GCSE Maths revision · Higher: probability and statistics

Histograms

Higher only

The idea

In a histogram the area of each bar is the frequency, so the height is frequency density $= \frac{\text{frequency}}{\text{class width}}$ .

Find each class width.
Frequency density $=$ frequency $\div$ class width.
To read a frequency back, multiply the bar's height by its width.

Watch out Plot frequency density, not frequency, on the vertical axis.

See it worked

Worked exampleNon-calculator3 marks

The table shows the times, $t$ minutes, that 55 people spent in a gallery.

Grouped frequency table: 0 to 10 minutes, 12; 10 to 15 minutes, 15; 15 to 25 minutes, 20; 25 to 45 minutes, 8.
Time (minutes)	Frequency
0 < t ≤ 10	12
10 < t ≤ 15	15
15 < t ≤ 25	20
25 < t ≤ 45	8

(a) A histogram is drawn. Work out the height of the bar for the class $10 < t \le 15$ .

(b) Work out the height of the bar for the class $25 < t \le 45$ .

(a) Frequency density $=$ frequency $\div$ class width $= 15 \div 5 = 3$ .
(b) Width 20: $8 \div 20 = 0.4$ .

Answer (a) 3; (b) 0.4

Your turn

Question 1Calculator allowed2 marks

A histogram class covers 15 < t $\le$ 25 minutes and contains 18 journeys, where t is the journey time. Work out the frequency density for this class.

Question 2Non-calculator5 marks

A histogram class 10 < t ≤ 30 has frequency 32. Its height is 8 cm. The next class 30 < t ≤ 45 has height 12 cm on the same scale.

(a) Find the frequency in the second class.

(b) Find the combined frequency of the two classes.

Question 3Calculator allowed3 marks

The table gives the classes and frequency densities for a histogram of times. All three classes have positive width.

Using the class midpoints, an estimate of the mean time is 20 seconds.

Work out $b$ .

Table of a histogram's classes. 0 < t ≤ 10: frequency density 1. 10 < t ≤ b: frequency density 2. b < t ≤ 40: frequency density 1.
Time (t seconds)	Frequency density
0 < t ≤ 10	1
10 < t ≤ b	2
b < t ≤ 40	1

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