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Representing and
Summarising Data
S1 Chapter 2
Mode / Median / Mean

These should be very familiar from GCSE. They
are all ‘measures of location.’
 Mode
is the most common observation
 Median is middle value when data put in order
 Mean is sum of observations divided by number of
observations.

Proper notation for mean:
 If x1, x2, x3,… xn are
x

x
n
the observations you have, then:
More accurately, x is used for a mean of a sample.
μ (‘mu’) is the symbol for the mean of a population
Using the Mean

All very simple, but need to be careful when doing
calculations with the mean.
x

x
If

Need to remember this when combining means!
Eg: Mean score for 25 kids in class 1 was 6.4 Mean
score for 30 kids in Class 2 was 7.2 What’s their
combined mean?

n
then we can also say:
 x  nx

When to use Mean / Mode / Median



Mode: For qualitative data, and quantative when there’s
only one or two modes. Useless when each value occurs
only once!
Median. For quantative data, and better than the mean
when there are extreme values, as it’s not affected.
Mean. For quantative data, and good because it uses all
the data.
Frequency Tables




Very often, data is presented in a freq. table, and we
need to be able to find averages from this.
Eg. Collar sizes of 95 shirts sold in a shop:
x
f
15
3
15.5
17
16
29
16.5
34
17
12
We now define the mean as:
fx

x
f
Mode is easy.
For median, it’s easiest to add cumulative frequency
Grouped Frequency


When we have continuous data, we can’t use discrete
classes – we have to put the data into groups.
This will mean that the detail is lost, so all our measures of
location now become estimates.

Mode – this is the ‘modal class’ – the class with highest freq.
fx

x
f
but now x is midpoint of class.

Mean – we can still use

Median – we estimate the median using interpolation.
Example 14 – Pine Cones

Length of
cone (mm)
freq.
30 - 31
2
2 x 30.5 = 61
32 - 33
25
25 x 32.5 = 812.5
34 - 36
30
30 x 35 = 1050
37 - 39
13
13 x 38 = 494
Totals
70
2717.5
f x mid
Note on classes:

These have been given to nearest mm, so we need to think
carefully about the class boundaries:
Class width
31.5
Lower class
boundary
32 – 33
Lower class
limit
33.5
Upper class
limit
Upper class
boundary
Example 14 – Pine Cones



Length of
cone (mm)
freq.
30 - 31
2
2 x 30.5 = 61
32 - 33
25
25 x 32.5 = 812.5
34 - 36
30
30 x 35 = 1050
37 - 39
13
13 x 38 = 494
Totals
70
2417.5
f x mid
Modal class = 34 – 36
Estimate for mean = 2417.5 ÷ 70 = 34.53mm
To find the median, we need to use interpolation:


We have 70 values, so, because n is large, we can say median
lies at 35th value.
We can see from table that this is in the 34 – 36 class… but
where?
Example 14 – Pine Cones


The principle we use for interpolation is this: the
proportion of the observations into the class, is the same
as the proportion of the measurement from the class
boundary.
ie, for this example….
33.5mm
m
36.5mm
27th
35th
57th
th in which
th to 57th observation
The median,
35
where
the
median
is.proportion
The
class
m, is interpolated
median
lies
as
goes
the
from
same
27
of the way
27
observation
is 33.5
(lower
class
boundary)
between
33.5
and
36.5,class
as it is
between 27 and 57.
and 57th is
36.5
(upper
boundary)
m  33.5
35  27

36.5  33.5 57  27
m  33 .5 8

3
30
8
m  33.5   3
30
m  33.5  0.8
Coding Data



Sometimes it’s useful to code data to make it more
manageable.
xa
The normal form of a coding is: y 
b
The mean / mode median of the coded data can be
interpreted for the original data – as in Example 16.