Measures of Centre – Mean, Median and Mode
Measures of Spread – Range
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The measures of centre are the statistical averages
The measures of centre are mean, median and mode/modal
class
Mean: the sum of the data divided by the number of data values
𝑚𝑒𝑎𝑛 =

𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
Median: the middle data value from a set of ordered data values.
𝑛+1
The position of this data value can be found using the rule 2 .
◦ From this we find:
 for an odd number of data values, the median is the middle data
value
 For an even number of data values, the median lies between the two
middle data values i.e. it is the mean of the two middle data values

Mode/Modal class: the most frequently occurring data
value/group/class
Measures of spread give a measure of the variation
in the data, or how wide the data is
 The measures of spread are range, interquartile
range and standard deviation
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Range: The full width of the data
𝑅𝑎𝑛𝑔𝑒 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒
Interquartile range and standard deviation will be
covered in Year 10
Determine the measures of centre and spread for the
following set of data:
2 4 4 5 6 6 7 7 7 8 9 9 10
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 84
𝑀𝑒𝑎𝑛 =
=
= 6.46
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 13
𝑛+1
14
𝑀𝑒𝑑𝑖𝑎𝑛 =
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 =
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 = 7𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
2
∴ 𝑀𝑒𝑑𝑖𝑎𝑛 = 7𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 = 7
𝑀𝑜𝑑𝑒 = 𝑣𝑎𝑙𝑢𝑒 𝑡ℎ𝑎𝑡 𝑎𝑝𝑝𝑒𝑎𝑟𝑠 𝑡ℎ𝑒 𝑚𝑜𝑠𝑡 = 7
𝑅𝑎𝑛𝑔𝑒 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 = 10 − 2 = 8
Determine the measures of centre and spread for the following
set of data:
3 4 4 5 5 6 6 7 7 7 9 9 10 11
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 95
𝑀𝑒𝑎𝑛 =
=
= 6.79
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 14
𝑛+1
15
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 =
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 = 7.5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
2
6+7
∴ 𝑀𝑒𝑑𝑖𝑎𝑛 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 7𝑡ℎ 𝑎𝑛𝑑 8𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 =
= 6.5
2
𝑀𝑒𝑑𝑖𝑎𝑛 =
𝑀𝑜𝑑𝑒 = 𝑣𝑎𝑙𝑢𝑒 𝑡ℎ𝑎𝑡 𝑎𝑝𝑝𝑒𝑎𝑟𝑠 𝑡ℎ𝑒 𝑚𝑜𝑠𝑡 = 7
𝑅𝑎𝑛𝑔𝑒 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 = 11 − 3 = 8
There are advantages and disadvantages of using mean and
median as a measure of centre (average) of the data
 When mean and median are compared to each other, they can
also give an indication of whether the data is symmetric or
skewed and which way the data is skewed.

Mean
Median
Advantage
Includes all the data values
and so considers all of the
information
Is not influenced by extreme
data values (outliers)
Disadvantage
Is influenced by extreme
data values (outliers)
Doesn’t consider the value of
the data values, only the
position of the values
Symmetric Shape
Approximately equal
Positively Skewed
Mean is higher than median
Negatively Skewed
Mean is lower than median
Example: Determine the measures of centre and spread for the
following set of data displayed in a frequency table
Number
Frequency
Product
Cumulative
Frequency
2
3
2×3= 6
3
3
7
3 × 7 = 21
3 + 7 = 10
4
9
36
10 + 9 = 19
5
4
20
19 + 4 = 23
6
3
18
26
7
2
14
28
8
1
8
29
9
1
9
30
10
1
10
31
Total
31
142
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To find the mean from a frequency table we need to add all the data values
first
This is made easier by the inclusion of a product column
The product column is the data value x frequency
Then the total of the product column is the sum of the data values
𝑀𝑒𝑎𝑛 =
2 × 3 + 3 × 7 + 4 × 9+. . . +10 × 1 142
=
= 4.58
31
31
To find the median from a frequency table, we need to know where the
middle data value would be
𝑛+1
We use the formula
to find the middle
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
2
◦

◦
◦
◦
31+1
𝑀𝑒𝑑𝑖𝑎𝑛 𝑣𝑎𝑙𝑢𝑒 = 2 = 16𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
We use the cumulative frequency column to find where the 16th data value
would be
𝑀𝑒𝑑𝑖𝑎𝑛=4
𝑀𝑜𝑑𝑒 = 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒 𝑡ℎ𝑎𝑡 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡ℎ𝑒 𝑚𝑜𝑠𝑡 = 4
𝑅𝑎𝑛𝑔𝑒 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒 = 10 − 2 = 8