problem solving involving mean

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Word Problems Involving the Mean of a Data Set

As we delve deeper into the world of statistics and probability, one specific term becomes very important to us: The mean. You've probably encountered this concept before -- even if you weren't aware of it. "Mean" essentially means the same thing as "average," and this number becomes very useful in a number of different fields. A baseball player's "batting average" tells you how skilled he or she is. The "average" return of an investment helps us understand how profitable it really is. As we can see, the mean is very useful in real-world applications. Let's tackle some word problems that show us how we can apply means to these real-world situations.

What is the mean?

We can find the mean of any data set with two easy steps:

• Add all the data points together
• Divide by the number of data points

Solving word problems involving means

Now let's put our knowledge to good use and solve a few word problems that involve means:

Consider the following data set: 4 , 8 , 20 , 25 , 32

If we add one more number ( x ) to this data set, we are left with a mean of 15. What is the value of this additional number?

First, let's try to add all of our numbers together: 4 + 8 + 20 + 25 + 32 + x

We don't know the value of x, so we can't complete this addition. But let's continue with the operation anyway:

If we add one more value (x) to our current data set, we get 6 total data points. This means that we need to divide the above sum by six:

4 + 8 + 20 + 25 + 32 + x + 6

We know that the end result is a mean of 15. So how can we solve this equation? Easy: We simplify it.

4 + 8 + 20 + 25 + 32 + x + 6 = 15

89 + x / 6 = 15

Now, all we need to do is multiply each side of the equation by 6:

6 × ( 89 + x ) / ( 6 ) = 15 × 6

89 + x = 90

The answer is 1.

We can even use our knowledge of means to help us set higher goals!

What happens if we know that we need an average of 90% or higher to get an "A" in math class?

What if our past test scores were 80%, 85%, 88%, and 95%?

We have one more test for the term. What score do we need to get an "A" average?

Right away, we know that our end result needs to be at least 90.

We also know that the total number of tests is 5.

With these values, we can find our minimum test score for an A average (x).

80 + 85 + 88 + 95 + x / 5

We can simplify this as 348 + x / 5

Now, all we need to do is multiply each side of the equation by 5:

5 × ( 348 + x ) / ( 5 ) = 90 × 5

Or: 348 + x = 450

Uh oh, looks like that value is 102. As we know, the highest possible score on a test is 100%. This means that it's theoretically impossible for us to get an A average this term -- unless, of course, we speak to our teacher about the possibility of doing additional work for extra credit. This highlights the importance of calculating means in order to plan effectively for the future.

Exercises such as this also show us how even a few low scores can seriously affect the overall average or mean.

The especially low score of 80% is 10 percentage points below our target grade, and this skews the data toward a lower mean. We can consider this value an outlier.

Topics related to the Word Problems Involving the Mean of a Data Set

Center and Variation of Data

Word Problems

Interpreting Data

Flashcards covering the Word Problems Involving the Mean of a Data Set

Statistics Flashcards

Common Core: High School - Statistics and Probability Flashcards

Practice tests covering the Word Problems Involving the Mean of a Data Set

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

Tutoring helps your student gain confidence when solving word problems involving means

Tutoring gives your student the opportunity to go over a few more examples of word problems involving means. Sometimes, all it takes to reach for confidence in these concepts is a little extra time outside of class. Tutors can even cater to your student's learning style during these sessions, whether they prefer verbal or visual cues. Speak with Varsity Tutors' Educational Directors today to learn more about the possibilities of tutoring. We'll match your student with a suitable tutor.

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Problems Based on Average

Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

How to solve average word problems?

To solve various problems we need to follow the uses of the formula for calculating arithmetic mean.

Average = (Sums of the observations)/(Number of observations)

Worked-out problems based on average:

1. The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.

Mean weight of 7 boys = 56 kg.

Total weight of 7 boys = (56 × 7) kg = 392 kg.

Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg

Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)

= (392 - 338) kg

Hence, the weight of the seventh boy is 54 kg.

2 . A cricketer has a mean score of 58 runs in nine innings. Find out how many runs are to be scored by him in the tenth innings to raise the mean score to 61.

Mean score of 9 innings = 58 runs.

Total score of 9 innings = (58 x 9) runs = 522 runs.

Required mean score of 10 innings = 61 runs.

Required total score of 10 innings = (61 x 10) runs = 610 runs.

Number of runs to be scored in the 10th innings 

= (total score of 10 innings) - (total score of 9 innings)

= (610 -522) = 88. 

Hence, the number of runs to be scored in the 10th innings = 88.

3.  The mean of five numbers is 28. If one of the numbers is excluded, the mean gets reduced by 2. Find the excluded number.

Mean of 5 numbers = 28.

Sum of these 5 numbers = (28 x 5) = 140.

Mean of the remaining 4 numbers = (28 - 2) =26.

Sum of these remaining 4 numbers = (26 × 4) = 104.

Excluded number

= (sum of the given 5 numbers) - (sum of the remaining 4 numbers)

= (140 - 104)

= 36.  Hence, the excluded number is 36.

4 . The mean weight of a class of 35 students is 45 kg. If the weight of the teacher be included, the mean weight increases by 500 g. Find the weight of the teacher.

Mean weight of 35 students = 45 kg.

Total weight of 35 students = (45 × 35) kg = 1575 kg.

Mean weight of 35 students and the teacher (45 + 0.5) kg = 45.5 kg.

Total weight of 35 students and the teacher = (45.5 × 36) kg = 1638 kg.

Weight of the teacher = (1638 - 1575) kg = 63 kg.

Hence, the weight of the teacher is 63 kg.

5. The average height of 30 boys was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the correct mean.

Calculated average height of 30 boys = 150 cm.

Incorrect sum of the heights of 30 boys

= (150 × 30)cm

Correct sum of the heights of 30 boys

= (incorrect sum) - (wrongly copied item) + (actual item)

= (4500 - 135 + 165) cm

Correct mean = correct sum/number of boys

= (4530/30) cm

Hence, the correct mean height is 151 cm.

6. The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.

Calculated mean of 16 items = 30.

Incorrect sum of these 16 items = (30 × 16) = 480.

Correct sum of these 16 items

= (incorrect sum) - (sum of incorrect items) + (sum of actual items)

= [480 - (22 + 18) + (32 + 28)]

Therefore, correct mean = 500/16 = 31.25.

Hence, the correct mean is 31.25.

7. The mean of 25 observations is 36. If the mean of the first observations is 32 and that of the last 13 observations is 39, find the 13th observation.

Mean of the first 13 observations = 32.

Sum of the first 13 observations = (32 × 13) = 416.

Mean of the last 13 observations = 39.

Sum of the last 13 observations = (39 × 13) = 507.

Mean of 25 observations = 36.

Sum of all the 25 observations = (36 × 25) = 900.

Therefore, the 13th observation = (416 + 507 - 900) = 23.

Hence, the 13th observation is 23.

8. The aggregate monthly expenditure of a family was $ 6240 during the first 3 months, $ 6780 during the next 4 months and $ 7236 during the last 5 months of a year. If the total saving during the year is $ 7080, find the average monthly income of the family.

Total expenditure during the year

= $[6240 × 3 + 6780 × 4 + 7236 × 5]

= $ [18720 + 27120 + 36180]

Total income during the year = $ (82020 + 7080) = $ 89100.

Average monthly income = (89100/12) = $7425.

Hence, the average monthly income of the family is $ 7425.

Arithmetic Mean

Word Problems on Arithmetic Mean

Properties of Arithmetic Mean

Properties Questions on Arithmetic Mean

• 9th Grade Math

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Statistics and probability

Course: statistics and probability   >   unit 3.

• Statistics intro: Mean, median, & mode
• Mean, median, & mode example

Mean, median, and mode

• Calculating the mean
• Calculating the median
• Choosing the "best" measure of center
• Mean, median, and mode review
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
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The mean , or arithmetic mean, is a concept usually introduced around 4th grade and revisited in successive grades.  Often referred to as the “average”, the mean is measure of central tendency.  In other words, the mean gives you a reference point for understanding a data set and comparing it to other data.

I prefer to introduce these statistical measurements separately, allowing students to master each skill before moving on.  On this page you will find free math worksheets devoted exclusively to finding the mean of a data set.

Like most of the worksheets at www.imathworksheets.com , these free Mean Worksheets start off with simple integers that most students can complete mentally, then progress to more difficult calculations involving story problems.  All of the Mean Worksheets come with step-by-step instructions and examples to guide students.  There’s also plenty of space for students to show their work so that you can see which step is challenging them.

Feel free to use these Mean worksheets to help you differentiate instruction.  The lower performers in your class can begin with Mean Worksheet #1, while your more ambitious students tackle the word problems in Worksheets #4, #5, and #6!

Mean Worksheet #1  – This is a 10 problem worksheet where students will find the mean of a simple set of data.  Only integers are included, and all of the means will be whole numbers.  Most of the data are single-digit numbers with a few simple two-digit numbers included.

Mean Worksheet 1 RTF Mean Worksheet 1 PDF Preview Mean Worksheet 1 in Your Browser View Answers

Mean Worksheet #2   – This is a 10 problem worksheet where students will find the mean of a simple set of data.  The data sets only include one and two digit integers.  There are some whole number means, but also some means  that go out to the tenths decimal place.

Mean Worksheet 2 RTF Mean Worksheet 2 PDF Preview Mean Worksheet 2 in Your Browser View Answers

Mean Worksheet #3   – This is a 10 problem worksheet where students will find the mean of a set of data.  Their will be some whole number means, as well as some means that go out to the tenths and hundredths place.

Mean Worksheet 3 RTF Mean Worksheet 3 PDF Preview Mean Worksheet 3 in Your Browser View Answers

Mean Worksheet #4   – This is a 10 problem worksheet featuring Mean Story Problems.  Students will calculate the mean from real-life, relevant word problems.  The data sets are mostly one and two digit numbers, and the means will all be whole numbers.

Mean Worksheet 4 RTF Mean Worksheet 4 PDF Preview Mean Worksheet 4 in Your Browser View Answers

Mean Worksheet #5   – This is a 10 problem worksheet featuring Mean Story Problems.  Students will calculate the mean from more real-life, relevant word problems.  The data sets are mostly one and two digit numbers, and most of the means will be go out to the tenths decimal place.

Mean Worksheet 5 RTF Mean Worksheet 5 PDF Preview Mean Worksheet 5 in Your Browser View Answers

Mean Worksheet #6   – This is a 7 problem worksheet featuring Mean Story Problems.  Each word problem includes a twist, or red herring to make students think a little harder about which numbers to include in the data set.  The answers are a mix of whole numbers an decimals.

Mean Worksheet 6 RTF Mean Worksheet 6 PDF Preview Mean Worksheet 6 in Your Browser View Answers

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• Averages and range

Mean in math

Here you will learn about the mean, including what the mean is and how to find the mean.

Students will first learn about the mean in math as part of statistics and probability in 6 th grade.

What is the mean in math?

The mean in math , specifically the arithmetic mean, is a type of average calculated by finding the total of the values and dividing the total by the number of values.

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}

For example,

Calculate the mean of 3, \, 8, \, 10, \, 11 and 13.

\text {Mean }=\cfrac{\text { total }}{\text { number of values }}=\cfrac{3+8+10+11+13}{5}=\cfrac{45}{5}=9

9 is the mean of the data set.

This value, also known as the population mean, is a measure of central tendency. It summarizes a data set (population) with a single point. Median and mode are also measures of central tendency.

While all three measure center in some way, they are not the same. Mean can be thought of as sharing equally between all data points.

It is also important to consider that the number of observations (number of data points) changes how much each data point affects the mean.

Now add the data point 10 to each data set and recalculate the mean.

Notice that the mean in data set A grew by 1.25, while the mean of data set B grew by 0.7.

Since there are less data points in A , adding (or taking away) a data point impacts the mean more than in a data set with more points, like data set B.

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Statistics and Probability (6.SP.A.3) Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

How to calculate the mean in math

In order to calculate the mean in math:

Find the sum of the data points.

Divide the sum by the number of data points.

Write down the answer.

[FREE] Averages and Range Check for Understanding (Grade 6)

Use this quiz to check your 6th grade students’ understanding of averages and range. 10+ questions with answers covering a range of 6th grade averages and range topics to identify areas of strength and support!

Mean in math examples

Example 1: finding the mean.

Calculate the mean value of this list of numbers:

2 \quad 7 \quad 9 \quad 10 \quad 12

2+7+9+10+12=40

2 Divide the sum by the number of data points.

There are 5 values in the data set. Divide the total by 5.

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{40}{5}=8

3 Write down the answer.

The mean is 8.

Example 2: finding the mean

Calculate the mean value of this set of numbers to the nearest tenth.

13 \quad 16 \quad 17 \quad 17 \quad 18 \quad 20

13+16+17+17+18+20=101

There are 6 values in the data set. Divide the total by 6.

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{101}{6}=16.8333…

16.8333… to the nearest tenth is 16.8.

16.8 is the mean.

Example 3: finding the mean

Calculate the mean value of this set of data to the nearest hundredth.

11 \quad 13 \quad 14 \quad 15 \quad 19 \quad 20 \quad 22

11+13+14+15+19+20+22=114

There are 7 values in the data set. Divide the total by 7.

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{114}{7}=16.2857…

16.2857… rounded to the nearest hundredth is 16.29.

16.29 is the mean.

Example 4: finding the mean

Calculate the mean value of this list of numbers.

101 \quad 102 \quad 105 \quad 106 \quad 108

101+102+105+106+108=522

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{522}{5}=104.4

104.4 is the mean.

How to solve a problem involving the mean in math

In order to solve a problem involving the mean in math:

Use the mean and number of values to find the total.

Find the sum of the known data points.

Subtract the sum of the known data points from the first total to find the missing data point.

Problem solving involving mean examples

Example 5: problem solving.

The mean of 4 values is 10.

Here are 3 of the values:

6 \quad 9 \quad 12

Find the 4^{th} value.

The mean of 4 values is 10. Multiply these together to find the total of the 4 numbers.

\text{Total of 4 values}=\text{mean} \times \text{number of values}=10\times 4=40

\text{Total of 3 values}=6+9+12=27

The 4^{th} value is 13.

Alternatively, you could use the equation for finding the mean. You could use x as the missing value.

Then rearrange and solve.

\begin{aligned} \text{Mean} &= \cfrac{\text{total}}{\text{number of values}}\\\\ 10&=\cfrac{6+9+12+x}{4}\\\\ 10 &= \cfrac{27+x}{4}\\\\ 40&=27+x\\\\ 13&=x \end{aligned}

Example 6: problem solving

The mean of 5 values is 14.

Here are 4 of the values:

5 \quad 11 \quad 13 \quad 19

Find the 5^{th} value:

The mean of 5 values is 14. Multiply these together to find the total of the 5 numbers.

\text{Total of 5 values}=\text{mean} \times \text{number of values}=14\times 5=70

\text{Total of 4 values}=5+11+13+19=48

The 5^{th} value is 22.

Alternatively, you could use the equation for finding the mean. You could use x as the missing value. Then rearrange and solve.

\begin{aligned} \text{Mean} &= \cfrac{\text{total}}{\text{number of values}}\\\\ 14&=\cfrac{5+11+13+19+x}{5}\\\\ 14 &= \cfrac{48+x}{5}\\\\ 70&=48+x\\\\ 22&=x\\\\ \end{aligned}

Teaching tips for mean in math

• Introduce mean with countable objects (like counters or connecting cubes). This will allow students to physically add all the data points together and then share them equally – matching the procedure for the mean. Doing a hands-on activity like this helps students make sense of what the mean represents and understand why the procedure to find mean works.
• Worksheets can be useful for teaching mean, but be sure to include a variety of question types – ones with mean missing, ones with a data point missing and a mixture of number types, such as fractions, decimals and integers.

Easy mistakes to make

• Confusing the mean and the median They are both average values, but do not represent the same average and are found in different ways. The median is the middle number (middle value) when the data set is arranged in ascending (or descending) order. The mean is a number that represents the data points equally shared and is found by adding all the values and then dividing by the number of data points.
• Forgetting the mean can be a decimal The mean does not have to be a whole number. It can be a decimal or a fraction. It may be a decimal which needs rounding.
• Thinking the mean has to be a number in the data set While the mean can be a number in the data set, often it is not. Adding the data points together and then dividing them by the number of data points allows for the mean to be a number outside of what is in the data set.

Related averages and range lessons

• Mean median mode
• Mode in math
• Range in math

Practice mean in math questions

1) Find the arithmetic mean of this set of values:

5 \quad 7 \quad 8 \quad 8 \quad 9

First calculate the sum of the numbers in the given set.

5+7+8+8+9=37

Then divide the sum by the number of data points.

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{5+7+8+8+9}{5}\\\\ \text{Mean}&= \cfrac{37}{5}\\\\ \text{Mean}&=7.4\\\\ \end{aligned}

2) Find the mean of this list of values:

3 \quad 5 \quad 6 \quad 9

First calculate the sum of the values.

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{3+5+6+9}{4}\\\\ \text{Mean}&= \cfrac{23}{4}\\\\ \text{Mean}&=5.75\\\\ \end{aligned}

3) Find the mean of this list of values. Round your answer to the nearest hundredth.

7 \quad 8 \quad 9 \quad 10 \quad 10 \quad 11

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{7+8+9+10+10+11}{6}\\\\ \text{Mean}&= \cfrac{55}{6}\\\\ \text{Mean}&=9.16666…\\\\ \text{Mean}&=9.17\\\\ \end{aligned}

4) Which data set has a mean of 6?

Since each of the given data sets have 6 data points, the total of the data points will be 6 \times 6 (the mean times the number of data points).

\begin{aligned} & 1+15+2+1+17+0=36 \\\\ & \text { Mean }=\cfrac{36}{6} \\\\ & \text { Mean }=6 \end{aligned} 

5) The mean of 4 numbers is 9.

Here are 3 of the numbers:

6 \quad 8 \quad 15

What is the 4^{th} number?

The total of 4 numbers is:

\text{Total of 4 values}=\text{mean} \times \text{number of values}=9\times 4=36

The total of 3 numbers is:

The difference between the totals is:

The 4^{th} number is 7.

6) The mean of 6 numbers is 12.

Here are 5 of the numbers:

7 \quad 9 \quad 11 \quad 13 \quad 18

What is the 6^{th} number?

\text{Total of 4 values}=\text{mean} \times \text{number of values}=12\times 6=72

The total of 5 numbers is:

7+9+11+13+18=58

The 6^{th} number is 14.

Mean in math FAQs

No, the context in which the sample was collected may include negative numbers. For example, temperatures or account balances.

Instead of counting all data points equally, the mean of a set is found by counting (or “weighing”) certain data points more than others.

A way to quantify the amount of variation around the mean within a data set or population.

In a normal distribution, all the measures of center are the same and are exactly at the center value. Thinking about the data in percentages, 68\% of the points are within one standard deviation of the mean, median and mode.

Geometric mean and harmonic mean are two other types of means. These are both addressed in upper level mathematics.

The next lessons are

• Frequency table
• Representing data
• Frequency graph
• Cumulative frequency

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Word Problems: When to Use Mean, Median, and Mode - Expii

Word problems: when to use mean, median, and mode, explanations (4).

(Video) Word Problems: When to Use Mean, Median, and Mode

by Anusha Rahman

This video by Anusha Rahman goes over three examples of when to use mean, median, and mode in word problems.

Example 1: Jessie’s math teacher said that their final class grade is based on the average of all exam grades. Jessie has gotten a 93,87,71, and 97 on her math exams. What measure of central tendency will she use to calculate the average? What is the average?

We know that when we see the word average , we're talking about the mean ! So, we will be using mean to solve this problem.

Mean=93+87+71+974 =3484 =87

So, we know that Jessie's average score on her math exams was 87!

Example 2: Meghan wants to get an accurate picture of household salary in her neighborhood. The salaries for people in her neighborhood are: 87,000;94,000;103,000;97,000; and 55,000. What measure of central tendency will most accurate describe the data?

At first glance, we can see that there is an outlier at 55,000. At outlier is going to mess with the mean, and mode (the frequency) is not going to give us a lot of information. So, we will use the median, or the middle value.

Step 1: Put the salaries in numerical order, from least to greatest.

55,000,87,000,94,000,97,000,103,000

Step 2: Cross of the numbers evenly from each side until you're left with a middle number.

In this case, the median is 94,000.

Example 3: At her birthday party, Charlotte asks everyone to go around and say their favorite Taylor Swift song. In order, people respond with:

All Too Well, Shake it Off, Black Space, New Romantics, All Too Well, New Romantics, All Too Well, and Mr Perfectly Fine

Charlotte wants to know which song is the most popular. What measure of central tendency should she use?

Because the question is asking for the most popular song, we know we are looking for the mode ! We also know that the mean and median cannot be used for nominal or qualitative data.

Our first step is to make a frequency table with the data.

SongFrequencyAll Too Well3Shake It Off1Blank Space1New Romantics2Mr. Perfectly Fine1

From this frequency table, we can see that All Too Well has the highest frequency, so that is the mode of the data set, and it is the most popular song!

When solving word problems with central tendency, it's important to understand the pros and cons of each type . Let's break them down.

PRO: asymmetrical data (with an outlier)

CON: doesn't work with qualitative or nominal data

PRO: symmetrical data (no outlier)

PRO: big data sets

CON: highly affected by outliers

PRO: working with qualitative or nominal data

PRO: very easy to look at and read graphs/charts

CON: very limited in what it tells you

Related Lessons

(video) understanding and calculating measures of central tendency.

by Nurse Killam

This video by Nurse Killam explains when to use mean, median, and mode.

She starts with the definition of a normal distribution . All of the measures of central tendency work best with a normal distribution.

The mean is the average of a data set. We find the mean by adding all of the values, and dividing the sum by the number of values. The mean gets more accurate with more values.

Mean Benefits

• Best measure for symmetrical distributions
• Influenced by all data
• Most reliable
• Good for interval and ratio data

Mean Limitations

• Works best with no outliers

The median is the middle value of a set. We find this by ordering the data, then finding the literal middle number. Even sized sets will have two numbers that share the middle. The median of these sets is the average of those two numbers.

Median Benefits

• Good for asymmetrical data
• Works for ordinal, interval, and ratio data

Median Limitations

• Does not account for extreme scores
• Not algebraically defined
• Not appropriate for nominal data

The mode is the value that occurs the most in a set. We find this by ordering the set and finding the number that repeats the most.

Mode Benefits

• The only appropriate measure for nominal data
• Can also be used for ordinal, interval, and ratio data

Mode Limitations

• Cannot be used if all scores are different
• There can be several modes
• Cannot be used for further calculations

Using the Measures of Central Tendency: Media, Median, and Mode

When you're faced with word problems where you're asking to look into the distribution of a dataset, it can feel intimidating to figure out whether to use mean, median, or mode. The good news is that because each measure is different, they each have their own advantages and disadvantages!

Image source: by Anusha Rahman

Let's go through some example problems.

Which measure of central tendency is the most effective way to analyze the following dataset? 4,7,9,10,11,18,2,20,15,350,14

When to Use Mean, Median, and Mode

Three extremely important measures of central tendency (part of statistics ) are mean , median , and mode . You can find the mean, median, and mode of most data sets. Sometimes, it is more advantageous to only focus on one or two of these measures of central tendency, depending on your data set. Let's find out when we should use the mean, median, or mode to accurately describe a data set.

The mean is the average value in a data set. We can calculate the mean using the following formula:

mean=sum of all valuestotal number of values

Advantages:

The mean is the best measure for symmetrical data distributions. It is affected by all of the data points, and the more data points you know, the more accurate the mean will be. The mean is helpful when working with interval and ratio data.

Limitations:

Outliers can really throw off the mean. For example, say we have the data set {1,4,5,6,6,7,8,8,9}. The average of this data set is 1+4+5+6+6+7+8+8+99=549=6. But, what if we had an outlier in the set? What if our new set was {1,4,5,6,6,7,8,8,9,4946}? When our new mean would be 1+4+5+6+6+7+8+8+9+494610=500010=500. See how much an outlier can throw off the mean?

The median is the value separating the first half of the data set with the second half. To find the median of a data set, first put the data set in order. Then, if there are an odd number of values, cross an equal amount of values off from the beginning of the list and the end of the list. The remaining data point in the middle of your list is the median! If there are an even number of values in your set, the median will be the average of the two middle values.

The median is helpful when describing asymmetrical data , and it also works for ordinal, ratio, or interval data. It is easy to find, especially if your data is in order. Also, it is not usually too affected by outliers.

Limitations

However, the fact that the median is not usually affected by outliers means that we can't use it to account for these extreme values. Also, the median is not helpful when we are working with nominal or qualitative data. Think about it ... how can you order a list of colors and find the middle color?

The mode of a data set is the value that occurs most frequently. We can find the mode by putting the data in order, and then identifying which value(s) repeat the most. Frequency tables , histograms , and frequency distributions can be helpful in identifying the mode.

Finding the mode can be especially helpful when we are working with qualitative data or nominal data , but it can also be used when we have ordinal, ratio, or interval data. It's easy to spot on a graph, frequency table, or even an ordered set of data. It is not usually affected by outliers.

There can be many modes, so finding the mode may not say much about how the data is distributed. On the other hand, if all the data points only occur once, there is no real mode. The mode is unstable, and can be easily influenced by certain changes in the data. Adding just one new data point can change the mode!

Mean, Mode, Median, Range Practice Questions

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Average Word Problems

These lessons help students to learn how to solve word problems involving average.

Related Pages Average Problems Average Speed Problems More Lessons for Algebra Math Worksheets

There are three main types of average problems commonly encountered in school algebra: Average (Arithmetic Mean), Weighted Average and Average Speed .

Average (Arithmetic Mean)

The following diagram shows the average (arithmetic mean) formula. Scroll down the page for more examples and solutions on how to use formula to solve average word problems.

Example: The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, the average of the remaining numbers is 15. What is the number that was removed?

Step 1: The removed number could be obtained by the difference between the sum of original 6 numbers and the sum of remaining 5 numbers i.e.

Number removed = sum of original 6 numbers – sum of remaining 5 numbers

Step 2: Using the formula

Sum of Terms = Average &times Number of Terms

Sum of original 6 numbers = 20 × 6 = 120 Sum of remaining 5 numbers = 15 × 5 = 75

Step 3: Using the formula from step 1

120 – 75 = 45

Answer: The number removed is 45.

Examples of average word problems

• The table below shows the total number of goals scored in each of 43 soccer matches in a regional tournament. What is the average number of goals scored per match, to the nearest 0.1 goal?
• Tom has taken 5 of the 8 equally weighted test in his U.S. History class this semester, and he has an average score of exactly 78.0 points. How many points does he need to earn on the 6th test to bring his average score to exactly 80.0 points?

How to solve word problems that involve finding the average of a group of numbers?

• All of the members of the Harvey family are very tall. Their heights are 81 inches, 78 inches, 71 inches, 75 inches and 70 inches. What is the average height of the 5 Harveys?
• There are 5 trees in Terry’s front yard. He measures each tree to find out how tall it is in inches and writes the measurement on a sheet of paper. This is Terry’s list: 98, 94, 41, 96, and 11. What is the average height of a tree in Terry’s front yard?

Word Problems with Averages

• Timothy’s average score on the first four tests was 76. On the next 5 tests, his average score was 85. What was his average score on all the 9 tests?
• Tracy mowed lawns for 2 hours and earned $7.40 per hour. Then, she was windows for 3 hours and earned $6.50 per hour. What were Tracy’s average earnings per hour for all 5 hours?
• After taking 3 quizzes, your average is 72 out of 100. What must your average be on the next five quizzes to increase your average to 77?

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Average word problems

These average word problems are based on the arithmetic mean usually called average or mean. To get the average, use the formula for average you see here . First, study the example below.

A word problem about finding the average

American Communications Network (ACN) of Troy, Michigan, also markets prepaid phone cards, which it refers to as “equity calling cards.” If ACN employs 214,302 persons in 32 locations, on the average, how many employees work at each location?

Divide 214302 by 32 to find on average the number of people working at each location.

214302/32 = 6696.9375

If rounded to the nearest one, on average, 6697 work at each of the 32 locations of American Communications Network (ACN)

More average word problems

Problem #1:

A student scores 85, 90, 80, 90, and 100 on 5 quizzes. What is the average score on these 5 quizzes?

Problem #2: The number of hours Noemy worked this week for the past 5 days is 8, 6, 5, 9, 8. What is Noemy's average hour?

Problem #3:

Mark works 3 jobs to make ends meet. His first job pays him 200/week, his second 300/week, and his third 100/week. What is Mark's average income/week per job?

More challenging average word problems

Problem #4:

The average age of a husband and a wife is 30. What is the wife's age is the husband's age is 40?

Problem #5: The average of three numbers is 6. When one number is removed from the list, the average is 5. What is the number that was removed from the list?

Problem #6:

A math student scored 75, 70, 85, 90, 100 on the first five tests he took . After he took his sixth math test, the average is now 85. What did he score on the sixth test?

Problem #7:

The average time a man spent watching TV daily for the past week is 4 hours. If we remove one of these days, the average time he spent watching TV is now 3.5. How many hours of TV time did the man have on the day we removed?

Problem #8: The average of four numbers is 6. When one number is added to the list, the average is again 6. What number what added to the list?

Problem #9: The average daily sales generated by a store for the past 20 days is 1250. What is the total amount of sales generated by the store for the past twenty days?

Problem #10: The average of two numbers is 5. One of the two numbers is four times as big as the other. What are these two numbers?

Problem #11: The recorded temperatures in a country during a period of 4 days was 29 degrees Celsius, 28 degrees Celsius, 25 degrees Celsius, and 26 degrees Celsius. Another country recorded temperatures of  25 degrees Celsius, 30 degrees Celsius, 27 degrees Celsius, and 24 degrees Celsius. Which country had an average temperature that was colder?

See below an easy but tricky average word problem

How to find the average

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Mean and Standard deviation Problems with Solutions

Mean and standard deviation problems along with their solutions at the bottom of the page are presented. Problems related to data sets as well as grouped data are discussed.

• Consider the following three data sets A, B and C. A = {9,10,11,7,13} B = {10,10,10,10,10} C = {1,1,10,19,19} a) Calculate the mean of each data set. b) Calculate the standard deviation of each data set. c) Which set has the largest standard deviation? d) Is it possible to answer question c) without calculations of the standard deviation?
• A given data set has a mean ? and a standard deviation ?. a) What are the new values of the mean and the standard deviation if the same constant k is added to each data value in the given set?Explain. b) What are the new values of the mean and the standard deviation if each data value of the set is multiplied by the same constant k?Explain.
• If the standard deviation of a given data set is equal to zero, what can we say about the data values included in the given data set?
• mean of Data set A = (9+10+11+7+13)/5 = 10 mean of Data set B = (10+10+10+10+10)/5 = 10 mean of Data set C = (1+1+10+19+19)/5 = 10
• Standard Deviation Data set A = √[ ( (9-10) 2 +(10-10) 2 +(11-10) 2 +(7-10) 2 +(13-10) 2 )/5 ] = 2 Standard Deviation Data set B = √[ ( (10-10) 2 +(10-10) 2 +(10-10) 2 +(10-10) 2 +(10-10) 2 )/5 ] = 0 Standard Deviation Data set C = √[ ( (1-10) 2 +(1-10) 2 +(10-10) 2 +(19-10) 2 +(19-10) 2 )/5 ] = 8.05
• Data set C has the largest standard deviation.
• Yes, since data Set C has data values that are further away from the mean compared to sets A and B.
• We limit the discussion to a data set with 3 values for simplicity, but the conclusions are true for any data set with quantitative data. Let x, y and z be the data values making a data set. The mean ? = (x + y + z) / 3 The standard deviation ? = √[ ((x - ?) 2 + (y - ?) 2 + (z - ?) 2 )/3 ] We now add a constant k to each data value and calculate the new mean ?'. ?' = ((x + k) + (y + k) + (z + k)) / 3 = (x + y + z) / 3 + 3k/3 = ? + k We now calculate the new mean standard deviation ?'. ?' = √[ ((x + k - ?') 2 +(y + k - ?') 2 +(z + k - ?') 2 )/3 ] Note that x + k - ?' = x + k - ? - k = x - ? also y + k - ?' = y + k - ? - k = y - ? and z + k - ?' = z + k - ? - k = z - ? Therefore ?' = √[ ((x - ?) 2 +(y - ?) 2 +(z - ?) 2 )/3 ] = ? If we add the same constant k to all data values included in a data set, we obtain a new data set whose mean is the mean of the original data set PLUS k. The standard deviation does not change.
• We now multiply all data values by a constant k and calculate the new mean ?' and the new standard deviation ?'. ?' = (kx + ky + kz) / 3 = k? ?' = √[ ((kx - k?) 2 +(ky - k?) 2 +(kz - k?) 2 )/3 ] = |k| ? If we multiply all data values included in a data set by a constant k, we obtain a new data set whose mean is the mean of the original data set TIMES k and standard deviation is the standard deviation of the original data set TIMES the absolute value of k.
• Again, we limit the discussion to a data set with 4 values for simplicity, but the conclusions are true for any data set with quantitative data. Let x, y, z and w be the data values making a data set with mean ?. The standard deviation ? = √[ ((x - ?) 2 + (y - ?) 2 + (z - ?) 2 + (w - ?) 2 )/3 ] Let ? = 0, hence √[ ((x - ?) 2 + (y - ?) 2 + (z - ?) 2 + (w - ?) 2 )/3 ] = 0 Which gives (x - ?) 2 + (y - ?) 2 + (z - ?) 2 + (w - ?) 2 = 0 All terms in the equation are positive and therefore, the above equation is equivalent to (x - ?) 2 = 0, (y - ?) 2 = 0, (z - ?) 2 = 0 and (w - ?) 2 = 0. Which gives x = y = z = w = ? : all data values in the set with ? = 0 are equal.
• Let x i be the i th salary and f i be the corresponding frequency. mean of grouped data = ? = (?x i *f i ) / ?f i = (3500*5 + 4000*8 + 4200*5 + 4300*2) /(5 + 8 + 5 + 2) = $3955 b) standard deviation of grouped data = √[ (?(x i -?) 2 *f i ) / ?f i ] = √[ (5*(3500-3955) 2 +8*(4000-3955) 2 +5*(4200-3955) 2 +2*(4300-3955) 2 ) /(20) ] = 282 (rounded to the nearest unit)

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The Mean Problem

There are four unknown numbers.

The mean of the first two numbers is 4. The mean of the first three numbers is 9.

The mean of all four numbers is 15.

If one of the four numbers is 2, what are the others?

PRACTICE PROBLEMS ON MEAN MEDIAN AND MODE

Problem 1 :

Find the (i) mean  (ii) median  (iii) mode for each of the following data sets :

a)  12, 17, 20, 24, 25, 30, 40

b)  8, 8, 8, 10, 11, 11, 12, 12, 16, 20, 20, 24

c)  7.9, 8.5, 9.1, 9.2, 9.9, 10.0, 11.1, 11.2, 11.2, 12.6, 12.9

d)  427, 423, 415, 405, 445, 433, 442, 415, 435, 448, 429, 427, 403, 430, 446, 440, 425, 424, 419, 428, 441

Problem 2 :

Consider the following two data sets :

Data set A : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12

Data set B : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 20

a)  Find the mean for both Data set A and Data set B.

b)  Find the median of both Data set A and Data set B.

c)  Explain why the mean of Data set A is less than the mean of Data set B.

d)  Explain why the median of Data set A is the same as the median of Data set B

Problem 3 :

The table given shows the result when 3 coins were tossed simultaneously 40 times. The number of heads appearing was recorded.

Calculate the :   a)  mean     b)  median     c)  mode

Problem 4 :

The following frequency table records the number of text messages sent in a day by 50 fifteen-years-olds

a)  For this data, find the : (i) mean   (ii)  median   (iii)  mode

b)  construct a column graph for the data and show the position of the measures of centre (mean, median and mode) on the horizontal axis.

c)  Describe the distribution of the data.

d)  why is the mean smaller than the median for this data ?

e)  which measure of centre would be the most suitable for this data set ?

Problem 5 :

The frequency column graph alongside gives the value of donations for an overseas aid organisation, collected in a particular street.

a)  construct the frequency table from the graph.

b)  Determine the total number of donations.

c)  For the donations find the :  (i)  mean   (ii)  median   (iii)  mode

d) which of the measures of central tendency can be found easily from the graph only ?

Problem 6 :

Hui breeds ducks. The number of ducklings surviving for each pair after one month is recorded in the table.

a)  Calculate the : (i)  mean   (ii)  median   (iii) mode

b)  Is the data skewed ?

c)  How does the skewness of the data affect the measures of the middle of the distribution ?

Answers 

(c)  the mean of A is less than the mean of B.

(d)   median is the same.

(3)  (a)   Mean  =  1.4     (b)   median  =  1  (c)     mode  =  1

(4)  

(a)   (i)   Mean  =  5.74  (ii)     median  =  7  (iii)   mode  =  8

(c)     bimodal data.

The mean takes into account the full range of numbers of text messages and is affected by extreme values. Also, the value which is lower than the median is well below it.

(e)   The median

(5)  

(b)   ∑f  =  30

(c)  (i)   Mean  =  $2.9  (ii)   median  =  $2  (iii)   mode  =  $2

(6)  

(a)  (i)  Mean  =  4.25    (ii)   median  =  5   (iii)   mode  =  5

c)   By observing the graph, the mean is less than the median and mode.

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Mean average - including problem solving and applying knowledge to different scenarios

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

22 February 2018

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."