There are three measures of central tendency that are commonly used to describe the average value of a set of data. These are the mode, the mean and the median.
● The mode is the most commonly occurring value.
● The mean is calculated by dividing the sum of the values by the number of values.
● The median is the value in the middle of an ordered set of data.
We use an average to summarise the values in a set of data. As a representative value, it should be fairly central to, and typical of, the values that it represents.
If we investigate the annual incomes of all the people in a region, then a single value (i.e.
an average income) would be a convenient number to represent our findings. However, choosing which average to use is something that needs to be thought about, as one measure may be more appropriate to use than the others.
Deciding which measure to use depends on many factors. Although the mean is the most familiar average, a shoemaker would prefer to know which shoe size is the most popular (i.e. the mode). A farmer may find the median number of eggs laid by their chickens to be the most useful because they could use it to identify which chickens are profitable and which are not. As for the average income in our chosen region, we must also consider whether to calculate an average for the workers and managers together or separately; and, if separately, then we need to decide who fits into which category.
PREREQUISITE KNOWLEDGE
Where it comes from What you should be able to do Check your skills IGCSE / O Level Mathematics Calculate the mean, median and mode
for individual and discrete data.
1 Find the mean, median and mode of the numbers 7.3, 3.9, 1.3, 6.6, 9.2, 4.7, 3.9 and 3.1.
Use a calculator efficiently and apply appropriate checks of accuracy.
2 Use a calculator to evaluate 6 1.7 8 1.9 11 2.1
6 8 11
× + × + ×
+ + and
then check that your answer is reasonable.
EXPLORE 2.1
Various sources tell us that the average person:
● laughs 10 times a day
● falls asleep in 7 minutes
● sheds 0.7 kg of skin each year
● grows 944 km of hair in a lifetime
● produces a sneeze that travels at 160 km/h
● has over 97 000 km of blood vessels in their body
● has a vocabulary between 5000 and 6000 words.
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The average adult male is 172.5 cm tall and weighs 80 kg.
The average adult female is 159 cm tall and weighs 68 kg.
What might each of these statements mean and how might they have been determined?
Have you ever met such an average person? How could this information be useful?
You can find a variety of continuously updated figures that yield interesting averages at http://www.worldometers.info/.
2.1 The mode and the modal class
As you will recall, a set of data may have more than one mode or no mode at all.
The following table shows the scores on 25 rolls of a die, where 2 is the mode because it has the highest frequency.
Score on die 1 2 3 4 5 6
Frequency ( )f 5 6 5 3 2 4
In a set of grouped data in which raw values cannot be seen, we can find the modal class, which is the class with the highest frequency density.
We saw how to calculate frequency density in Chapter 1, Section 1.3.
REWIND
The modal class does not contain the most pencils but it does contain the greatest number of pencils per centimetre.
TIP WORKED EXAMPLE 2.1
Find the modal class of the 270 pencil lengths, given to the nearest centimetre in the following table.
Length ( cm)x No. pencils ( )f
4–7 100
8–10 90
11–12 80
Answer
The modal class is 11–12 cm (or, more accurately,
<x<
10.5 12.5 cm).
Length ( cm)x No. pencils ( )f Width (cm)
Frequency density
<
x
3.5< 7.5 100 4 100÷ =4 25
<
x
7.5< 10.5 90 3 90 3 30÷ =
<
x
10.5< 12.5 80 2 80 2÷ =40
0 10 20 30 40
3.5
Length (cm) 7.5 Frequency density (pencils per cm)
10.5 12.5
Class boundaries, class widths and frequency density calculations are shown in the table.
Although the histogram shown is not needed to answer this question, it is useful to see that, in this case, the modal class is the one with the tallest column and the greatest frequency density, even though it has the lowest frequency.
In histograms, the modal class has the greatest column height. If there is no modal class then all classes have the same frequency density.
KEY POINT 2.1
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Two classes of data have interval widths in the ratio 3:2. Given that there is no modal class and that the frequency of the first class is 48, find the frequency of the second class.
Answer
Let the frequency of the second class be x.
x x
x 48: 3:2
48 2 3 32
=
=
=
The second class has a frequency of 32.
Or, let the frequency of the second class be x.
x x 2
48 3 32
=
=
‘No modal class’ means that the frequency densities of the two classes are equal, so class frequencies are in the same ratio as interval widths.
Alternatively, frequencies are proportional to interval widths.
WORKED EXAMPLE 2.2
EXERCISE 2A
1 Find the mode(s) of the following sets of numbers.
a 12, 15, 11, 7, 4, 10, 32, 14, 6, 13, 19, 3 b 19, 21, 23, 16, 35, 8, 21, 16, 13, 17, 12, 19, 14, 9 2 Which of the eleven words in this sentence is the mode?
3 Identify the mode of x and of y in the following tables.
x 4 5 6 7 8
f 1 5 5 6 4
y –4 –3 –2 –1 0
f 27 28 29 27 25
4 Find the modal class for x and for y in the following tables.
x 0– 4– 14–20
f 5 9 8
y 3–6 7–11 12–20
f 66 80 134
5 A small company sells glass, which it cuts to size to fit into window frames. How could the company benefit from knowing the modal size of glass its customers purchase?
6 Four classes of continuous data are recorded as 1–7, 8–16, 17–20 and 21–25. The class 1–7 has a frequency of 84 and there is no modal class. Find the total frequency of the other three classes.
7 Data about the times, in seconds, taken to run 100 metres by n adults are given in the following table.
Time ( s)x 13.6<x<15.4 15.4<x<17.4 17.4<x<19.8
No. adults ( )f a b 27
By first investigating the possible values of a and of b, find the largest possible value of n, given that the modal class contains the slowest runners.
PS
In the special case where all classes have equal widths, frequency densities are proportional to frequencies, so the modal class is the class with the highest frequency.
TIP
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8 Three classes of continuous data are given as 0– 4, 4–10 and 10–18. The frequency densities of the classes 0– 4 and 10–18 are in the ratio 4 : 3 and the total frequency of these two classes is 120. Find the least possible frequency of the modal class, given that the modal class is 4–10.
PS
2.2 The mean
The mean is referred to more precisely as the arithmetic mean and it is the most commonly known average. The sum of a set of data values can be found from the mean. Suppose, for example, that 12 values have a mean of 7.5 :
Mean sum of values number of values
= , so 7.5 sum of values
= 12 and sum of values=7.5 12× =90. You will soon be performing calculations involving the mean, so here we introduce notation that is used in place of the word definition used above.
We use the upper case Greek letter ‘sigma’, written Σ, to represent ‘sum’ and x to represent the mean, where x represents our data values.
The notation used for ungrouped and for grouped data are shown on separate rows in the following table.
Sum Data values Frequency of data values
Number of data values
Sum of data values
Mean
Ungrouped Σ x – n Σx x= Σx
n
Grouped Σ x f Σf Σxf or Σfx = Σ
x Σxf f
xf fx
Σ = Σ indicates the sum of the products of each value and its frequency. For example, the sum of five 10s and six 20s is (10 5) (20 6) × + × =
× =(5 10) (6 20) 170.× + × = TIP
Five labourers, whose mean mass is 70.2 kg, wish to go to the top of a building in a lift with some cement. Find the greatest mass of cement they can take if the lift has a maximum weight allowance of 500 kg.
Answer x x n
70.2 5 351kg Σ = ×
= ×
=
y y
351 500
149 + =
=
The greatest mass of cement they can take is 149 kg.
We first rearrange the formula x x
= Σn to find the sum of the labourers’ masses.
We now form an equation using y to represent the greatest possible mass of cement.
WORKED EXAMPLE 2.3
= Σ
n f for grouped data.
TIP
For ungrouped data, the mean is x= Σx
n. For grouped data, x xf
= Σf
Σ or fx f Σ
Σ . KEY POINT 2.2
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