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Standard Deviation
Consider the following sets of data:
3,9,12,15,19,20
0
Range = 20-3 = 17
20
Mean = (3+9+12+15+19+20)/6=13
11,12,13,13,14,15
0
20
Range = 15-11 = 4
Mean = (11+12+13+13+14+15)/6 = 13
Note: The two sets of data have the same mean i.e. 13 but are
very different.
0
20
0
20
A measure of spread which uses
all the data is the standard deviation.
When the standard deviation is low it means the scores
are close to the mean.
0
20
Mean
When the standard deviation is high it means the scores
are spread out from the mean
0
20
Mean
Exercise 1
Look at the three sets of scores below and place the
standard deviations for these scores in order, low to high
1
Mean
2
Mean
3
Mean
Calculating the Standard Deviation of a set of scores
The standard deviation or “root, mean, square deviation” is a
measure of how far all the scores differ from the mean. It can be
calculated from first principles or by the application of a formula.
Consider the scores listed earlier:
3,9,12,15,19,20
Mean =(3+9+12+15+19+20) / 6 = 13
We now construct a table to see how far each score differs
from the mean.
Score
3
9
12
15
19
20
The mean square
deviation is 206  6
=34.7
Deviation
(Deviation )2
(3-13) = -10
(9-13) = -4
(12-13) = -1
(15-13) = 2
(19-13) = 6
(20-13) = 7
100
16
1
4
36
49
Total
206
The standard deviation is
34.7  5.9
Standard Deviation by Formula
2

(
x
)

2
x


n
s 
n 1









All of the values can be found using a scientific calculator.
You do not have to learn this formula as it is given
on the exam paper cover.
Using the Sharp
EL-531VH
You need to be in STAT mode.
MODE
2nd F
DRG
The calculator display shows
MODE
0-2
Press
1
?
You are now in STATS mode and the calculator display shows
Stat x
0
The next task is to enter the data.
We will use the example already covered:
We will enter the numbers:
3,9,12,15,19,20
All the STATS keys are in green on the calculator.
3
M+
9
M+
12
M+
15
M+
19
M+
20
We can now pick out any values from the keyboard using
M+
RCL
The values we require for the formula are laid out in the following
key positions on the calculator:
8
7
9
x
sx
4
5
6
1
2
3
n
0
x
7
 x2
+/-
These values are obtained by pressing
RCL
first.
The values can now be obtained and entered into the formula:
 x2
=
1220.
 x
=
78.
n
=
6.
2

(
x
)

2
x


n
s 
n 1










782
 1220 
6
s 
6 1









1220  1014

5
206

 41.2  6.42
5
Note: There is a slight difference in answer between the two
methods. The formula uses n-1 instead of n. This is because
the data is treated as a sample.