Topic 7 of 12

Mean, median, and data displays

Step 8 · Ratios and negative numbers topics

Sum up a set of numbers with one typical value, and picture the whole set with a dot plot or a histogram.

Seeing the data

Seven friends count the books they read last month: 2, 3, 3, 4, 5, 5 and 13. A dot plot shows the whole set at once. It is a number line with one dot above it for each value.

A dot plot of the seven values. Most are between 2 and 5. One value, 13, sits far from the rest.

Mean

The mean is what each person would get if the total were shared out equally. Add all the values, then divide by how many there are.

\text{mean} = \frac{2 + 3 + 3 + 4 + 5 + 5 + 13}{7} = \frac{35}{7} = 5

Median

The median is the middle value when the values are in order. There are 7 values here, so the middle one is the 4th. The median is 4.

With an even number of values there are two middle values. The median is the number halfway between them, which is their mean.

Example: Median of an even number of values

Find the median of 8, 3, 10, 5, 7, 12.

Put the values in order: 3, 5, 7, 8, 10, 12.
There are 6 values, so the two middle ones are the 3rd and 4th: 7 and 8.
Find the number halfway between them: $\frac{7 + 8}{2} = 7.5$ .

Answer7.5

Which one to use

The friend who read 13 books pulled the mean up to 5, which is more than most of the group read. The median, 4, was not affected. When a few values are far from the rest, the median usually describes a typical value better. When the values are spread fairly evenly, the mean and the median are close together and either one works.

Histograms

With a lot of data, a dot plot gets crowded. A histogram groups the values into equal ranges, such as 0 to 4, 5 to 9 and 10 to 14, and draws one bar for each range. The height of a bar shows how many values fall in that range. The bars touch each other, because the ranges run along the number line with no gaps.

Try it yourself

Work each one on paper first, then open the answer to check.

Find the mean of 4, 8, 6, 10 and 7.
Show answer
7, because the total is 35 and $35 \div 5 = 7$ .
Find the median of 12, 5, 9, 3 and 7.
Show answer
7, the middle value when they are put in order: 3, 5, 7, 9, 12.
Find the median of 6, 2, 9 and 4.
Show answer
5, halfway between the two middle values. In order they are 2, 4, 6, 9.
Five houses sold for 200, 210, 220, 230 and 900 thousand dollars. Find the mean and the median. Which describes a typical house better?
Show answer
The mean is 352 thousand ( $1760 \div 5$ ) and the median is 220 thousand. The median is better here, because one very expensive house pulls the mean up.
A dot plot shows the values 1, 1, 2, 2, 2, 3 and 5. How many dots are above 2, and what is the median?
Show answer
3 dots are above 2. The median is 2, the 4th of the 7 values.
The mean of four numbers is 6. Three of them are 4, 5 and 8. What is the fourth?
Show answer
7, because the total must be $4 \times 6 = 24$ , and $24 - 4 - 5 - 8 = 7$ .