Mean, Median, Mode: How To Find Them, With Examples

What Are Mean, Median And Mode?

Mean, median and mode are called measures of central tendency, or measures of center. Each one sums up a whole list of numbers with a single "typical" value, but they define "typical" in different ways.

Range is often taught alongside them. It is not a measure of center. It measures spread: the largest value minus the smallest.

How To Find The Mean

The mean is the sum of all the values divided by the number of values. It is what most people mean when they say "average."

Mean = sum of all values ÷ number of values

Example: find the mean of 4, 8, 6, 5, 3, 7, 9

1. Add the values: 4 + 8 + 6 + 5 + 3 + 7 + 9 = 42
2. Count the values: there are 7
3. Divide: 42 ÷ 7 = 6

The mean does not have to be one of the numbers in the list, and it does not have to be a whole number. The mean of 1, 2 and 4 is 7 ÷ 3 ≈ 2.33.

Finding A Missing Value From The Mean

You can work backward if you know the mean. Multiply the mean by the number of values to get the total, then subtract the values you know.

Example: your first four quiz scores are 80, 85, 90 and 75. What do you need on the fifth quiz to average 84?

1. Total needed: 84 × 5 = 420
2. Points so far: 80 + 85 + 90 + 75 = 330
3. Score needed: 420 – 330 = 90

A grade calculator does this kind of "what do I need" math for whole classes.

Weighted Mean

A weighted mean gives some values more importance than others. You multiply each value by its weight, add the results, then divide by the total weight.

Example: tests count 40%, homework 40% and the final 20%. Your scores are 90, 80 and 70.

(90 × 0.4) + (80 × 0.4) + (70 × 0.2) = 36 + 32 + 14 = 82

The plain mean of those scores would be 80, so the weights pulled the result up. GPA works the same way, with credit hours as weights. The GPA calculator handles that math.

How To Find The Median

The median is the middle value when the data is sorted from smallest to largest. Half the values sit below it and half sit above it.

Median With An Odd Number Of Values

When the count is odd, there is exactly one middle value.

Example: find the median of 9, 3, 7, 5, 4

1. Sort the values: 3, 4, 5, 7, 9
2. There are 5 values, so the middle one is the 3rd.
3. Median = 5

A quick way to find the position: add 1 to the count and divide by 2. For 5 values, (5 + 1) ÷ 2 = 3, so take the 3rd value.

Median With An Even Number Of Values

When the count is even, there are two middle values. The median is the mean of those two.

Example: find the median of 15, 3, 8, 20, 7, 12

1. Sort: 3, 7, 8, 12, 15, 20
2. There are 6 values, so the middle two are the 3rd and 4th: 8 and 12.
3. Median = (8 + 12) ÷ 2 = 10

With an even count, the median may not be one of the numbers in the list. That is normal.

How To Find The Mode

The mode is the value that appears most often. It is the only measure of center you can use with non-number data, like favorite colors or shoe brands.

Example: find the mode of 2, 3, 3, 5, 7, 7, 9

Both 3 and 7 appear twice, more than any other value. So this data set has two modes: 3 and 7.

A data set can have:

If every value appears once, most teachers say there is "no mode." Some textbooks instead say every value is a mode. Check which rule your class uses.

Range And Midrange

Range measures how spread out the data is, while midrange is a quick, rough measure of center. Both use only the largest and smallest values.

• Range = maximum – minimum
• Midrange = (maximum + minimum) ÷ 2

Example: for 3, 7, 8, 12, 15, 20

• Range = 20 – 3 = 17
• Midrange = (20 + 3) ÷ 2 = 11.5

Range is easy to find but sensitive to extreme values. For a better measure of spread that uses every value, use the standard deviation. The standard deviation calculator shows each step.

Mean, Median And Mode Examples, Start To Finish

These full examples find all four values (mean, median, mode and range) for one data set at a time.

Example 1: A Data Set With Repeats

Find the mean, median, mode and range of 13, 16, 12, 14, 19, 12, 14, 13, 14

1. Sort: 12, 12, 13, 13, 14, 14, 14, 16, 19
2. Mean: the sum is 127, and there are 9 values. 127 ÷ 9 ≈ 14.11
3. Median: 9 values, so the 5th value: 14
4. Mode: 14 appears three times, more than any other value: 14
5. Range: 19 – 12 = 7

Example 2: Test Scores

Five students score 72, 85, 90, 85 and 78.

1. Sort: 72, 78, 85, 85, 90
2. Mean: 410 ÷ 5 = 82
3. Median: the 3rd value is 85
4. Mode:85 appears twice
5. Range: 90 – 72 = 18

Here the median and mode are both 85, while the mean is lower at 82. The low score of 72 pulls the mean down more than it moves the median.

Example 3: Data In A Frequency Table

Sometimes data comes in a table that shows how often each value appears.

1. Mean: multiply each size by its count and add: (7 × 3) + (8 × 5) + (9 × 8) + (10 × 4) + (11 × 2) = 21 + 40 + 72 + 40 + 22 = 195. Then 195 ÷ 22 ≈ 8.86
2. Median: 22 values, so average the 11th and 12th. Sizes 7 and 8 cover the first 8 students, and size 9 covers students 9 through 16. Both the 11th and 12th are size 9, so the median is 9
3. Mode: size 9 has the highest count: 9
4. Range: 11 – 7 = 4

A shoe store would care most about the mode, since it shows which size to stock the most.

How Outliers Change The Mean, Median And Mode

An outlier is a value far away from the rest of the data. It can pull the mean a long way, while the median and mode barely move.

Example: seven homes on a street sell for (in thousands of dollars) 40, 42, 45, 47, 48, 50 and 250.

One expensive home pushed the mean up by almost $30,000, but the median moved only $1,000. That is why the median is often a better "typical" value when the data has outliers.

Skewed Data: Which Way Does The Mean Move?

In a skewed data set, the mean is pulled toward the long tail of the data, while the median stays closer to the middle of the pack.

Income is the classic example: a few very high earners pull the mean well above what most households earn, so income reports usually focus on the median.

When To Use The Mean, Median Or Mode

Each measure answers a slightly different question. Pick the one that fits your data and what you want to know.

A good habit is to calculate more than one. If the mean and median are close, the data is probably balanced. If they are far apart, look for outliers or skew before you report a single "average." For bigger data sets, the statistics calculator also finds quartiles and variance.

When Students Learn Mean, Median And Mode

In schools that follow the Common Core State Standards, measures of center are a grade 6 topic. Students learn that a measure of center sums up a whole data set with one number, then use the mean and median to describe real data.

Common Mistakes To Avoid

Most mistakes with mean, median and mode come from skipping the sort step or mixing up the definitions.

Practice Problems With Answers

Try these, then check below.

1. Find the mean of 10, 15, 20, 25, 30.
2. Find the median of 8, 2, 11, 5, 14, 3.
3. Find the mode of 4, 6, 6, 8, 9, 9, 9, 10.
4. Find the range of 21, 35, 17, 42, 28.
5. Your first three test scores are 88, 92 and 79. What do you need on the fourth to average 87?

Answers

1. 100 ÷ 5 = 20
2. Sorted: 2, 3, 5, 8, 11, 14. Middle two are 5 and 8, so the median is 6.5
3. 9 (it appears three times)
4. 42 – 17 = 25
5. 87 × 4 = 348, and 348 – 259 = 89

You can check any of these with the mean, median and mode calculator.

Summary

The mean is the average, the median is the middle value once the data is sorted, and the mode is the most common value. Range, the largest minus the smallest value, measures spread.

Use the mean for balanced data, the median for skewed data or outliers, and the mode for categories. Always sort before finding the median.

Next step: collect a small data set of your own, like the number of minutes you spend on homework each day for a week, and find all four values. Then check them with the mean median mode calculator.

Frequently Asked Questions