S 14

Histograms with Unequal Class Intervals

Subject Content Reference: S3.2h

GCSE Maths Statistics & Probability

A histogram with unequal class intervals uses the notion of frequency density . . learnlearn

Example

Frequency Density = Frequency ÷ Class Interval

The following table shows the time (in minutes) taken for 100 athletes to complete a marathon. Draw up a histogram andfrom 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 . .

Step 2: Draw in the bars . .

Step 3: Shade in the parts of the histogram between21/2 and 3 hours (150 and 180 mins) . .

. . and calculate the frequency represented

= 1/2(16)

= 8 + 28 + 14

= 50 answer

It’s important to realise that the area of each rectanglerepresents the frequency (frequency density x time)

+ 2/5(35)+ 28

Fre

quen

cy D

ensi

ty

Time in minutes

1

2

3

50 100 150 200 250

Exercise 1

1) The following table shows the time taken for members of a cycling club to finish a 100 km charity ride:

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:

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

Exercise 1 (cont’d)

2) The following table shows the distance covered by cars in a 24 hour race:

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:

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