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10.2 Calculating statistics from grouped or continuous data

10.2 Calculating statistics from grouped or continuous data

Sets of data with lots of values may be written Mass (kg) 16–20 21–25 26–30 31–35 36–40 41–45
in grouped frequency tables.This example
Frequency 12 14 20 30 17 7
shows the masses of 100 girls,
to the nearest kilogram.
The masses are divided into six classes. This shows the overall shape of the distribution.

You do not know the separate mass of each girl, so you cannot find the
mode, the median, the mean or the range. You can estimate the median, Mass is a continuous variable.
the mean and the range. The next worked example shows how to do this. For example, the class 21–25 kg
includes girls with mass from 20.5
You cannot find the mode, but you can find the modal class. This is
kg to 25.5 kg.
the class with the largest frequency.

Worked example 10.2

Look at the table of girls’ masses, above.
a Find the modal class. b Estimate the median height. c Estimate the mean. d Estimate the range.

a The modal class is 31–35 kg. This class has the largest frequency (30).
b An estimate of the median is 31 kg. There are 100 girls. The median mass is between the 50th and
the 51st, arranged in order. 12 + 14 + 20 = 46 girls have mass
up 30.5 kg. 46 + 30 = 76 girls have a mass up to 35.5 kg. A
reasonable estimate of the median is in the class 31–35 kg.
c An estimate of the mean is 30.4 kg. Find the midpoint (the halfway point) of each class and multiply
by the frequency.
An estimate of the Mass Midpoint, m Frequency, fm × f
mean is 3035 ÷ 100 16–20 18 12 216
= 30.4 kg to 1 d.p. 21–25 23 14 322
d The range is between 20 and 30 kg.
For grouped data we can only find the largest and 26–30 28 20 560
smallest possible value of the range: 31–35 33 30 990
largest range = 45.5 – 15.5 = 30 (largest value from 36–40 38 17 646
class 41–45 – smallest value from class 16–20)
smallest range = 40.5 – 20.5 = 20 (smallest value from 41–45 43 7 301
class 41–45 – largest value from class 16–20) Total 100 3035

✦ Exercise 10.2

1 These are the marks for 40 students in an examination.
Mark 11–20 21–30 31–40 41–50
a What is the modal class?
Frequency 8 16 9 7
b Explain why the midpoint of the first class is 15.5.
c Estimate the mean mark.

2 These are the times for 50 runners to complete Time (minutes) 20– 25– 30– 35– 40– 45–
a race.
Frequency 5 8 22 12 3 0
a Write down the modal class.
b Explain why the midpoint of the first class is 22.5.
20– means ‘20 minutes or
c Estimate the median time.
more but less than 25’.
d Estimate the mean time.
10 Processing and presenting data 105