Year 6AC9M6ST01

Mean, Median, Mode and Range

Compare data distributions in terms of mode, range and shape. Note: <b>mean and median</b> are introduced in Year 7 (AC9M7ST01), so those parts are extension.

Read this first

These four numbers each say something different about a set of data. Three describe the middle. One describes the spread.

One high score pulls the mean up. Four of the five you scored below it.

Words you need

Mean: Add them all up, then share equally. Also called the average.
Median: Put them in order. The median is the middle one.
Mode: The one that appears most often.
Range: Biggest take away smallest. It shows the spread.

Worked example

e.g. Five you score 6, 7, 7, 8 and 12. Find the mean, median, mode and range.

Put them in order first. They already are: 6, 7, 7, 8, 12.
Mean. Add them: $6+7+7+8+12 = 40$ . Share by 5: $40 \div 5 = 8$ .
Median. The middle one of five is the third. That is 7.
Mode. 7 appears twice. Everything else appears once. The mode is 7.
Range. $12 - 6 = 6$ .

The mean is 8 but the median is 7. The single score of 12 drags the mean up. When one value is far from the rest, the median often describes the group better.

Practice

The pictures fade as you go. By the last few you are on your own.

1Find the mode of this data.with a picture

Answer5. It appears twice and everything else appears once.

2Find the range of this data.with a picture

Answer8. Biggest take away smallest:

10 - 2 = 8

3Find the mean of 4, 6, 8, 10 and 12.with a hint

Need a hint?

Add them all first, then divide by how many there are.

Answer8. They add to 40, and

40 \div 5 = 8

4Find the median of 9, 3, 7, 1 and 5.with a hint

Need a hint?

Put them in order first. That step is easy to forget.

Answer5. In order they are 1, 3, 5, 7, 9. The middle one is 5.

5Find the mean of 2, 5, 5 and 8.on your own

Answer5. They add to 20, and

20 \div 4 = 5

6Find the median of 4, 8, 2 and 6.on your own

Answer5. In order: 2, 4, 6, 8. With four values, take the middle two and find their mean:

(4+6) \div 2 = 5

7Find the range of 15, 3, 9 and 21.on your own

Answer18.

21 - 3 = 18

8Five people earn

\$30

\$32

\$34

\$36

and

\$200

. Which describes the group better, the mean or the median?on your own

AnswerThe median, which is

\$34

. The mean is

\$66.40

, but four of the five people earn far less than that. One very large value pulls the mean away from the group.

Think about it

Why can the mean be a number that is not in the data at all?

One good answerThe mean shares the total out equally. That fair share need not match anyone's actual value. Four people with 1, 2, 3 and 6 have a mean of 3, and only one of them has 3.

When is the median more useful than the mean?

One good answerWhen one or two values are far from the rest. House prices are a good example: a few very expensive houses pull the mean up, so the median gives a better sense of a typical price.

Can a set of data have no mode?

One good answerYes. If every value appears exactly once, nothing repeats, so there is no mode. A set can also have more than one mode.

Two classes both have a mean score of 70. Does that mean they did equally well?

One good answerNot necessarily. One class might have every score near 70, while the other has half at 40 and half at 100. The range would show that difference straight away.

Common mistakes

Finding the median without ordering the data first.

The median is the middle of the ordered list. Skipping the sort gives the wrong answer nearly every time.

Dividing by the wrong number when finding the mean.

Divide by how many values there are, not by the largest one.

Thinking the range is a middle value.

The range measures spread, not the middle. It is biggest take away smallest.

Assuming the mean is always the best average.

One unusual value can drag it a long way. Check whether the median describes the group better.