Mean, Median, Mode, and Range: How to Find and Choose Them

Mean, median, mode, and range are usually taught together, and most homework asks you to find all four for one list of numbers. Finding them is not hard. The harder question is which number best describes the data, so one data set is worked all the way through below.

1. What these numbers are for

The Common Core grade 6 standards describe the big idea: a measure of center summarizes all the values in a data set with a single number, while a measure of variation describes how the values vary with a single number. Mean, median, and mode are measures of center. Range is the simplest measure of variation.

The same standards also stress where data comes from. A statistical question is one that expects variability in the answers. Asking one person's age is not a statistical question, but asking how old the students in a whole school are is one, because the ages will differ.

2. The data set

Seven students recorded how many minutes they spent reading on one evening:

4, 5, 5, 6, 7, 8, 30

The numbers are already in order from least to greatest. If yours are not, sort them first. Several of the steps below depend on it.

3. Mean: add and divide

The mean is the total divided by the number of values.

• Total: 4 + 5 + 5 + 6 + 7 + 8 + 30 = 65
• Number of values: 7
• Mean: 65 / 7 = about 9.29 minutes

4. Median: the middle value

The median is the middle number once the list is in order. With 7 values, the middle one is the 4th: 4, 5, 5, 6, 7, 8, 30.

The median is 6 minutes.

If there is an even number of values, there is no single middle. Take the two middle values and find their mean. For example, in 4, 5, 5, 6, 7, 8 (six values), the middle two are 5 and 6, so the median is (5 + 6) / 2 = 5.5.

5. Mode: the most frequent value

The mode is the value that appears most often. Here, 5 appears twice and every other value appears once, so the mode is 5 minutes.

A data set can have one mode, more than one mode, or no mode at all if every value appears the same number of times.

6. Range: largest minus smallest

The range is the difference between the greatest and least values: 30 - 4 = 26 minutes.

A common slip is writing the range as "4 to 30." In most homework, the range is a single number, the difference.

7. Summary for this data set

8. What one unusual value does

Look at the mean again. Six of the seven students read between 4 and 8 minutes, yet the mean is above 9. The single value of 30 pulls it up.

Remove the 30 and recompute with the remaining six values (4, 5, 5, 6, 7, 8):

• Mean: 35 / 6 = about 5.83
• Median: 5.5
• Range: 8 - 4 = 4

Removing one value dropped the mean by almost 3.5 minutes and shrank the range from 26 to 4, but moved the median only half a minute. That is the main difference between the two most common measures of center: the mean responds to every value, including extreme ones, while the median depends only on the middle of the ordered list.

9. Choosing the right measure

The grade 6 standards ask students to relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. In practice:

• Use the median when a few values are far from the rest, as in the reading example. It describes a typical student better than the mean here.
• Use the mean when the data has no extreme values and you want every value to count, such as averaging several similar quiz scores.
• Use the mode for the most common category or value, such as the most popular shoe size. It is the only one of the three that works for non-number data like favorite colors.
• Report a measure of variation too. A center alone hides how spread out the data is. The grade 6 standards go beyond range to the interquartile range and mean absolute deviation, which are less affected by a single extreme value.

10. Comparing two groups

The grade 7 standards build on this by asking students to use measures of center and variability to draw informal comparative inferences about two populations. For example, if one class has a median reading time of 6 minutes and another has a median of 15 minutes, and both groups have similar spread, the second class clearly tends to read longer. If the spreads are huge compared with the difference in centers, the comparison is much less convincing.

11. Key takeaways

• Sort the data before finding the median, mode, or range.
• The mean is the total divided by the count; the median is the middle value, or the mean of the two middle values when the count is even.
• The mode is the most frequent value, and the range is the greatest value minus the least.
• One extreme value can move the mean a lot while barely changing the median.
• Choose the measure that fits the data and the question, and report spread along with the center.