Averages and Spread

Introduction

A list of a thousand numbers is hard to understand. A few well-chosen summary numbers can tell the story. The most common are the averages, and it is important to know that "average" can mean different things.

The three averages

Take the number of minutes five friends spent reading yesterday:

4, 5, 5, 6, 50

• Mean: add everything up and divide by how many there are. (4 + 5 + 5 + 6 + 50) ÷ 5 = 70 ÷ 5 = 14.
• Median: the middle value when the numbers are in order. With five numbers, the middle one is the third: 5.
• Mode: the value that appears most often: 5.

Notice the difference. The mean is 14, but four of the five friends read for six minutes or less. One friend who read for 50 minutes pulled the mean up. The median, 5, describes a typical friend better.

For an even number of values, the median is the mean of the two middle ones. For 2, 4, 6, 8 it is (4 + 6) ÷ 2 = 5.

Which average to use?

• Use the mean when values are fairly balanced, with no extreme ones, such as test scores in a class.
• Use the median when there are outliers (unusually high or low values) or the data is lopsided, such as incomes or house prices. A few billionaires would pull the mean income of a town up hugely, while the median would barely move.
• Use the mode for categories or the most common value, such as the most popular shoe size.

When someone says "the average", ask which one.

Spread: how different are the values?

Two groups can have the same mean and look very different:

Group A: 49, 50, 50, 51
Group B: 10, 30, 70, 90

Both have a mean of 50. Group A is tightly clustered. Group B is very spread out.

• Range = largest minus smallest. For A it is 2, and for B it is 80.
• Standard deviation is a more detailed measure of typical distance from the mean. A small standard deviation means the values stay near the mean. Software calculates it for you.
• Quartiles split the ordered data into four equal parts, and the interquartile range is the middle half. They are less affected by outliers.

Always report both a typical value and a measure of spread.

Outliers

An outlier is a value far from the rest. Do not just delete it. Ask:

1. Is it a mistake, such as an extra zero? If so, fix or remove it, and note that you did.
2. Is it real? Then it may be the most interesting result.

In a spreadsheet

Spreadsheet programs have simple formulas. If your numbers are in cells A1 to A5:

=AVERAGE(A1:A5)
=MEDIAN(A1:A5)
=MODE(A1:A5)
=MAX(A1:A5)-MIN(A1:A5)

These give the mean, median, mode and range. The same free spreadsheet tools in your browser or computer can do all of this.

Try it yourself

Collect eight numbers from your own life, for example daily steps. Calculate the mean, median and range. Then change the biggest number to something ten times larger, and see which summary changed the most.

Summary

• The mean adds and divides, the median is the middle value and the mode is the most common one.
• Outliers pull the mean, so the median is better for lopsided data.
• Report a typical value and the spread, such as the range, because averages alone can hide a lot.