Data: averages and charts – Year 7 maths lesson to download as PDF

Example 1

Find the mode, median, mean and range of 2, 3, 3, 4, 5, 5, 5, 9.

Mode: 5 (it appears three times). Median: the middle two of 8 values are 4 and 5, so the median is 4.5. Mean: \(36\div8=4.5\). Range: \(9-2=7\).

Example 2

Find the median of 3, 7, 1, 9, 5.

In order: 1, 3, 5, 7, 9. The middle value is 5.

Example 3

Shoe sizes in a class of 20: size 3 (2 pupils), size 4 (5), size 5 (8), size 6 (4), size 7 (1). Find the mode, mean and median.

Mode: 5. Total: \(3\times2+4\times5+5\times8+6\times4+7\times1=6+20+40+24+7=97\). Mean: \(97\div20=4.85\). The 10th and 11th values are both 5 (cumulative frequencies 2, 7, 15), so the median is 5.

Example 4

Temperatures were 12, 13, 11, 14 and 40. Which average is most typical?

The mean is \(90\div5=18\), which is higher than four of the five values. The median is 13, which is more typical.

Example 5

40 pupils were asked how they travel to school. 16 walk. Find the angle for walking, and check the other angles in the chart.

Each pupil is \(360\div40=9^{\circ}\). Walking: \(16\times9=144^{\circ}\). Bus 108°, car 72° and cycle 36° complete the chart: \(144+108+72+36=360\).

Example 6

A bar chart shows how many pupils chose each fruit. How do you read it?

Find the bar, then read its height on the vertical axis. Always check the scale.

Example 7

Class A has a mean of 62 and a range of 30. Class B has a mean of 60 and a range of 12. Compare them.

Class A did slightly better on average, because its mean is higher. Class B was more consistent, because its range is smaller.

Worked example in context

The heights (in cm) of five players are 160, 165, 170, 172 and 193. Find the mean and the median. Which is better to describe a typical height?

Mean: \(860\div5=172\). Median: 170. The outlier 193 pulls the mean up a little, but both are close, so either is a good description.

Example 1

The heights of five plants in centimetres are 12, 15, 11, 15 and 22. A sixth plant is measured and the mean of all six becomes 15 cm. How tall is the sixth plant?

The mean of six plants is 15 cm, so their total height is \(6\times15=90\) cm. The first five plants add up to \(12+15+11+15+22=75\) cm. The sixth plant must make up the difference: \(90-75=15\) cm. This “total first” method works for every missing-value question about the mean: multiply the mean by the number of values to find the total, then subtract the values you know.

Example 2

Two shops sell sandwiches. Shop A’s daily sales over a week have a mean of 40 and a range of 6. Shop B’s have a mean of 40 and a range of 25. Compare the two shops.

The means are the same, so on average both shops sell 40 sandwiches a day. The ranges are very different. Shop A’s sales only vary by 6, so they are consistent and easy to plan for. Shop B’s sales vary by 25, so some days it sells many more and other days many fewer. A good comparison always mentions an average and the spread, and it uses the context: “Shop A is more reliable, so it is easier to know how many sandwiches to make.”

Example 3

A pie chart shows how 72 people voted. The angle for “yes” is 130°. How many people voted yes?

The whole circle of 360° stands for all 72 people, so each person is worth \(360\div72=5°\). The “yes” sector is 130°, so \(130\div5=26\) people voted yes. Always find the angle for one item first; it turns every pie chart question into a simple division.