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DATA ANALYSIS
DATA ANALYSIS

Measures of Central Tendency
 MEAN
 MODE
 MEDIAN

Measures of Central Tendency
MEAN
Sum of the Scores

Number of scores
x
x 
n
MEDIAN
is the middle score when the
scores are arranged in
numerical order
MODE
is the most frequently
occurring score in the data
[It is possible to have no mode
or more than one mode]
Example 1- Find the mean, mode and
median of the following set of
scores
3, 5, 2, 7, 8, 8, 9, 10
3  5  2  7  8  8  9  10
Mean =
8
52

8
= 6.5
Example 1 Continued
Arrange the
scores in order
2, 3, 5, 7, 8, 8, 9, 10
7  8  7 .5
Median =
2
Note that when you have an even number of scores
the median is the mean of the two middle scores
Mode =
8
Example 2 - Find the mean, mode and median of the
following set of scores.
67, 88, 43, 76, 75, 82, 71
67  88  43  76  75  82  71
Mean =
7
502

7
= 71.7143
Example 2 Continued
Arrange the
scores in order
43, 67, 71, 75, 76, 82, 88
Median = 75
No Mode
All scores occur the same
number of times, so there is
no mode in this case
Example 3 - Find the mean, mode and median of the
following set of scores.
12, 18, 23, 16, 15, 12, 11, 16,
20, 14, 12, 22, 16, 14, 22
Mean = 12  18  23  16  15  12  11  16  20  14  12  22  16  14  22
15
243

15
= 16.2
Example 3 Continued
Arrange the
scores in order
11, 12, 12, 12, 14, 14, 15, 16, 16, 16, 18,
20, 22, 22, 23
Median = 16
Mode
When we have two modes,
the data is bi-modal
= 12 and 16
MEASURES OF SPREAD

Range

Interquartile Range

Standard Deviation
MEASURES OF SPREAD


Range
= Highest score – Lowest score
Interquartile
= Upper Quartile – Lowest Quartile
Range
The interquartile range looks at the middle 50% of
the scores and measures the range in this set.
The upper quartile is the median of the top 50% of
the scores whilst the lower quartile is the median
of the bottom half of the scores.
Standard Deviation
The standard deviation measures the deviations
from the mean.
 x  x 
s
n
2
x

2
s
x
n
2
or
The advantage of using standard deviation as a
measure of spread is that it uses all scores.
Example 1- Find the range, interquartile range
and standard deviation of the
following set of scores.
3, 5, 2, 7, 8, 8, 9, 10
Range = Highest score – Lowest score
Range = 10 – 2
= 8
Remember to
subtract
Example 1 continued - Finding the interquartile range
Arrange the
scores in order
2, 3, 5, 7, 8, 8, 9, 10
Divide the scores
into two sets
Find the middle
score of each half
Interquartile Range = 8.5 – 4
= 4.5
Example 1 continued - Finding the standard deviation
Mean = 6.5
2, 3, 5, 7, 8, 8, 9, 10
 x  x 
s
6.5 from
n
each score
Using
2
subtract
-4.5, -3.5, -1.5, 0.5, 1.5, 1.5, 2.5, 3.5
Square each of these values
20.25, 12.25, 2.25, 0.25, 2.25, 2.25, 6.25, 12.25
 x  x 
s
n
2
 58  7.25  2.69
8
That method is too complicated and hence is
very rarely used.
Let’s consider the second formula for
calculating standard deviation.
x

2
s
x
n
2
Example 1 continued
Mean = 6.5
2, 3, 5, 7, 8, 8, 9, 10
Using
square
2
x

2
s
 x each score
n
4, 9, 25, 49, 64, 64, 81, 100
396  6.52
x

2 
s
x
8
n
2
 49.5  42.25
 2.69
Generally we do not calculate the
standard deviation in this manner. We
use our calculator in statistics mode.
The symbol used to represent standard
deviation varies from calculator to
calculator.
Some examples are
x n
and

Example 2- Find the range, interquartile range and
standard deviation of the following set of
scores.
67, 88, 43, 76, 75, 82, 71
Range = Highest score – Lowest score
Range = 88 – 43
= 45
Remember to
subtract
Example 2 continued
Arrange the
scores in order
43, 67, 71, 75, 76, 82, 88
Divide the scores
into two sets
Find the middle
score of each half
Interquartile Range = 82 – 67
= 15
Example 2 continued
Mean = 71.71
43, 67, 71, 75, 76, 82, 88
Using
square
2
x

2
s
 x each score
n
1849, 4489, 5041, 5625, 5776, 6724, 7744
37248  71.712
x

2 
s
x
7
n
2
 5321.14  5142.94
 13.4
Frequency Tables
Frequency tables are a good way to present
data.
The first column is the score, the second
column shows the frequency or number of
times the score in the first column occurred.
Formulas used with frequency tables.
Mean =
Standard Deviation =
 fx
f
 fx
f
2
x
2
Frequency Tables
Example - Calculate the mean, mode, range and
standard deviation for the data in the table.
Score
Frequency
fx
15
16
17
18
19
20
3
5
7
9
5
1
45
80
119
162
95
20
Total
30
521
Mean =
 fx
f
521

30
 17.37
Frequency Tables - Finding the standard deviation
2
Score
Frequency
fx
fx
15
16
17
18
19
20
3
5
7
9
5
1
45
80
119
162
95
20
675
1280
2023
2916
1805
400
Total
30
521
9099
Standard

Deviation
 fx
f
2
 x  9099  17.37 2
30
 1 .3
2