Chapter 55

Measures of central tendency

and dispersion

55.1 Measures of central tendency

A single value, whichis representative of a set of values,
may be used to give an indication of the general size of
the members in a set, the word ‘average’oftenbeing
used to indicate the single value.
The statistical term used for ‘average’ is the arithmetic
mean or just themean.
Other measures of central tendency may be used and
these include themedianand themodalvalues.

55.2 Mean, median and mode for

discrete data

Mean

Thearithmeticmean valueis foundbyadding together
the values of the members of a set and dividing by the
number of members in the set. Thus, the mean of the set
of numbers:{ 4 , 5 , 6 , 9 }is:

4 + 5 + 6 + 9

4

,i.e. 6

In general, the mean of the set:{x 1 ,x 2 ,x 3 ,...,xn}is

x=

x 1 +x 2 +x 3 +···+xn

n

,written as

∑

x

n

where

∑
is theGreek letter ‘sigma’ and means ‘thesum
of’, andx(calledx-bar) is used to signify a mean value.

Median

The median valueoften gives a better indication

of the general size of a set containing extreme val-

ues. The set:{ 7 , 5 , 74 , 10 }has a mean value of 24,

which is not really representative of any of the val-

ues of the members of the set. The median value is

obtained by:

(a) rankingthe set in ascending order of magni-

tude, and

(b) selecting the valueof themiddle memberfor sets

containing an odd number of members, or finding

the value of the mean of the two middle members

for sets containing an even number of members.

For example, the set: { 7 , 5 , 74 , 10 } is ranked as

{ 5 , 7 , 10 , 74 }, and since it contains an even number of

members (four in this case), the mean of 7 and 10 is

taken, giving a median value of 8.5. Similarly, the set:

{ 3 , 81 , 15 , 7 , 14 }is ranked as{ 3 , 7 , 14 , 15 , 81 }and the

median value is the value of the middle member, i.e. 14.

Mode

Themodal value,ormode, is the most commonly

occurring value in a set. If two values occur with

the same frequency, the set is ‘bi-modal’. The set:

{ 5 , 6 , 8 , 2 , 5 , 4 , 6 , 5 , 3 }has a model value of 5, since the

member having a value of 5 occurs three times.

Problem 1. Determine the mean, median and

mode for the set:

{ 2 , 3 , 7 , 5 , 5 , 13 , 1 , 7 , 4 , 8 , 3 , 4 , 3 }