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The mean and standard deviation
of grouped
data The Normal
Distribution Page Areas in the tail of the normal
distribution table Box
and Whisker diagrams with MS Excel The Business Section |
Means, Standard
Deviations and Coefficients of Variation of Grouped Data The page on the standard deviation is fine as far as it goes: it deals with simple means and standard deviations of data sets. This brief additional page provides a discussion of the mean and standard deviations of grouped data. Since the amount of dispersion can also be an important
issue, we introduce the Coefficient of Variation here, too. These ideas are all introduced by working with the heights
and shoe sizes of a very large sample of British schoolchildren. Grouped Data By grouped data we mean the situation where we have, for
example, sampled many people or situations and it is convenient to classify
our data into groups and sub groups: all will become clear with the examples
we use her
where x is the x variable f is the frequency of responses The best way to appreciate these formulae is to use them:
let’s do that now. Example Here are the heights of almost 30,000 British
schoolchildren, ages 7 to 16 years of ag
i
Work out the mean and standard deviations of the data
for a
all children b
children aged 7 years c
children aged 10 years d
aged 16 years
ii
comment on your results
Note: you can go to http://censusatschool.ntu.ac.uk/tableheight.asp
and download these and additional data for the heights of The original class intervals are those given in the left
hand column: we have use the mid point of these class intervals to represent
our ‘x’ values: notice that the final class of >200 is open ended and we
have chosen to limit that class interval to 210 cms
… argue if you disagree and you can always check your version against this on Answers i We’ll give the answer to part
‘a’ of the question in full and then to the other parts in outline only.
b, c and d answers are all contained in a summary table, along with the results for part a, repeated for convenience:
ii The results we have can be interpreted as follows: since we are working with the
heights of children, we must expect the average heights to increase year by
year: that is, the average height of 7 year old children should be less than
the average height of a 10 year old and that should be less than the average
height of a 16 year old. as for the standard deviations, we can see that they are increasing in line with the increases in average heights. An explanation of the increasing standard deviations could be that the children are growing at different rates: this means that whilst an overall age group is increasing, there will be a few children who are growing at, say 0.5 cm a year whilst others are growing at 1 cm, 2 cms and so on. The effect of these different growth rates is to increase the standard deviations as the dispersion of the age groups will be greater and greater as the age increases. We should see this effect on the following graph:
Coefficient of
Variation The coefficient of variation shows us the extent of the
dispersion, or variation, in a data set by comparing the standard deviation
with the mean:
We said in part ii of the worked example that the heights of the 16 year age group was more disperse than the other age groups … the coefficient of variation will extend this part of the discussion for us. Here are the coefficients of variation for all of the data sets we have:
So, we can see that even though the averages and standard
deviations are changing as we have already seen, in fact the dispersion, or
variability, of the data sets is reducing as the children get older. Your Turn From http://censusatschool.ntu.ac.uk/tablefootsize.asp
we have found the corresponding heights and shoe sizes of almost 60,000
British schoolchildren, as per the table below. Choosing any shoe size range you like, calculate the a
means b
standard deviations c coefficients of variation
© where appropriate
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