p-values and statistical significance, reporting p-values, our learners also ask, what is p-value in statistical hypothesis.

Few statistical estimates are as significant as the p-value. The p-value or probability value is a number, calculated from a statistical test , that describes how likely your results would have occurred if the null hypothesis were true. A P-value less than 0.5 is statistically significant, while a value higher than 0.5 indicates the null hypothesis is true; hence it is not statistically significant. So, what is P-Value exactly, and why is it so important?

In statistical hypothesis testing , P-Value or probability value can be defined as the measure of the probability that a real-valued test statistic is at least as extreme as the value actually obtained. P-value shows how likely it is that your set of observations could have occurred under the null hypothesis. P-Values are used in statistical hypothesis testing to determine whether to reject the null hypothesis. The smaller the p-value, the stronger the likelihood that you should reject the null hypothesis. 

Your Data Analytics Career is Around The Corner!

P-values are expressed as decimals and can be converted into percentage. For example, a p-value of 0.0237 is 2.37%, which means there's a 2.37% chance of your results being random or having happened by chance. The smaller the P-value, the more significant your results are. 

In a hypothesis test, you can compare the p value from your test with the alpha level selected while running the test. Now, let’s try to understand what is P-Value vs Alpha level.    

A P-value indicates the probability of getting an effect no less than that actually observed in the sample data.

An alpha level will tell you the probability of wrongly rejecting a true null hypothesis. The level is selected by the researcher and obtained by subtracting your confidence level from 100%. For instance, if you are 95% confident in your research, the alpha level will be 5% (0.05).

When you run the hypothesis test, if you get:

• A small p value (<=0.05), you should reject the null hypothesis
• A large p value (>0.05), you should not reject the null hypothesis

In addition to the P-value, you can use other values given by your test to determine if your null hypothesis is true. 

For example, if you run an F-test to compare two variances in Excel, you will obtain a p-value, an f-critical value, and a f-value. Compare the f-value with f-critical value. If f-critical value is lower, you should reject the null hypothesis. 

P-Values are usually calculated using p-value tables or spreadsheets, or calculated automatically using statistical software like R, SPSS, etc. 

Depending on the test statistic and degrees of freedom (subtracting no. of independent variables from no. of observations) of your test, you can find out from the tables how frequently you can expect the test statistic to be under the null hypothesis. 

How to calculate P-value depends on which statistical test you’re using to test your hypothesis.  

• Every statistical test uses different assumptions and generates different statistics. Select the test method that best suits your data and matches the effect or relationship being tested.
• The number of independent variables included in your test determines how big or small the test statistic should be in order to generate the same p-value. 

Regardless of what statistical test you are using, the p-value will always denote the same thing – how frequently you can expect to get a test statistic as extreme or even more extreme than the one given by your test. 

In the P-Value approach to hypothesis testing, a calculated probability is used to decide if there’s evidence to reject the null hypothesis, also known as the conjecture. The conjecture is the initial claim about a data population, while the alternative hypothesis ascertains if the observed population parameter differs from the population parameter value according to the conjecture. 

Effectively, the significance level is declared in advance to determine how small the P-value needs to be such that the null hypothesis is rejected.  The levels of significance vary from one researcher to another; so it can get difficult for readers to compare results from two different tests. That is when P-value makes things easier. 

Readers could interpret the statistical significance by referring to the reported P-value of the hypothesis test. This is known as the P-value approach to hypothesis testing. Using this, readers could decide for themselves whether the p value represents a statistically significant difference.  

The level of statistical significance is usually represented as a P-value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis. 

• A P-Value < or = 0.05 is considered statistically significant. It denotes strong evidence against the null hypothesis, since there is below 5% probability of the null being correct. So, we reject the null hypothesis and accept the alternative hypothesis.
• But if P-Value is lower than your threshold of significance, though the null hypothesis can be rejected, it does not mean that there is 95% probability of the alternative hypothesis being true. 
• A P-Value >0.05 is not statistically significant. It denotes strong evidence for the null hypothesis being true. Thus, we retain the null hypothesis and reject the alternative hypothesis. We cannot accept null hypothesis; we can only reject or not reject it. 

A statistically significant result does not prove a research hypothesis to be correct. Instead, it provides support for or provides evidence for the hypothesis. 

• You should report exact P-Values upto two or three decimal places. 
• For P-values less than .001, report as p < .001. 
• Do not use 0 before the decimal point as it cannot equal1. Write p = .001, and not p = 0.001
• Make sure p is always italicized and there is space on either side of the = sign. 
• It is impossible to get P = .000, and should be written as p < .001

An investor says that the performance of their investment portfolio is equivalent to that of the Standard & Poor’s (S&P) 500 Index. He performs a two-tailed test to determine this. 

The null hypothesis here says that the portfolio’s returns are equivalent to the returns of S&P 500, while the alternative hypothesis says that the returns of the portfolio and the returns of the S&P 500 are not equivalent.  

The p-value hypothesis test gives a measure of how much evidence is present to reject the null hypothesis. The smaller the p value, the higher the evidence against null hypothesis. 

Therefore, if the investor gets a P value of .001, it indicates strong evidence against null hypothesis. So he confidently deduces that the portfolio’s returns and the S&P 500’s returns are not equivalent.

1. What does P-value mean?

P-Value or probability value is a number that denotes the likelihood of your data having occurred under the null hypothesis of your statistical test. 

2. What does p 0.05 mean?

A P-value less than 0.05 is deemed to be statistically significant, meaning the null hypothesis should be rejected in such a case. A P-Value greater than 0.05 is not considered to be statistically significant, meaning the null hypothesis should not be rejected. 

3. What is P-value and how is it calculated?

The p-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.  

P-values are usually automatically calculated using statistical software. They can also be calculated using p-value tables for the relevant statistical test. P values are calculated based on the null distribution of the test statistic. In case the test statistic is far from the mean of the null distribution, the p-value obtained is small. It indicates that the test statistic is unlikely to have occurred under the null hypothesis. 

4. What is p-value in research?

P values are used in hypothesis testing to help determine whether the null hypothesis should be rejected. It plays a major role when results of research are discussed. Hypothesis testing is a statistical methodology frequently used in medical and clinical research studies. 

5. Why is the p-value significant?

Statistical significance is a term that researchers use to say that it is not likely that their observations could have occurred if the null hypothesis were true. The level of statistical significance is usually represented as a P-value or probability value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis. 

6. What is null hypothesis and what is p-value?

A null hypothesis is a kind of statistical hypothesis that suggests that there is no statistical significance in a set of given observations. It says there is no relationship between your variables.   

P-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.   

P-Value is used to determine the significance of observational data. Whenever researchers notice an apparent relation between two variables, a P-Value calculation helps ascertain if the observed relationship happened as a result of chance. Learn more about statistical analysis and data analytics and fast track your career with our Professional Certificate Program In Data Analytics .  

Data Science & Business Analytics Courses Duration and Fees

Data Science & Business Analytics programs typically range from a few weeks to several months, with fees varying based on program and institution.

Recommended Reads

Free eBook: Top Programming Languages For A Data Scientist

The Difference Between Data Mining and Statistics

A Comprehensive Look at Percentile in Statistics

Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer

Process Capability Analysis: Minitab with Statistics Training

A Complete Guide on Hypothesis Testing in Statistics

Get Affiliated Certifications with Live Class programs

Post graduate program in data analytics.

• Post Graduate Program certificate and Alumni Association membership
• Exclusive hackathons and Ask me Anything sessions by IBM

Data Analyst

• Industry-recognized Data Analyst Master’s certificate from Simplilearn
• Dedicated live sessions by faculty of industry experts
• PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.

• Machine Learning Tutorial
• Data Analysis Tutorial
• Python - Data visualization tutorial
• Machine Learning Projects
• Machine Learning Interview Questions
• Machine Learning Mathematics
• Deep Learning Tutorial
• Deep Learning Project
• Deep Learning Interview Questions
• Computer Vision Tutorial
• Computer Vision Projects
• NLP Project
• NLP Interview Questions
• Statistics with Python
• 100 Days of Machine Learning
• Data Analysis with Python

Introduction to Data Analysis

• What is Data Analysis?
• Data Analytics and its type
• How to Install Numpy on Windows?
• How to Install Pandas in Python?
• How to Install Matplotlib on python?
• How to Install Python Tensorflow in Windows?

Data Analysis Libraries

• Pandas Tutorial
• NumPy Tutorial - Python Library
• Data Analysis with SciPy
• Introduction to TensorFlow

Data Visulization Libraries

• Matplotlib Tutorial
• Python Seaborn Tutorial
• Plotly tutorial
• Introduction to Bokeh in Python

Exploratory Data Analysis (EDA)

• Univariate, Bivariate and Multivariate data and its analysis
• Measures of Central Tendency in Statistics
• Measures of spread - Range, Variance, and Standard Deviation
• Interquartile Range and Quartile Deviation using NumPy and SciPy
• Anova Formula
• Skewness of Statistical Data
• How to Calculate Skewness and Kurtosis in Python?
• Difference Between Skewness and Kurtosis
• Histogram | Meaning, Example, Types and Steps to Draw
• Interpretations of Histogram
• Quantile Quantile plots
• What is Univariate, Bivariate & Multivariate Analysis in Data Visualisation?
• Using pandas crosstab to create a bar plot
• Exploring Correlation in Python
• Mathematics | Covariance and Correlation
• Factor Analysis | Data Analysis
• Data Mining - Cluster Analysis
• MANOVA Test in R Programming
• Python - Central Limit Theorem
• Probability Distribution Function
• Probability Density Estimation & Maximum Likelihood Estimation
• Exponential Distribution in R Programming - dexp(), pexp(), qexp(), and rexp() Functions
• Mathematics | Probability Distributions Set 4 (Binomial Distribution)
• Poisson Distribution - Definition, Formula, Table and Examples

P-Value: Comprehensive Guide to Understand, Apply, and Interpret

• Z-Score in Statistics
• How to Calculate Point Estimates in R?
• Confidence Interval
• Chi-square test in Machine Learning
• Understanding Hypothesis Testing

Data Preprocessing

• ML | Data Preprocessing in Python
• ML | Overview of Data Cleaning
• ML | Handling Missing Values
• Detect and Remove the Outliers using Python

Data Transformation

• Data Normalization Machine Learning
• Sampling distribution Using Python

Time Series Data Analysis

• Data Mining - Time-Series, Symbolic and Biological Sequences Data
• Basic DateTime Operations in Python
• Time Series Analysis & Visualization in Python
• How to deal with missing values in a Timeseries in Python?
• How to calculate MOVING AVERAGE in a Pandas DataFrame?
• What is a trend in time series?
• How to Perform an Augmented Dickey-Fuller Test in R
• AutoCorrelation

Case Studies and Projects

• Top 8 Free Dataset Sources to Use for Data Science Projects
• Step by Step Predictive Analysis - Machine Learning
• 6 Tips for Creating Effective Data Visualizations

A p-value is a statistical metric used to assess a hypothesis by comparing it with observed data.

This article delves into the concept of p-value, its calculation, interpretation, and significance. It also explores the factors that influence p-value and highlights its limitations.

Table of Content

• What is P-value?

How P-value is calculated?

How to interpret p-value, p-value in hypothesis testing, implementing p-value in python, applications of p-value, what is the p-value.

The p-value, or probability value, is a statistical measure used in hypothesis testing to assess the strength of evidence against a null hypothesis. It represents the probability of obtaining results as extreme as, or more extreme than, the observed results under the assumption that the null hypothesis is true.

In simpler words, it is used to reject or support the null hypothesis during hypothesis testing. In data science, it gives valuable insights on the statistical significance of an independent variable in predicting the dependent variable. 

Calculating the p-value typically involves the following steps:

• Formulate the Null Hypothesis (H0) : Clearly state the null hypothesis, which typically states that there is no significant relationship or effect between the variables.
• Choose an Alternative Hypothesis (H1) : Define the alternative hypothesis, which proposes the existence of a significant relationship or effect between the variables.
• Determine the Test Statistic : Calculate the test statistic, which is a measure of the discrepancy between the observed data and the expected values under the null hypothesis. The choice of test statistic depends on the type of data and the specific research question.
• Identify the Distribution of the Test Statistic : Determine the appropriate sampling distribution for the test statistic under the null hypothesis. This distribution represents the expected values of the test statistic if the null hypothesis is true.
• Calculate the Critical-value : Based on the observed test statistic and the sampling distribution, find the probability of obtaining the observed test statistic or a more extreme one, assuming the null hypothesis is true.
• Interpret the results: Compare the critical-value with t-statistic. If the t-statistic is larger than the critical value, it provides evidence to reject the null hypothesis, and vice-versa.

Its interpretation depends on the specific test and the context of the analysis. Several popular methods for calculating test statistics that are utilized in p-value calculations.

In general, a small p-value indicates that the observed data is unlikely to have occurred by random chance alone, which leads to the rejection of the null hypothesis. However, it’s crucial to choose the appropriate test based on the nature of the data and the research question, as well as to interpret the p-value in the context of the specific test being used.

The table given below shows the importance of p-value and shows the various kinds of errors that occur during hypothesis testing.

Type I error: Incorrect rejection of the null hypothesis. It is denoted by α (significance level). Type II error: Incorrect acceptance of the null hypothesis. It is denoted by β (power level)

Let’s consider an example to illustrate the process of calculating a p-value for Two Sample T-Test:

A researcher wants to investigate whether there is a significant difference in mean height between males and females in a population of university students.

Suppose we have the following data:

Starting with interpreting the process of calculating p-value

Step 1 : Formulate the Null Hypothesis (H0):

H0: There is no significant difference in mean height between males and females.

Step 2 : Choose an Alternative Hypothesis (H1):

H1: There is a significant difference in mean height between males and females.

Step 3 : Determine the Test Statistic:

The appropriate test statistic for this scenario is the two-sample t-test, which compares the means of two independent groups.

The t-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

• s1 = First sample’s standard deviation
• s2 = Second sample’s standard deviation
• n1 = First sample’s sample size
• n2 = Second sample’s sample size

So, the calculated two-sample t-test statistic (t) is approximately 5.13.

Step 4 : Identify the Distribution of the Test Statistic:

The t-distribution is used for the two-sample t-test . The degrees of freedom for the t-distribution are determined by the sample sizes of the two groups.

 The t-distribution is a probability distribution with tails that are thicker than those of the normal distribution.

• where, n1 is total number of values for 1st category.
• n2 is total number of values for 2nd category.

The degrees of freedom (63) represent the variability available in the data to estimate the population parameters. In the context of the two-sample t-test, higher degrees of freedom provide a more precise estimate of the population variance, influencing the shape and characteristics of the t-distribution.

T-Statistic

The t-distribution is symmetric and bell-shaped, similar to the normal distribution. As the degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution. Practically, it affects the critical values used to determine statistical significance and confidence intervals.

Step 5 : Calculate Critical Value.

To find the critical t-value with a t-statistic of 5.13 and 63 degrees of freedom, we can either consult a t-table or use statistical software.

We can use scipy.stats module in Python to find the critical t-value using below code.

Comparing with T-Statistic:

The larger t-statistic suggests that the observed difference between the sample means is unlikely to have occurred by random chance alone. Therefore, we reject the null hypothesis.

• p ≤ (α = 0.05) : Reject the null hypothesis. There is sufficient evidence to conclude that the observed effect or relationship is statistically significant, meaning it is unlikely to have occurred by chance alone.
• p > (α = 0.05) : reject alternate hypothesis (or accept null hypothesis). The observed effect or relationship does not provide enough evidence to reject the null hypothesis. This does not necessarily mean there is no effect; it simply means the sample data does not provide strong enough evidence to rule out the possibility that the effect is due to chance.

In case the significance level is not specified, consider the below general inferences while interpreting your results. 

• If p > .10: not significant
• If p ≤ .10: slightly significant
• If p ≤ .05: significant
• If p ≤ .001: highly significant

Graphically, the p-value is located at the tails of any confidence interval. [As shown in fig 1]

Fig 1: Graphical Representation 

What influences p-value?

The p-value in hypothesis testing is influenced by several factors:

• Sample Size : Larger sample sizes tend to yield smaller p-values, increasing the likelihood of detecting significant effects.
• Effect Size: A larger effect size results in smaller p-values, making it easier to detect a significant relationship.
• Variability in the Data : Greater variability often leads to larger p-values, making it harder to identify significant effects.
• Significance Level : A lower chosen significance level increases the threshold for considering p-values as significant.
• Choice of Test: Different statistical tests may yield different p-values for the same data.
• Assumptions of the Test : Violations of test assumptions can impact p-values.

Understanding these factors is crucial for interpreting p-values accurately and making informed decisions in hypothesis testing.

Significance of P-value

• The p-value provides a quantitative measure of the strength of the evidence against the null hypothesis.
• Decision-Making in Hypothesis Testing
• P-value serves as a guide for interpreting the results of a statistical test. A small p-value suggests that the observed effect or relationship is statistically significant, but it does not necessarily mean that it is practically or clinically meaningful.

Limitations of P-value

• The p-value is not a direct measure of the effect size, which represents the magnitude of the observed relationship or difference between variables. A small p-value does not necessarily mean that the effect size is large or practically meaningful.
• Influenced by Various Factors

The p-value is a crucial concept in statistical hypothesis testing, serving as a guide for making decisions about the significance of the observed relationship or effect between variables.

Let’s consider a scenario where a tutor believes that the average exam score of their students is equal to the national average (85). The tutor collects a sample of exam scores from their students and performs a one-sample t-test to compare it to the population mean (85).

• The code performs a one-sample t-test to compare the mean of a sample data set to a hypothesized population mean.
• It utilizes the scipy.stats library to calculate the t-statistic and p-value. SciPy is a Python library that provides efficient numerical routines for scientific computing.
• The p-value is compared to a significance level (alpha) to determine whether to reject the null hypothesis.

Since, 0.7059>0.05 , we would conclude to fail to reject the null hypothesis. This means that, based on the sample data, there isn’t enough evidence to claim a significant difference in the exam scores of the tutor’s students compared to the national average. The tutor would accept the null hypothesis, suggesting that the average exam score of their students is statistically consistent with the national average.

• During Forward and Backward propagation: When fitting a model (say a Multiple Linear Regression model), we use the p-value in order to find the most significant variables that contribute significantly in predicting the output.
• Effects of various drug medicines: It is highly used in the field of medical research in determining whether the constituents of any drug will have the desired effect on humans or not. P-value is a very strong statistical tool used in hypothesis testing. It provides a plethora of valuable information while making an important decision like making a business intelligence inference or determining whether a drug should be used on humans or not, etc. For any doubt/query, comment below.

The p-value is a crucial concept in statistical hypothesis testing, providing a quantitative measure of the strength of evidence against the null hypothesis. It guides decision-making by comparing the p-value to a chosen significance level, typically 0.05. A small p-value indicates strong evidence against the null hypothesis, suggesting a statistically significant relationship or effect. However, the p-value is influenced by various factors and should be interpreted alongside other considerations, such as effect size and context.

Frequently Based Questions (FAQs)

Why is p-value greater than 1.

A p-value is a probability, and probabilities must be between 0 and 1. Therefore, a p-value greater than 1 is not possible.

What does P 0.01 mean?

It means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It represents a 1% chance of observing the test statistic or a more extreme one under the null hypothesis.

Is 0.9 a good p-value?

A good p-value is typically less than or equal to 0.05, indicating that the null hypothesis is likely false and the observed relationship or effect is statistically significant.

What is p-value in a model?

It is a measure of the statistical significance of a parameter in the model. It represents the probability of obtaining the observed value of the parameter or a more extreme one, assuming the null hypothesis is true.

Why is p-value so low?

A low p-value means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It suggests that the observed relationship or effect is statistically significant and not due to random sampling variation.

How Can You Use P-value to Compare Two Different Results of a Hypothesis Test?

Compare p-values: Lower p-value indicates stronger evidence against null hypothesis, favoring results with smaller p-values in hypothesis testing.

Please Login to comment...

Similar reads.

• Machine Learning

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List
• v.6(3); 2002

Statistics review 3: Hypothesis testing and P values

Elise whitley.

1 Lecturer in Medical Statistics, University of Bristol, Bristol, UK.

Jonathan Ball

2 Lecturer in Intensive Care Medicine, St George's Hospital Medical School, London, UK.

The present review introduces the general philosophy behind hypothesis (significance) testing and calculation of P values. Guidelines for the interpretation of P values are also provided in the context of a published example, along with some of the common pitfalls. Examples of specific statistical tests will be covered in future reviews.

Introduction

The previous review in this series described how to use confidence intervals to draw inferences about a population from a representative sample. A common next step in data analysis is calculation of P values, also known as hypothesis testing. Hypothesis testing is generally used when some comparison is to be made. This comparison may be a single observed value versus some hypothesized quantity (e.g. the number of babies born in a single delivery to mothers undergoing fertility treatment as compared with typical singleton birth), or it may be a comparison of two or more groups (e.g. mortality rates in intensive care unit patients who require renal replacement therapy versus those who do not). The choice of which statistical test to use depends on the format of the data and the study design. Examples of some of the more common techniques will be covered in subsequent reviews. However, the philosophy behind these statistical tests and the interpretation of the resulting P values are always the same, and it is these ideas that are covered in the present review.

The null hypothesis

A typical research question is most easily expressed in terms of there being some difference between groups. For example, 'In patients with acute myocardial infarction (AMI), does the administration of intravenous nitrate (as compared with none) reduce mortality?' To answer this question, the most appropriate study design would be a randomized controlled trial comparing AMI patients who receive intravenous nitrate with control patients. The challenge then is to interpret the results of that study. Even if there is no real effect of intravenous nitrate on mortality, sampling variation means that it is extremely unlikely that exactly the same proportion of patients in each group will die. Thus, any observed difference between the two groups may be due to the treatment or it may simply be a coincidence, in other words due to chance. The aim of hypothesis testing is to establish which of these explanations is most likely. Note that statistical analyses can never prove the truth of a hypothesis, but rather merely provide evidence to support or refute it.

To do this, the research question is more formally expressed in terms of there being no difference. This is known as the null hypothesis. In the current example the null hypothesis would be expressed as, 'The administration of intravenous nitrate has no effect on mortality in AMI patients.'

In hypothesis testing any observed differences between two (or more) groups are interpreted within the context of this null hypothesis. More formally, hypothesis testing explores how likely it is that the observed difference would be seen by chance alone if the null hypothesis were true.

What is a P value?

There is a wide range of statistical tests available, depending on the nature of the investigation. However, the end result of any statistical test is a P value. The ' P ' stands for probability, and measures how likely it is that any observed difference between groups is due to chance. In other words, the P value is the probability of seeing the observed difference, or greater, just by chance if the null hypothesis is true. Being a probability, P can take any value between 0 and 1. Values close to 0 indicate that the observed difference is unlikely to be due to chance, whereas a P value close to 1 suggests there is no difference between groups other than that due to random variation. The interpretation of a P value is not always straightforward and several important factors must be taken into account, as outlined below. Put simply, however, the P value measures the strength of evidence against the null hypothesis.

Note that the aim of hypothesis testing is not to 'accept' or 'reject' the null hypothesis. Rather, it is simply to gauge how likely it is that the observed difference is genuine if the null hypothesis is true.

Interpreting P values

Continuing with the previous example, a number of trials of intravenous nitrates in patients with AMI have been carried out. In 1988 an overview of those that had been conducted at that time was performed in order to synthesize all the available evidence [ 1 ]. The results from six trials of intravenous nitrate are given in Table ​ Table1 1 .

Results from six trials of intravenous nitrates in acute myocardial infarction patients

In the first trial (Chiche), 50 patients were randomly assigned to receive intravenous nitrate and 45 were randomly assigned to the control group. At the end of follow up, three of the 50 patients given intravenous nitrate had died versus eight in the control group. The calculation and interpretation of odds ratios will be covered in a future review. However, the interpretation in this context is that the odds ratio approximately represents the risk of dying in the nitrate group as compared with that in the control group. The odds ratio can take any positive value (above 0); in this context, values less than 1 indicate a protective effect of intravenous nitrate (a reduction in risk of death in patients administered intravenous nitrate), whereas an odds ratio greater than 1 points to a harmful effect (i.e. an increase in risk of death in patients administered intravenous nitrate). An odds ratio close to 1 is consistent with no effect of intravenous nitrate (i.e. no difference between the two groups). Interpretation of the confidence intervals is just as described in Statistics review 2, with the first confidence interval (Chiche) indicating that the true odds ratio in the population from which the trial subjects were drawn is likely to be between 0.09 and 1.13.

Initially ignoring the confidence intervals, five of the six trials summarized in Table ​ Table1 1 have odds ratios that are consistent with a protective effect of intravenous nitrate (odds ratio <1). These range from a risk reduction of 17% (Flaherty) to one of 76% (Bussman). In other words, in the Bussman trial the risk of dying in the nitrate group is about one-quarter of that in the control group. The remaining trial (Jaffe) has an odds ratio of 2.04, suggesting that the effect of intravenous nitrate might be harmful, with a doubling of risk in patients given this treatment as compared with those in the control group.

The P values shown in the final column of Table ​ Table1 1 give an indication of how likely it is that these differences are simply due to chance. The P value for the first trial (Chiche) indicates that the probability of observing an odds ratio of 0.33 or more extreme, if the null hypothesis is true, is 0.08. In other words, if there is genuinely no effect of intravenous nitrate on the mortality of patients with AMI, then 8 out of 100 such trials would show a risk reduction of 66% or more just by chance. Equivalently, 2 out of 25 would show such a chance effect. The question of whether this is sufficiently unlikely to suggest that there is a real effect is highly subjective. However, it is unlikely that the management of critically ill patients would be altered on the basis of this evidence alone, and an isolated result such as this would probably be interpreted as being consistent with no effect. Similarly the P value for the Bussman trial indicates that 1 in 100 trials would have an odds ratio of 0.24 or more extreme by chance alone; this is a smaller probability than in the previous trial but, in isolation, perhaps still not sufficiently unlikely to alter clinical care in practice. The P value of 0.70 in the Flaherty trial suggests that the observed odds ratio of 0.83 is very likely to be a chance finding.

Comparing the P values across different trials there are two main features of interest. The first is that the size of the P value is related, to some extent, to the size of the trial (and, in this context, the proportion of deaths). For example, the odds ratios in the Lis and Jugdutt trials are reasonably similar, both of which are consistent with an approximate halving of risk in patients given intravenous nitrate, but the P value for the larger Jugdutt trial is substantially smaller than that for the Lis trial. This pattern tends to be apparent in general, with larger studies giving rise to smaller P values. The second feature relates to how the P values change with the size of the observed effect. The Chiche and Flaherty trials have broadly similar numbers of patients (in fact, the numbers are somewhat higher in the Flaherty trial) but the smaller P value occurs in the Chiche study, which suggests that the effect of intravenous nitrate is much larger than that in the Flaherty study (67% versus 17% reduction in mortality). Again, this pattern will tend to hold in general, with more extreme effects corresponding to smaller P values. Both of these properties are discussed in considerably more detail in the next review, on sample size/power calculations.

There are two additional points to note when interpreting P values. It was common in the past for researchers to classify results as statistically 'significant' or 'non-significant', based on whether the P value was smaller than some prespecified cut point, commonly 0.05. This practice is now becoming increasingly obsolete, and the use of exact P values is much preferred. This is partly for practical reasons, because the increasing use of statistical software renders calculation of exact P values increasingly simple as compared with the past when tabulated values were used. However, there is also a more pragmatic reason for this shift. The use of a cut-off for statistical significance based on a purely arbitrary value such as 0.05 tends to lead to a misleading conclusion of accepting or rejecting the null hypothesis, in other words of concluding that a 'statistically significant' result is real in some sense. Recall that a P value of 0.05 means that one out of 20 studies would result in a difference at least as big as that observed just by chance. Thus, a researcher who accepts a 'significant' result as real will be wrong 5% of the time (this is sometimes known as a type I error). Similarly, dismissing an apparently 'non-significant' finding as a null result may also be incorrect (sometimes known as a type II error), particularly in a small study, in which the lack of statistical significance may simply be due to the small sample size rather than to any real lack of clinical effect (see the next review for details). Both of these scenarios have serious implications in terms of practical identification of risk factors and treatment of disease. The presentation of exact P values allows the researcher to make an educated judgement as to whether the observed effect is likely to be due to chance and this, taken in the context of other available evidence, will result in a far more informed conclusion being reached.

Finally, P values give no indication as to the clinical importance of an observed effect. For example, suppose a new drug for lowering blood pressure is tested against standard treatment, and the resulting P value is extremely small. This indicates that the difference is unlikely to be due to chance, but decisions on whether to prescribe the new drug will depend on many other factors, including the cost of the new treatment, any potential contraindications or side effects, and so on. In particular, just as a small study may fail to detect a genuine effect, a very large study may result in a very small P value based on a small difference of effect that is unlikely to be important when translated into clinical practice.

P values and confidence intervals

Although P values provide a measure of the strength of an association, there is a great deal of additional information to be obtained from confidence intervals. Recall that a confidence interval gives a range of values within which it is likely that the true population value lies. Consider the confidence intervals shown in Table ​ Table1. 1 . The odds ratio for the Chiche study is 0.33, suggesting that the effect of intravenous nitrate is to reduce mortality by two thirds. However, the confidence interval indicates that the true effect is likely to be somewhere between a reduction of 91% and an increase of 13%. The results from that study show that there may be a substantial reduction in mortality due to intravenous nitrate, but equally it is not possible to rule out an important increase in mortality. Clearly, if the latter were the case then it would be extremely dangerous to administer intravenous nitrate to patients with AMI.

The confidence interval for the Bussman study (0.08, 0.74) provides a rather more positive picture. It indicates that, although the reduction in mortality may be as little as 26%, there is little evidence to suggest that the effect of intravenous nitrate may be harmful. Administration of intravenous nitrate therefore appears more reasonable based on the results of that study, although the P value indicates a 1 in 100 probability that this may be a chance finding and so the result in isolation might not be sufficient evidence to change clinical practice.

The overview of those trials was carried out because the results did not appear to be consistent, largely because the individual trials were generally too small to provide reliable estimates of effect. A pooled analysis of the data from all of the nitrate trials shown in Table ​ Table1 1 (and including one other trial with no deaths) was therefore conducted to obtain a more robust estimate of effect (for details of the methods used, see Yusuf et al . [ 1 ]). The odds ratios and 95% confidence intervals for the individual trials in Table ​ Table1 1 are shown in Fig. ​ Fig.1. 1 . The odds ratio for each trial is represented by a box, the size of which is proportional to the amount of statistical information available for that estimate, and the 95% confidence interval is indicated by a horizontal line. The solid vertical line indicates an odds ratio of 1.0; in other words it shows the line of 'no effect'. The combined odds ratio from all six trials is indicated by the dashed vertical line, and its associated 95% confidence interval by the diamond at the bottom.

Individual and combined odds ratios and 95% confidence intervals for six intravenous nitrate trials.

This pooled analysis resulted in an estimated overall odds ratio of 0.53 with a 95% confidence interval of (0.36, 0.75), suggesting a true reduction in mortality of somewhere between one-quarter and two-thirds. Examination of the confidence intervals from individual studies shows a high degree of overlap with the pooled confidence interval, and so all of the evidence appears to be consistent with this pooled estimate; this includes the evidence from the Jaffe study, which, at first glance, appears to suggest a harmful effect. The P value for the pooled analysis was 0.0002, which indicates that the result is extremely unlikely to have been due to chance.

Note that, since that meta-analysis was reported, treatment of AMI patients has changed dramatically with the introduction of thrombolysis. In addition, the Fourth International Study of Infarct Survival (ISIS-4) [ 2 ], which randomized over 58,000 patients with suspected AMI, found no evidence to suggest that mortality was reduced in those given oral nitrates. Thus, in practice the indications for intravenous nitrates in patients with AMI are restricted to symptom and blood pressure control.

Specific methods for comparing two or more means or proportions will be introduced in subsequent reviews. In general, these will tend to focus on the calculation of P values. However, there is still much to be learned from examination of confidence intervals in this context. For example, when comparing the risk for developing secondary infection following trauma in patients with or without a history of chronic alcohol abuse, it may be enlightening to compare the confidence intervals for the two groups and to examine the extent to which they do or do not overlap. Alternatively, it is possible to calculate a confidence interval for the difference in two means or the difference or ratio of proportions directly. This can also give a useful indication of the likely effect of chronic alcohol abuse, in particular by exploring the extent to which the range of likely values includes or excludes 0 or 1, the respective expected values of a difference or ratio if there is no effect of chronic alcohol abuse, or in other words under the null hypothesis.

Although P values provide a measure of the strength of an association, an estimate of the size of any effect along with an associated confidence interval is always required for meaningful interpretation of results. P values and confidence intervals are frequently calculated using similar quantities (see subsequent reviews for details), and so it is not surprising that the two are closely related. In particular, larger studies will in general result in narrower confidence intervals and smaller P values, and this should be taken into account when interpreting the results from statistical analyses. Both P values and confidence intervals have an important role to play in understanding data analyses, and both should be presented wherever possible.

Key messages

A P value is the probability that an observed effect is simply due to chance; it therefore provides a measure of the strength of an association. A P value does not provide any measure of the size of an effect, and cannot be used in isolation to inform clinical judgement.

P values are affected both by the magnitude of the effect and by the size of the study from which they are derived, and should therefore be interpreted with caution. In particular, a large P value does not always indicate that there is no association and, similarly, a small P value does not necessarily signify an important clinical effect.

Subdividing P values into 'significant' and 'non-significant' is poor statistical practice and should be avoided. Exact P values should always be presented, along with estimates of effect and associated confidence intervals.

Abbreviations

AMI=acute myocardial infarction.

Competing interests

None declared.

This article is the third in an ongoing, educational review series on medical statistics in critical care. Previous articles have covered ‘presenting and summarising data’ [ 3 ] and ‘samples and populations’ [ 4 ]. Future topics to be covered include power calculations, comparison of means, comparison of proportions, and analysis of survival data to name but a few. If there is a medical statistics topic you would like explained contact us on moc.murofcc@lairotide .

• Yusuf S, Collins R, MacMahon S, Peto R. Effect of intravenous nitrates on mortality in acute myocardial infarction: an overview of the randomised trials. Lancet. 1988; 1 :1088–1092. [ PubMed ] [ Google Scholar ]
• Anonymous ISIS-4: a randomised factorial trial assessing early oral captopril, oral mononitrate, and intravenous magnesium sulphate in 58,050 patients with suspected acute myocardial infarction. Lancet. 2002; 345 :669–685. [ PubMed ] [ Google Scholar ]
• Whitley E, Ball J. Statistics review 1: Presenting and summarising data. Crit Care. 2002; 6 :66–71. [ PMC free article ] [ PubMed ] [ Google Scholar ]
• Whitley E, Ball J. Statistics review 2: Samples and populations. Crit Care. 2002; 6 (2) [ PMC free article ] [ PubMed ] [ Google Scholar ]

• Math Article

In Statistics, the researcher checks the significance of the observed result, which is known as test static . For this test, a hypothesis test is also utilized. The P-value  or probability value concept is used everywhere in statistical analysis. It determines the statistical significance and the measure of significance testing. In this article, let us discuss its definition, formula, table, interpretation and how to use P-value to find the significance level etc. in detail.

Table of Contents:

P-value Definition

The P-value is known as the probability value. It is defined as the probability of getting a result that is either the same or more extreme than the actual observations. The P-value is known as the level of marginal significance within the hypothesis testing that represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance at which the null hypothesis would be rejected. If the P-value is small, then there is stronger evidence in favour of the alternative hypothesis.

P-value Table

The P-value table shows the hypothesis interpretations:

Generally, the level of statistical significance is often expressed in p-value and the range between 0 and 1. The smaller the p-value, the stronger the evidence and hence, the result should be statistically significant. Hence, the rejection of the null hypothesis is highly possible, as the p-value becomes smaller.

Let us look at an example to better comprehend the concept of P-value.

Let’s say a researcher flips a coin ten times with the null hypothesis that it is fair. The total number of heads is the test statistic, which is two-tailed. Assume the researcher notices alternating heads and tails on each flip (HTHTHTHTHT). As this is the predicted number of heads, the test statistic is 5 and the p-value is 1 (totally unexceptional).

Assume that the test statistic for this research was the “number of alternations” (i.e., the number of times H followed T or T followed H), which is two-tailed once again. This would result in a test statistic of 9, which is extremely high and has a p-value of 1/2 8 = 1/256, or roughly 0.0039. This would be regarded as extremely significant, much beyond the 0.05 level. These findings suggest that the data set is exceedingly improbable to have happened by random in terms of one test statistic, yet they do not imply that the coin is biased towards heads or tails.

The data have a high p-value according to the first test statistic, indicating that the number of heads observed is not impossible. The data have a low p-value according to the second test statistic, indicating that the pattern of flips observed is extremely unlikely. There is no “alternative hypothesis,” (therefore only the null hypothesis can be rejected), and such evidence could have a variety of explanations – the data could be falsified, or the coin could have been flipped by a magician who purposefully swapped outcomes.

This example shows that the p-value is entirely dependent on the test statistic used and that p-values can only be used to reject a null hypothesis, not to explore an alternate hypothesis.

P-value Formula

We Know that P-value is a statistical measure, that helps to determine whether the hypothesis is correct or not. P-value is a number that lies between 0 and 1. The level of significance(α) is a predefined threshold that should be set by the researcher. It is generally fixed as 0.05. The formula for the calculation for P-value is

Step 1: Find out the test static Z is

P0 = assumed population proportion in the null hypothesis

N = sample size

Step 2: Look at the Z-table to find the corresponding level of P from the z value obtained.

P-Value Example

An example to find the P-value is given here.

Question: A statistician wants to test the hypothesis H 0 : μ = 120 using the alternative hypothesis Hα: μ > 120 and assuming that α = 0.05. For that, he took the sample values as

n =40, σ = 32.17 and x̄ = 105.37. Determine the conclusion for this hypothesis?

We know that,

Now substitute the given values

Now, using the test static formula, we get

t = (105.37 – 120) / 5.0865

Therefore, t = -2.8762

Using the Z-Score table , we can find the value of P(t>-2.8762)

From the table, we get

P (t<-2.8762) = P(t>2.8762) = 0.003

If P(t>-2.8762) =1- 0.003 =0.997

P- value =0.997 > 0.05

Therefore, from the conclusion, if p>0.05, the null hypothesis is accepted or fails to reject.

Hence, the conclusion is “fails to reject H 0. ”

Frequently Asked Questions on P-Value

What is meant by p-value.

The p-value is defined as the probability of obtaining the result at least as extreme as the observed result of a statistical hypothesis test, assuming that the null hypothesis is true.

What does a smaller P-value represent?

The smaller the p-value, the greater the statistical significance of the observed difference, which results in the rejection of the null hypothesis in favour of alternative hypotheses.

What does the p-value greater than 0.05 represent?

If the p-value is greater than 0.05, then the result is not statistically significant.

Can the p-value be greater than 1?

P-value means probability value, which tells you the probability of achieving the result under a certain hypothesis. Since it is a probability, its value ranges between 0 and 1, and it cannot exceed 1.

What does the p-value less than 0.05 represent?

If the p-value is less than 0.05, then the result is statistically significant, and hence we can reject the null hypothesis in favour of the alternative hypothesis.

• Share Share