1. S 14 Histograms with Unequal Class Intervals
GCSE Maths Statistics & Probability
S 14
Histograms with Unequal Class Intervals
Subject Content Reference: S3.2h

2. Frequency Density = Frequency ÷ Class Interval
A histogram with unequal class intervals uses the notion of frequency density . .
Frequency Density = Frequency ÷ Class Interval
Example
The following table shows the time (in minutes) taken for 100 athletes to complete a marathon. Draw up a histogram and
from it, estimate the number of athletes who took between 21/2 and 3 hours to finish:
Time (t) in minutes
125 ≤ t < 140
140 ≤ t < 160
160 ≤ t < 170
170 ≤ t < 195
195 ≤ t < 235
235 ≤ t < 285
Frequency 6 16 28 35 10 5
Frequency Density
0.4 (6 ÷ 15)
0.8 (16 ÷ 20)
2.8 (28 ÷ 10)
1.4 (35 ÷ 25)
0.25 (10 ÷ 40)
0.1 (5 ÷ 50)
Step 1:
Draw the axes for time and frequency density . .
Frequency Density
Time in minutes 1 2 3 50
100
150
200
250
Step 2:
Draw in the bars . .
Step 3:
Shade in the parts of the histogram between
21/2 and 3 hours (150 and 180 mins) . .
. . and calculate the frequency represented
= 1/2(16)
+ 28
+ 2/5(35) =
= 50
answer
It’s important to realise that the area of each rectangle
represents the frequency (frequency density x time)

3. Exercise 1
1) The following table shows the time taken for members of a cycling club to finish a 100 km charity ride:
Time (t) in minutes
120 ≤ t < 140
140 ≤ t < 155
155 ≤ t < 170
170 ≤ t < 195
195 ≤ t < 205
205 ≤ t < 285
Frequency 6 30 45 55 28 16
Frequency Density
a) Complete the table by calculating frequency densities:
b) Draw a histogram to show this information:
c) Use the histogram to estimate the number of cyclists completing the ride between 21/2 and 31/2 hours:

4. Exercise 1 (cont’d)
2) The following table shows the distance covered by cars in a 24 hour race:
Distance (d) in 1000 km
2.25 ≤ d < 2.4
2.4 ≤ d < 2.5
2.5 ≤ d < 2.75
2.75 ≤ d < 2.95
2.95 ≤ d < 3.25
3.25 ≤ d < 3.5
Frequency 3 5 15 18 12 8
Frequency Density
a) Complete the table by calculating frequency densities:
b) Draw a histogram to show this information:
c) Use the histogram to estimate the number of cars covering between 2650 and 3150 km: