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The Normal
Distribution Page Areas in the tail of the normal
distribution table The mean and standard deviation
of grouped
data Box
and Whisker diagrams with MS Excel The Business Section |
Standard Deviation Introduction This page looks at the standard deviation:
a lot of people, students and practitioners alike, are scared of them; but by
the end of this page we are confident that you will be a master of them! The standard deviation is a scary name
for what is really only the average of the average … honestly,
it’s really not much more than that. More than that, if we take the
calculation of the standard deviation step by step, we’ll see that anyone can do it. No, really. The normal distribution on the other
hand is just a fact of life: our world is filled with examples that fit into
the normal distribution: the heights of men and women, blood pressures in
healthy people, the lengths of string on reels of string that comes out of a
string factory. The standard deviation is not
necessarily of any use if all we do is calculate it: however, we need to know
what it is and how to calculate it when we look at the normal distribution.
This page, therefore, takes us through the standard deviation first: what it
is, how to calculate it and what it means. Once we’ve understood the standard
deviation, we will look at the normal distribution of data. Finally, we’ll
put both the standard deviation together as we work on a variety of examples
and situations. A subsequent page deals with the means,
standard deviations and coefficients of variation of grouped data: see the
link in the menu on the left of this page. Averages Perhaps the most common thing we do to
data once we have collected it and put it into a table or spreadsheet is to
work out the average value of it. We know that we can calculate the
arithmetic mean, the median and the mode as different versions of what we can
call an average. Once we’ve calculated an average figure, we can talk about
average earnings, average marks in an exam and average height of sixteen year
olds. Nothing wrong with that: we all do that and it’s fine. The average does have a problem,
though, which is that if there are some exceptionally small or large values
in our data set, they can have an undue influence on the values in the middle
of our data set. For example, look at the averages of the data in this table:
See, value 1 in set 1 is exceptionally
large and it has dragged the average (arithmetic mean in this case) up to
56.6 for set 1 whereas set 2 has no exceptional value (we call these
exceptional values outliers, by the way) and so the average is much smaller
and much more representative of all of the values in the data set. However,
apart from the outlier, set 1 has poorer results than set 2 A major
disadvantage of the mean is that it is sensitive to outlying points http://bmj.com/collections/statsbk/2.shtml Imagine that the table above shows the
exam results for two groups of students: on the basis of the averages, set 1 would
be classed as the better of the two groups, right? Still, it’s only because
of the performance of one very talented student that their average is so much
better than the other set. Graphically, we can see the influence
of this outlier on set 1 and we can also see how stable set 2’s data are:
The formula for calculation the
average, in this case the arithmetic mean was, is and always will be
where
∑ is the Greek capital letter
Sigma that means total of or sum of x is the values of the variable ‘x’ n is the number of values or data
points For set 1, we use the arithmetic mean
formula as follows:
The formula works with set 2 works in
the same way … the average for set 2, remember, is 50.6 There is a way to sort out the problem
where we know the average of a data set but we don’t yet know some of the other
characteristics of that set: whether all of the values are near to the
average, some are near to the average, most are near but one or two are far
away or outliers … it’s the standard deviation. The standard deviation The standard deviation is a statistic that tells
you how tightly all the various examples are clustered around the mean in a
set of data. When the examples are pretty tightly bunched together… the
standard deviation is small. When the examples are spread apart … that tells
you have a relatively large standard deviation. http://www.robertniles.com/stats/stdev.shtml The standard deviation is really the
average of the average: more than that, though, it tells us instantly whether
we are dealing with data like set 1 or set 2: to prove that before we do any
calculations, the standard deviation for set 1 is 19.92, for set 2 it is
5.37; because the standard deviation for set 1 is so large, it tells us that
set 1’s data are more spread out than set 2’s data. And that’s true! The standard deviation actually comes
in two parts:
The variance is a measure of dispersion
of values based on their deviation from the mean. In general, the formula for
the variance is
where
∑ is the Greek capital letter
Sigma that mean total of or sum of x is the values of the variable ‘x’ n is the number of values or data
points This formula calculates the variance
for sample data, or where the number of values is less than 30. If we are
working with population data, or more than 30 values, the formula would
change and become
In fact, if n is bigger than 30, the
two formulae tend to give the same, or almost the same, values. The Variance for Sets 1 and 2 We’ll work out the variance for set 1
and you can work out the variance for set 2:
Notice we have to calculate Confirmation: The value of the variance for set 2 is
28.80, did you get that? If you disagree, check your workings carefully and
correct your mistakes. What does the Variance Mean? We now know that the variances for the
two data sets are 396.80 and 28.80 respectively: set 1’s variance is almost
14 times bigger than set 2’s variance. That tells us that set 1’s data is
wider spread, more dispersed, than set 2’s data. That’s all it means: nothing
complicated! However, the variance is really just a
number that isn’t really real! Look at the calculation and see that the
variance is the average of the differences between each value and the average
SQUARED: for set 1, the variance of 396.80 is really the average of the
differences SQUARED. This is where the standard deviation comes in … yep, the standard deviation is actually helpful! Definition of the standard deviation The simple definition of the standard
deviation is that it is the square root of the variance: that’s true but
since we might want to use the standard deviation alone, without the
variance, it’s best to think of it as the average of the average. The formula for the standard deviation
looks as dreadful as the formula for the variance but by working through some
examples we’ll turn it into mincemeat!!
That’s it! The value of 19.92 is much
more meaningful than 396.80 because it is now equivalent to the raw data … it
is 19.92 marks and not 396.80 square marks. Your Turn: calculate the standard
deviation for set 2 now Did you get 5.37? You should have:
check your workings if you didn’t. Comparing the standard deviations now,
we can see that set 1’s standard deviation is 19.92 and set 2’s is 5.37:
19.92 is a bit less than four times bigger than set 2’s standard deviation.
So, we know that the set 1 data set is more variable or spread or disperse
than set 2. Please note: there is no single, absolute,
value of the SD that tells us whether data is dispersed: all we can do is
compare SDs from different data sets, combine an SD with the normal
distribution and so on and then we can begin to determine dispersion. Your Turn Calculate
for each of set 3 and set 4 from the table below; and say which set of
data is the more dispersed.
Did you get these? The answers you are looking
for are
Set 4 is the more disperse of the two sets
of data because both its variance and standard deviations are higher than
those for set 3. Basic Rule of the Variance and the standard deviation If the Variance of data set A is
greater than the Variance of data set B, then the standard deviation of data set A
will be larger than the standard deviation of data set B Want to make this look fancy to impress
your friends? if VarA > VarB
then SDA > SDB; and the converse is true or even if VarA > VarB
then σA > σB;
and the converse is true Variance and standard deviation Template Here is a template that you can use
whenever you want to calculate some variances and/or standard deviations.
Just fill in your raw data and then do the arithmetic … a spreadsheet is
brilliant at doing this kind of work for you!
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