The standard deviation measures the spread of the data about the
mean value. It is useful in comparing sets of data which
may have the same mean but a different range. For example, the mean of the
following two is the same: 15, 15, 15, 14, 16 and 2, 7, 14, 22, 30. However, the
second is clearly more spread out. If a set has a low standard deviation, the
values are not spread out too much.
Just like when working out the mean, the method is
different if the data is given to you in groups.
NonGrouped Data
Nongrouped data is just a list of values. The standard
deviation is given by the formula:
s means 'standard
deviation'. S means 'the sum of'.
means 'the mean'
Example
Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12,
14 First work out the mean: 10.222 Now,
subtract the mean individually from each of the numbers given and square the
result. This is equivalent to the (x 
)² step. x refers to the values given in the question.
x 
4 
9 
11 
12 
17 
5 
8 
12 
14 
(x 
)^{2} 
38.7 
1.49 
0.60 
3.16 
45.9 
27.3 
4.94 
3.16 
14.3 
Now add up these results (this is the 'sigma' in the
formula): 139.55 Divide by n. n is the number of values, so in this case is
9. This gives us: 15.51 And finally, square root this: 3.94
The
standard deviation can usually be calculated much more easily with a calculator
and this may be acceptable in some exams. On my calculator, you go into the
standard deviation mode (mode '.'). Then type in the first value, press 'data',
type in the second value, press 'data'. Do this until you have typed in all the
values, then press the standard deviation button (it will probably have a lower
case sigma on it). Check your calculator's manual to see how to calculate it on
yours.
NB: If you have a set of numbers (e.g. 1, 5, 2, 7, 3, 5 and 3),
if each number is increased by the same amount (e.g. to 3, 7, 4, 9, 5, 7 and 5),
the standard deviation will be the same and the mean will have increased by the
amount each of the numbers were increased by (2 in this case). This is because
the standard deviation measures the spread of the data. Increasing each of the
numbers by 2 does not make the numbers any more spread out, it just shifts them
all along.
Grouped Data
When dealing with grouped data, such as the following:
x 
f 
4 
9 
5 
14 
6 
22 
7 
11 
8 
17 
the formula for standard deviation becomes:
Try working out the standard deviation of the above data. You
should get an answer of 1.32 .
You may be given the data in the form of groups, such as:
Number 
Frequency 
3.5  4.5 
9 
4.5  5.5 
14 
5.5  6.5 
22 
6.5  7.5 
11 
7.5  8.5 
17 
In such a circumstance, x is the midpoint of groups.
Copyright © Matthew Pinkney 2003
