The arithmetic mean is equal to the sum of values in a distribution divided by the
total number of values. For the following 10 numbers,
2 15 9 2 18 14 0 6 11 3,
the arithmetic mean is:
(2+15+9+2+18+14+0+6+11+3)/10=80/10=8.^
The mean is sometimes called the first moment. This terminology stems from mechanics
where the first moment corresponds to the centre of gravity of a distribution of mass. The
mean corresponds to the centre of a distribution.
The three measures of central tendency, mode median and mean will suffice for the
majority of situations you are likely to encounter. There are however two situations when
an arithmetic mean may not be appropriate. When all the values in a distribution do not
have equal importance or when we want to compute an overall mean from two samples
combined. Values may not have equal importance if, for example, some values have been
measured more precisely. In these circumstances we should give relatively more weight
to the more precise values.
When combining values from two or more samples the arithmetic mean would be
misleading unless the samples to be combined were of equal size. Each sample that is
combined should be weighted by the number of observations in the sample. This is
because a sample mean’s reliability is in proportion to the number of values in the
sample. Smaller samples are less reliable than larger samples and should therefore be
given less weight in calculation of an overall mean. Consider one sample with 10
observations,
2 15 9 2 18 14 0 6 11 3,
and a second sample with 5 observations,
17 6 21 16 15.
The arithmetic mean for sample one is 80/10=8, and for sample two is 75/5=15. You
may think that the overall mean is simply the average of both sample means i.e.,
(15+8)/2=11.5. However, this is incorrect because equal weight is given to both samples
when sample one has twice as many observations as sample two.
The weighted mean for the two samples is, the sum of, each sample mean multiplied
by its appropriate weight, all divided by the sum of the weights.
This value of 10.3 is the same value you would obtain if you treated the 15 observations
as one sample. Combining the two sample means without weighting them resulted in a
higher value of 11.5 compared with the weighted mean of 10.3. The overall mean was
pulled upwards by the relatively larger mean of the smaller sample.
The trimmed mean may be used with large samples and is similar to the arithmetic
mean but has some of the smallest and largest values removed before calculation. Usually
the bottom and top 5 per cent of values are removed and the mean is calculated on the
remaining 90 per cent of values. The effect is to minimize the influence of extreme
outlier observations in calculation of the mean.
The geometric mean is useful for calculating averages of rates. Suppose a new house
dwindles in value to 95 per cent of its original value during the first year. In the following
Initial data analysis 65