PSY145: Research Methods
Statistics Supplement
Quantifying variability: Variance and Standard Deviation
NOTE: Relevant reading for this topic can be found in Gravetter and Wallnau: chapter 4
1. Measures of variability: an example of reading tests
An education researcher persuades a primary school to co-operate with a study of a new
reading scheme. The children are divided randomly into two groups. One group receives a
conventional (control) scheme for teaching reading and the other the experimental scheme. At
the end of the year the researcher comes and tests the children on reading ability. The reading
test gives a whole number score in the range 0 to 24 and the data for the 30 participants in
each group are plotted in the bar charts below.
5
Control Group
4
3
2
1
0
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
9
10 11 12 13 14 15 16 17 18 19 20
5
Experimental Group
4
3
2
1
0
2
3
4
5
6
7
8
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The mean performance score for the experimental condition is 11.3 and for the control
condition is 12.1, so the average performance with the two schemes is more or less the same
(the experimental scheme is very slightly worse).
As you see, the distributions are reasonably symmetrical in each case, so the mean should be
a good measure of central tendency. However, you would be ignoring an important aspect of
the data if you were to conclude that the experimental scheme had little effect. Looking at the
two distributions, there is clearly a difference between them which the mean scores do not
capture. Children on the experimental scheme are much more variable in their performance.
Perhaps more able children benefit but less able ones suffer under the new scheme.
How can we measure this variability? The simplest method is to compute the range of scores
form minimum to maximum. This is 2-20 (18) for the experimental group and 7-17 (10) for the
control group. The more usual and statistically superior measure is the variance, or its square
root, known as the standard deviation.
It is conventional to refer to the mean and standard deviation of a population as  and  , and
to describe the mean and standard deviation of a sample as  and s. Let us consider first the
measurement of the sample variance, s2, and its standard deviation, s.
1.1 Sample Variance and Standard Deviation
One way to express the variability of a set of data might be to work out the average variation of
the data points (each X value) around the mean  . Unfortunately, the positive and negative
differences each side of the mean would cancel each other out and so we can’t directly work
out the ‘average’ deviation in this way. However, if we take the square of each deviation
(x  x) 2 this problem goes away (try it on a calculator - the square of a negative number is
always a positive number). Therefore we take the squared deviation, and average that instead.
Hence, the variance appears as:
 (x x)
2
n
where n is the number of observations in the sample and

means ‘sum what follows’, in this
case the square of each deviation from the mean. The top part of the above equation is know
as the sum of squares (i.e. the sum of squared deviations from the mean). When we divide the
sum of squares by the number of data points in our sample (n) to get an average of the squared
deviations from the mean we call the result the variance or mean square (i.e. the mean of
squared deviations from the mean).
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Actually, experience shows that the variance of a sample (s2) under-estimates the variance of
the population (2) from which it was drawn unless it is calculated with a divisor of n-1 rather
than n. This is because samples are (by definition) smaller than the population from which they
were drawn. Because they have relatively few data points in them there is a fairly low
probability that they will contain the most extreme (and therefore unusual) values in the
population. Using n-1 compensates for this under-estimation of variability because the sum of
squares is divided by a smaller number and consequently the resulting mean square is that little
bit larger (representing more variation). Don’t worry if you don’t really understand this
explanation. All you need to know is that the correct formula for calculating the variance of a
sample is:
 (xx)
2
s2 
n 1
Because the ‘average squared deviation from the mean’ is difficult to visualise we usually
convert the variance back into something more meaningful by taking its square root. In other
words, we ‘un-square’ it to get a figure which is more descriptive of the ‘average (or ‘standard’)
deviation from the mean’. This figure is known as the standard deviation. The formula for the
standard deviation (s ) of a sample is:
 (xx)
2
s
n 1
Using this formula the standard deviations for the experimental and control groups in the
example above compute to 3.82 and 2.55 respectively. With this information we would be able
see that the variability in the experimental group was larger than for the control group even if we
hadn’t seen the bar graphs of the distributions of the scores for each group. This is important
because experimental reports published in journals rarely give plots of the distribution of data.
Normally only the mean and standard deviation for each group is given and you have to draw
your conclusions about variability from those.
1.2 Population Variance and Standard Deviation
It is more usual for psychologists to work with sample data. However, there may be occasions
when you need to describe the variability in a whole population of scores. The equations for
calculating variance (2) and standard deviation () for a population are identical to those for a
sample, except that the mean is represented by the symbol , the number of data points is N
rather than n, and because there is no problem with under-estimation (there is no need for
estimation when dealing with population data) there is no need to subtract 1. So, the formula
for population variance is:
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(x )
  N
2
2
and for population standard deviation is:

 (x )
2

N
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