hypothesis testing t distribution calculator

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

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

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

Teach yourself statistics

T Distribution Calculator

The t distribution calculator makes it easy to compute the cumulative probability associated with a t score or with a sample mean. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about Student's t distribution, go to Stat Trek's tutorial on the t distribution .

• In the dropdown box, select the statistic of interest.
• Enter a value for degrees of freedom.
• Enter a value for all but one of the remaining textboxes.
• Click the Calculate button to compute a value for the blank textbox.

Note : Both the t distribution and the standard normal distribution assume that observations are normally distributed in the population . And as sample size increases, the t distribution becomes increasingly similar to the standard normal distribution. So when would a researcher choose the t distribution over the standard normal distribution? A common rule of thumb is to choose the t distribution when (1) the sample size is small and/or (2) the population standard deviation is unknown.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

Which statistic should I use - the t score or the mean score"?

The t distribution calculator accepts two statistics as input: a t score or a sample mean. Choose the option that is easiest. Here are some things to consider.

t = [ x - μ ] / [ s / sqrt( n ) ]

• If you choose to work with the sample mean, you can avoid the "transformation" step. But you will need to provide additional input in the form of the population mean and/or the sample standard deviation.

For an example that uses the t score, see Sample Problem 1 . For an example that uses the sample mean, see Sample Problem 2

What are degrees of freedom?

Degrees of freedom can be described as the number of scores that are free to vary. For example, suppose you tossed three dice. The total score adds up to 12. If you rolled a 3 on the first die and a 5 on the second, then you know that the third die must be a 4 (otherwise, the total would not add up to 12). In this example, 2 die are free to vary while the third is not. Therefore, there are 2 degrees of freedom.

In many situations, the degrees of freedom are equal to the number of observations minus one. Thus, if the sample size were 20, there would be 20 observations; and the degrees of freedom would be 20 minus 1 or 19.

What is a standard deviation?

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. It is a measure of the average distance of individual observations from the group mean.

What is a t statistic?

A t statistic is a statistic whose values are given by

t = [ x - μ> ] / [ s / sqrt( n ) ]

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and t is the t statistic.

What is a population mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A population mean is the mean score of a population .

What is a sample mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A sample mean is the mean score of a sample .

What is a probability?

A probability is a number expressing the chances that a specific event will occur. This number can take on any value from 0 to 1. A probability of 0 means that there is zero chance that the event will occur; a probability of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify the uncertainty associated with the event. For example, the probability of a coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent of the time, the coin flip would result in Heads; and fifty percent of the time, it would result in Tails.

What is a cumulative probability?

A cumulative probability is a sum of probabilities. In connection with the t distribution calculator, a cumulative probability refers to the probability that a t score or a raw score will be less than or equal to a specified value.

Suppose, for example, that we sample 100 first-graders. If we ask about the probability that the average first grader weighs exactly 70 pounds, we are asking about a simple probability - not a cumulative probability.

But if we ask about the probability that average weight is less than or equal to 70 pounds, we are really asking about a sum of probabilities (i.e., the probability that the average weight is exactly 70 pounds plus the probability that it is 69 pounds plus the probability that it is 68 pounds, etc.). Thus, we are asking about a cumulative probability.

Note: The t distribution calculator only reports cumulative probabilities (e.g., the probability that a t score is less than or equal to a specified value.)

Sample Problems

• Compute a t statistic, assuming that the breaking strength for the customer's chains is 19,800 pounds.
• Determine the cumulative probability for that t statistic.
• The t statistic is equal to -0.4276.
• The number of degrees of freedom is equal to 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

Now, we are ready to use the T Distribution Calculator . Since we have already computed the t statistic, we select "t score" from the drop-down box. Then, we enter the t statistic (-0.4276) and the degrees of freedom (13) into the calculator, and hit the Calculate button. The calculator reports that the cumulative probability is 0.338.

Therefore, there is a 33.8% chance that the average breaking strength for the customer's chains will be no more than 19,800 pounds.

• The population mean is 20,000.
• The standard deviation is 1750.
• The sample mean, for which we want to find a cumulative probability, is 19,800.
• The number of degrees of freedom is 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

First, we select "mean score" from the dropdown box in the T Distribution Calculator . Then, we plug our known inputs (degrees of freedom, sample mean, standard deviation, and population mean) into the T Distribution Calculator and hit the Calculate button. The calculator reports that the cumulative probability is 0.338. Thus, there is a 33.8% probability that the average breaking strength for the customer's chains will be 19,800 pounds or less.

Note: This is the same answer that we found in Example 1. However, the approach that we followed in this example may be a little bit easier than the approach that we used in the previous example, since this approach does not require manual computation of a t statistic.

• The cumulative probability is 0.90.
• The standard deviation is 11.
• The sample mean is 115.
• The number of degrees of freedom is 14. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 15 - 1 or 14.)

First, we select "mean score" from the dropdown box in the T Distribution Calculator . Then, we plug the known inputs (cumulative probability, standard deviation, sample mean, and degrees of freedom) into the calculator and hit the Calculate button. The calculator reports that the population mean is 111.2.

Thus, if the principal's assessment of the IQ of his faculty is correct, we conclude that the average IQ of a teacher in the district is 111.2.

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T-Distribution Probability Calculator

Instructions: Use this T-Distribution Probability Calculator to Compute t-distribution probabilities using the form below. Please type the number of degrees of freedom associated to the t-distribution, and provide details about the event you want to compute the probability for:

More about the t distribution probability

The t-distribution is a type of continuous probability distribution that takes random values on the whole real line. The main properties of the t-distribution are:

• It is continuous (and as a consequence, the probability of getting any single, specific outcome is zero)
• It is "bell shaped", in the same way the normal curves are bell-shaped
• It is determined by one parameter: the number of degrees of freedom (df). For one sample, the number of degrees of freedom is df = n - 1, where n is the sample size
• It is symmetric with respect to 0
• The t-distribution "converges" to the standard normal distribution as the number of degrees of freedom (df) converges to infinity (+∞)

In order to compute probabilities associated to the t-distribution we can either use specialized software such as Excel, etc, or we can use t-distribution tables (normally available at college statistics textbooks. The use of the t-distribution arises when performing hypothesis testing (for the case when the population standard deviation is not known).

How do you this this t-distribution calculator

This calculator allows you have both a T distribution calculator for two tailed, as well as for left-tailed and two-tailed. This is achieved by selecting the right type of event in the form above.

What is the meaning Area of t distribution calculator

The are of the t-distribution, the same as all probability distributions, represents the probability that a random even occurs in the region for which the area is computed. In other words, in this case Area = Probability.

What is the T distribution calculator accuracy

It really depends on which tool do you use to do the calculation. For example, if you use t-tables to compute t-distribution probabilities, you will find that those typically have accuracy of 4 digits.

Now, when you use our calculator (or Excel), you get about 15 digits accuracy. You may think that is great and it is better, and it actually is, only you have to be carefully when answering tests and quizzes, where they may want you to round intermediate and final steps, which could lead to answers that can be different than the answer expected by the system.

More calculators related to the t-distribution

This is calculator for the T-Distribution. If you need to work with the normal distribution, you may be also interested in our z-distribution calculator . Also, a normal probability for samples could come in handy to deal with things like the Central Limit Theorem and things like that.

Application: Calculation of a probability

Question : Compute the probability of the event (-1, 0.8) for the t-distribution with 34 degrees of freedom.

We need to compute \(\Pr(-1 \leq T \leq 0.8)\), where \(T\) has a t-distribution with \(df = 34\) degrees of freedom. Therefore, the probability is computed as:

Therefore, based on the information provided, it is concluded that \( \Pr(-1 \leq T \leq 0.8) = 0.6232\)

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T Statistic Calculator (T-Value)

Introduction.

The T-statistic, also known as the T-value, is a statistical measure used to assess whether the means of two groups are significantly different from each other. It is a fundamental tool in hypothesis testing, helping researchers determine whether the differences observed in a sample are likely due to random chance or if they represent a real effect. The T-statistic is particularly useful when dealing with small sample sizes where the distribution of data may not be perfectly normal.

In this article, we will explore the T-Statistic Calculator, its formula, how to use it, provide an example, and answer some frequently asked questions to help you understand this crucial statistical concept.

The formula for calculating the T-statistic depends on the context of the analysis. There are two main scenarios: when you have two independent sample groups or when you have one sample group and you want to compare it to a known population mean. Here are the formulas for both scenarios:

1. Independent Sample T-Test:

For comparing the means of two independent sample groups, the T-statistic formula is as follows:

T = (x̄₁ – x̄₂) / (s√((1/n₁) + (1/n₂)))

• x̄₁ and x̄₂ are the sample means of the two groups.
• s is the pooled standard deviation of the two groups.
• n₁ and n₂ are the sample sizes of the two groups.

2. One-Sample T-Test:

For comparing the mean of a single sample group to a known population mean, the T-statistic formula is as follows:

T = (x̄ – μ) / (s / √n)

• x̄ is the sample mean.
• μ is the known population mean.
• s is the sample standard deviation.
• n is the sample size.

How to Use?

Using the T-Statistic Calculator is relatively straightforward:

• Identify the type of analysis you are conducting: independent sample T-test or one-sample T-test.
• For the independent sample T-test, you need data from two separate groups.
• For the one-sample T-test, you need data from a single group and a known population mean for comparison.
• For the independent sample T-test, input the means, standard deviations, and sample sizes of both groups.
• For the one-sample T-test, input the sample mean, known population mean, sample standard deviation, and sample size.
• Click the “Calculate” button.
• The calculator will provide you with the T-statistic value.
• Compare the calculated T-statistic to a critical value from the T-distribution table or use it to calculate a p-value.

Let’s walk through an example of a one-sample T-test using the T-Statistic Calculator:

Suppose you are a manufacturer of light bulbs, and you claim that your bulbs last, on average, 1200 hours. You want to test this claim using a sample of 30 light bulbs, and you find that the sample has a mean lifespan of 1150 hours with a standard deviation of 100 hours.

• Identify the type of analysis: one-sample T-test.
• Sample mean ( x̄ ): 1150 hours
• Known population mean ( μ ): 1200 hours
• Sample standard deviation ( s ): 100 hours
• Sample size ( n ): 30 bulbs
• Enter the values into the T-Statistic Calculator.
• Click “Calculate.”
• The calculator provides you with the T-statistic value, let’s say it’s -2.0.
• You can now compare this T-statistic to a critical value or calculate a p-value. In this case, you might find that the T-statistic corresponds to a p-value of 0.029.

Q1: What is the T-distribution?

The T-distribution is a probability distribution used in hypothesis testing when the sample size is small, and the population standard deviation is unknown. It resembles a normal distribution but has heavier tails.

Q2: What is a p-value, and how is it related to the T-statistic?

The p-value is a probability that measures the evidence against a null hypothesis. In T-testing, a smaller p-value suggests stronger evidence against the null hypothesis. You can calculate the p-value using the T-statistic and degrees of freedom.

Q3: What is the significance level (alpha) in hypothesis testing?

The significance level (alpha) is the threshold value used to determine the statistical significance of results. Common choices for alpha are 0.05 and 0.01. If the p-value is less than alpha, you reject the null hypothesis.

Conclusion:

The T-Statistic Calculator, also known as the T-value calculator, is an essential tool in statistical analysis, helping researchers assess the significance of differences between sample means or compare a sample mean to a known population mean. Understanding how to calculate and interpret the T-statistic is crucial for making informed decisions in various fields, including science, engineering, and business. By following the provided formula and guidelines, you can use this calculator to perform hypothesis tests and draw meaningful conclusions from your data.

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1. Given Data :

First enter the data in by pressing STAT and selecting 1:Edit by pressing ENTER. The result is shown below

2. Given Statistics.

Press STAT, then RIGHT ARROW twice to TESTS. Press 2 to select 2:T-Test.

Student's t-distribution calculator with graph generator

Critical value calculator - student's t-distribution.

This statistical calculator allows you to calculate the critical value corresponding to the Student's t-distribution, you can also see the result in a graph through our online graph generator and if you wish you can download the graph. Just enter the significance value (alpha), degrees of freedom, and left, right, or both tails.

Critical value result

P-value calculator - student's t distribution.

Use our online statistical calculator to calculate the p-value of the Student's t-distribution. You just need to enter the t-value and degrees of freedom and specify the tail. In addition to the p-value, you can get and download the graph created with our graph generator

p-value result

One sample t-test calculator.

The one sample t-test is a statistical hypothesis test calculator, use our calculator to check if you get a statistically significant result or not. To obtain it, fill in the corresponding fields and you will obtain the value of the t-score, p-value, critical value, and the degrees of freedom. You can also download a graph that will display your results in the form of the Student's t-distribution.

T-score result

Two sample t-test calculator.

To determine whether or not the means of two groups are equal, you can use our two-sample t-test calculator that applies the t-test. The results are displayed in a Student's t-distribution plot that you can download. To complete the form, you must include information for both groups, including the mean, standard deviation, sample size, significance level,and whether the test is left, right, or two-tailed.

Common questions related to the Student's t-distribution

In this section, we will try to address the most frequently asked questions about the Student's t-distribution. To give you a fundamental and complementary understanding, we will try to dive into the underlying ideas of the t-distribution. The approach we want to take is to answer the most common questions from students with relevant information. Let's tackle problems simply and offer short and understandable solutions.

Questions related to the student's t-distribution

The formula in relation to the probability density function (pdf) for Student's t-distribution, is given as follows:

Where: π is the pi (approximately 3.14), ν correspond to the degrees of freedom, and Γ is the Euler Gamma function.

A distribution of mean estimates derived from samples taken from a population is what is, by definition, the Student's t-distribution. The t-distribution, commonly known as the Student's t-distribution, is a type of symmetric bell-shaped distribution, it has a lower height but a wider spread than the normal distribution. It is symmetric around 0, but the t-distribution has a wider spread than the typical normal distribution curve, or put another way, the t-distribution has a high standard deviation. The variability of individual observations around their mean is measured by a standard deviation. The degrees of freedom (df) are n - 1. So, df is equal to n – 1, where n is the sample size. The degrees of freedom affect the shape of each t distribution curve.

When the sample size is less than 30 and the population standard deviation is unknown, the t-distribution is utilized in hypothesis testing. It is helpful when the sample size is relatively small or the population standard deviation is unknown. It resembles the normal distribution more closely as sample size grows.

A statistical metric known as the standard deviation is used to quantify the distances between each observation and the mean in a set of data. The standard deviation calculates the degree of dispersion or variability. In other words, it's used to calculate how much a random variable deviates from the mean.

The t-value and t-score have the same meanings. It is one of the relative position measurements. By definition, a value of t defines the location of a continuous random variable, X, in relation to the number of standard deviations from the mean.

The significance level is a point in the normal distribution that must be understood in order to either reject or fail to reject the null hypothesis and to assess whether or not the results are statistically significant. If you decide to make use of our t distribution calculator , you must enter the alpha value corresponding to the significance level. The most common alpha values are 0.1, 0.05 or 0.01. Generally, the most common confidence intervals are: 90%, 95% and 99% (1 − α is the confidence level).

The p-value is a probability with a value ranging from 0 to 1. It is used to test a hypothesis. As an example, in some experiment, we choose the significance level value as 0.05, in this case, the alternative hypothesis is more likely to be supported by stronger evidence when the p-value is less than 0.05 (p-value < 0.05), in case the p-value is high (p-value > 0.05), the probability of accepting the null hypothesis is also high.

The z and t distributions are symmetric and bell-shaped. However, what most characterizes the t distribution are its tails, since they are heavier than in the normal distribution. Furthermore, it can be seen that there are more values in the t-distribution located at the ends of the tail instead of the center of the distribution. You must have the population standard deviation to use the standard normal or z distribution. On the other hand, one of the important conditions for adopting the t distribution is that the population variance is unknown

The t-test , it is a parametric comparison test, is used if the means of two samples are compared using a hypothesis test, if they are independent, from two separate samples, or dependent, a sample evaluated at two different times. The procedure is carried out to evaluate if the differences between the means are significant, determining that they are not due to chance.

To interpret the results of a t-test, you can compare the t-score to the critical value and consider the p-value. A high t-score and low p-value indicate that there is a statistically significant difference between the two means, while a low t-score and high p-value indicate that the difference is not statistically significant. The degrees of freedom and the significance level (alpha) also play a role in determining the critical value and the p-value.

A one sample t-test is a statistical procedure used to test whether the mean of a single sample is significantly different from a hypothesized mean. It is used to determine whether the sample comes from a population with a mean that is different from the hypothesized mean. To perform a one sample t-test using a calculator, you need to input the following information: The sample data, including the mean and standard deviation. The hypothesized mean. The significance level (alpha). The type of tail (left, right, or two-tailed). The calculator will then calculate the t-score and p-value based on this information, and will also provide the critical value and degrees of freedom. To interpret the results, you can compare the t-score to the critical value and consider the p-value. If the t-score is greater than the critical value and the p-value is less than the significance level, you can reject the null hypothesis and conclude that the sample mean is significantly different from the hypothesized mean. If the t-score is less than the critical value or the p-value is greater than the significance level, you cannot reject the null hypothesis and must conclude that the sample mean is not significantly different from the hypothesized mean.

A two-sample t-test is a statistical procedure used to determine whether there is a significant difference between the means of two groups. It is often used to compare the means of two groups in order to determine whether a difference exists between them. For example, a researcher might use a two-sample t-test to determine whether there is a significant difference in the average scores on a test between males and females, or between two different treatment groups in a medical study. The t-test is based on the t-statistic , which is calculated from the sample data and represents the difference between the two groups in relation to the variation within the groups. The t-test is used to determine whether this difference is statistically significant, meaning that it is unlikely to have occurred by chance.

For optimal use, please visit DATAtab on your desktop PC!

Metric Variables:

Ordinal variables:, nominal variables:, hypothesis test calculator.

Do you want to calculate a hypothesis test such as a t-test , Chi Square test or an ANOVA ? You can do that easily here in the browser.

If you want to use your own data just clear the upper table

• Clear the table in the Hypothesis test calculator.
• Copy your data into the table.
• Select the variables.

In the hypothesis test calculator you can calculate e.g. a t-test, a chi-square test, a binomial test or an analysis of variance. If you need a more detailed explanation, you can find more information in the tutorials.

In order to use the hypothesis test calculator, you must first formulate your hypothesis and collect your data. DATAtab will then suggest the hypothesis test you need based on the data entered into the statistics calculator.

p value calculator

With the p value calculator you can calculate the p value for different tests. There is a wide range of methods for this. Just click on the variables you want to evaluate above and DATAtab will give you the tests you can use.

For example, if you select a metric and a categorical variable, the Independent t-Test calculator is automatically selected. If your data is not normally distributed, simply use the Mann-Whitney U-test calculator.

H0 and H1 calculator

With the h0 and h1 calculator for the different hypothesis test you can calculate the p-value which gives you an indication if you can reject the H0 or not.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

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​Student t-Value Calculator Online

Student t-value calculator.

In order to calculate the Student T Value for any degrees of freedom and given probability. The calculator will return Student T Values for one tail (right) and two tailed probabilities. Please input degrees of freedom and probability level and then click “CALCULATE”

One Sample T Test Calculator

Enter sample data, reporting results in apa style, one sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators.

T-Test Calculator

Compare the means of two samples using a single-sample or two-sample t-test below.

• Single Sample
• Two Sample (Unpaired)

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How to do a t-test, types of t-tests, how to calculate t using a one-sample t-test, how to calculate t using a student’s t-test, how to calculate t using welch’s t-test, find the p-value.

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A t-test calculates how significant the difference between the means of two groups are. The results let you know if those differences could have occurred by chance, or rather, whether the difference is statistically significant.

A t-test uses the test statistic, sometimes called a t-value or t-score, the t-distribution values, and the degrees of freedom to calculate the statistical significance of the difference.

Since a t-test is a parametric test, it relies on assumptions about the process that generated the underlying data. In particular, the likelihood or unlikelihood that the t-test provides for a difference being due to chance depends on the assumption that the data are normally distributed and each data point’s values are independent of one another.

Depending on how plausible those assumptions are, the analysis that follows will be more or less useful. If your data is continuous and comes from a relatively large random sample from some population, the central limit theorem implies that these assumptions will likely be approximately satisfied.

The first part of doing a t-test is determining which type of t-test you need to do.

There are three different types of t-tests:

• one-sample t-test: used to compare the mean of a sample to the known mean of a population
• two-sample t-test: used to compare the mean of two different independent samples
• paired t-test: used to compare the mean of two different samples after an intervention or change

A one-sample t-test, or single-sample test, is used to compare a sample mean to a population mean when the null hypothesis is that the sample mean is equal to the population mean.

Those who first encounter this test often wonder why they would use it, since the population mean is often not known (and the data is often collected to determine the population mean in the first place).

It often does make sense to use a one-sample t-test if you have a particular interest in whether a sample’s mean is different from some reference value that is determined to be substantively important for other reasons.

For example, let’s suppose that 5 micrograms of lead per liter of blood is the maximum safe amount, according to most medical references. Then, you may well consider doing a one-sample t-test to examine whether the average blood lead level of a sample of individuals was above that medically acceptable limit.

One-Sample T-Test Formula

To calculate the t value using a one-sample t-test, use the following formula:

Where: x̄ = sample mean μ = population mean s = sample standard deviation n = sample size

Thus, the test statistic t is equal to the difference between the sample mean x̄ and the population mean μ , divided by the standard error s / √n .

A Student’s t-test is used for test statistics that follow a Student’s t-distribution under the null hypothesis that two populations have equal means.

The name “Student” refers to the pseudonym of the author who first proposed the test in an academic journal, and does not refer to the fact it is one of the most commonly taught tests in statistics courses (although the latter is also true).

The Student’s t-test assumes that the variances of two populations are equal and asks whether their means differ significantly.

This is a type of two-sample test used to compare two sample means, where a large t-value suggests that the samples are very different, and a small t-value suggests that they are similar.

Similar to the one-sample t-test, individuals who first encounter this test may wonder about the plausibility of its assumptions. In particular, you might question how the variances in two samples could possibly be equal if the means are different.

In some contexts (for example, the industrial experiments that motivated Student’s efforts), there might be substantive reasons to assume equal variances. More informally, if you calculate the standard deviations in each sample and sees that they are close, you might proceed to calculate Student’s t-test.

More formally, some analysts would recommend that you initially conduct an F-test to determine whether variances are different, and then proceed to consider the means. But many analysts would also simply not make the equal variances assumption and proceed directly to Welch’s t-test.

Student’s T-Test Formula

The formula for a Student’s t-test is:

Given the formula to calculate the pooled standard deviation s p :

Where: x̄ 1 = first sample mean x̄ 2 = second sample mean n 1 = first sample size n 2 = second sample size s 1 = first sample standard deviation s 2 = second sample standard deviation n 1 + n 2 – 2 = degrees of freedom ν

In a Student’s t-test, the test statistic t is equal to the difference between sample means x̄ 1 and x̄ 2 , divided by the pooled standard deviation s p times the square root of 1 divided by the first sample size n 1 plus 1 divided by the second sample size n 2 .

The pooled standard deviation s p is equal to the first sample size n 1 minus 1 times the first sample standard deviation s 1 plus the second sample size n 2 minus 1 times the second sample standard deviation s 2 , divided by the degrees of freedom, in this case the sum of the sample sizes minus two.

It is called the “pooled” standard deviation because it combines or “pools” the data between both samples to determine the overall variability of the data.

This formula can be broken down into a few simple steps.

Step One: Calculate the Degrees of Freedom

Step two: calculate the pooled standard deviation, step three: calculate the test statistic.

Recall that the Student’s t-test assumes that the variances of two populations are equal. As was mentioned above, this is often a questionable assumption, and ultimately unverifiable.

In this case, you can use Welch’s t-test, which is sometimes also called an unequal variances t-test or an “unpooled” t-test. Like before, the null hypothesis with this test is that two populations have equal means.

Welch’s T-Test Formula

The formula for Welch’s t-test is:

Degrees of Freedom Formula

To find the degrees of freedom when using Welch’s t-test, use the Satterthwaite formula:

The next step is to find the p-value for the test statistic. The p-value is a measure of how “surprising” or “unlikely” some statistic would be given the particular assumptions that the analyst makes.

In the case of these t-tests for differences in means, the p-value is the probability of calculating a t-statistic that is as large or larger than what was actually calculated from the observed data if, in fact, the population means were identical.

More generally, a p-value is used to determine whether to reject the null hypothesis. In formal hypothesis testing, you would specify beforehand the p-value that would lead you to conclude that the two samples came from different populations.

What is the Right P-Value?

These standards differ by field and disciplines a lot, for example, in social and biological sciences, a p-value of 0.05 or smaller (implying 5% or lower chance of observing the data under the null hypothesis) is common, although in some cases 0.1 or 0.01 might be the standard.

In the physical sciences, it is not uncommon to pre-specify a “6 sigma” standard for certain kinds of evidence, which requires an astronomically small p-value.

How to Calculate the P-Value

To calculate the p-value from a t-statistic, use a t-table and locate the degrees of freedom in the leftmost column. Then, locate the desired p-value in the heading row, 0.05 is most commonly used for a 95% confidence level.

Then, find the intersection of the row and column to find the critical value.

Drawing Conclusions Using the P-Value

If the calculated t-value is larger than the critical value, then you can reject the null hypothesis. If it is less than the critical value, then you fail to reject the null hypothesis.

The t-distribution is related to the normal distribution; indeed, it can be thought of as the normal distribution’s “heavy-tailed” cousin. The degrees of freedom in the t-distribution determines how heavy the tails are, with fewer degrees of freedom resulting in greater departures from normality.

As the degrees of freedom increase, it becomes harder and harder to tell the differences between the associated t-distribution and the normal distribution.

Because of this fact, experienced statistical analysts are often able to approximately estimate the p-value of a particular t-statistic through their familiarity with the normal distribution.

A t-statistic of 2 or greater is typically enough to confirm statistical significance in the social and biological contexts.

Recommended Statistics Resources

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

• Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
• Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
• Enter data for the test, based on the format you chose in Step 1.
• Click Calculate Now and View the results. All options will perform a two-tailed test .

Performing t tests? We can help.

Sign up for more information on how to perform t tests and other common statistical analyses.

Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

• Exactly two groups
• Sample is normally distributed
• Independent observations
• Unequal or equal variance?
• Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

• Clear guidance to pick the right t test and detailed results summaries
• Custom, publication quality t test graphics, violin plots, and more
• More t test options, including normality testing as well as nested and multiple t tests
• Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

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9.4: Distribution Needed for Hypothesis Testing

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• Page ID 773

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Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's \(t\)-distribution. (Remember, use a Student's \(t\)-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually \(n\) is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for means :

\[\bar{X} - N\left(\mu_{x}, \frac{\sigma_{x}}{\sqrt{n}}\right)\]

The population parameter is \(\mu\). The estimated value (point estimate) for \(\mu\) is \(\bar{x}\), the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

\[P' - N\left(p, \sqrt{\frac{p-q}{n}}\right)\]

The population parameter is \(p\). The estimated value (point estimate) for \(p\) is \(p′\). \(p' = \frac{x}{n}\) where \(x\) is the number of successes and n is the sample size.

Assumptions

When you perform a hypothesis test of a single population mean \(\mu\) using a Student's \(t\)-distribution (often called a \(t\)-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a \(t\)-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean \(\mu\) using a normal distribution (often called a \(z\)-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion \(p\), you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are: there are a certain number \(n\) of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success \(p\). The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities \(np\) and \(nq\) must both be greater than five \((np > 5\) and \(nq > 5)\). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with \(\mu = p\) and \(\sigma = \sqrt{\frac{pq}{n}}\). Remember that \(q = 1 – p\).

In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.

When testing for a single population mean:

• A Student's \(t\)-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
• The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.

When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions: \(np > 5\) and \(nq > 5\) where \(n\) is the sample size, \(p\) is the probability of a success, and \(q\) is the probability of a failure.

Formula Review

If there is no given preconceived \(\alpha\), then use \(\alpha = 0.05\).

Types of Hypothesis Tests

• Single population mean, known population variance (or standard deviation): Normal test .
• Single population mean, unknown population variance (or standard deviation): Student's \(t\)-test .
• Single population proportion: Normal test .
• For a single population mean , we may use a normal distribution with the following mean and standard deviation. Means: \(\mu = \mu_{\bar{x}}\) and \(\\sigma_{\bar{x}} = \frac{\sigma_{x}}{\sqrt{n}}\)
• A single population proportion , we may use a normal distribution with the following mean and standard deviation. Proportions: \(\mu = p\) and \(\sigma = \sqrt{\frac{pq}{n}}\).
• It is continuous and assumes any real values.
• The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
• It approaches the standard normal distribution as \(n\) gets larger.
• There is a "family" of \(t\)-distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.

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Student T-Value Calculator

You can use this T-Value Calculator to calculate the Student's t-value based on the significance level and the degrees of freedom in the standard deviation.

How to use the calculator

• Enter the degrees of freedom (df)
• Enter the significance level alpha (α is a number between 0 and 1)
• Click the "Calculate" button to calculate the Student's t-critical value.

Degrees of Freedom (df) :

Significance Level (α) :

• Currently 4.76/5

Rating: 4.8 /5 (821 votes)

Hypothesis Test Graph Generator

Note: After clicking "Draw here", you can click the "Copy to Clipboard" button (in Internet Explorer), or right-click on the graph and choose Copy. In your Word processor, choose Paste-Special from the Edit menu, and select "Bitmap" from the choices

Note: This creates the graph based on the shape of the normal curve, which is a reasonable approximation to the t-distribution for a large sample size. These graphs are not appropriate if you are doing a t-distribution with small sample size (less than 30).