Three examples where simple arithmetic means are inappropriate

Box 40.2 Three examples where simple arithmetic means are inappropriate 

If moans of samples are themselves rheaned, an error can arise if the sarnpies are of different size. For example, the arithmetic mean of the means in the table shown left is 7. but this does not take account of the different reliabilities of each mean due to their sample sizes. The correct weighted mean is obtained by multiplying each mean, by its sample size (n) (a. weight ) and dividing the sum of these values by the total number of observations, i.o. in the case shown, (24-r 49 ( 8)/12 - 6.75. 

If the measurement scale is not linear, arithmetic means may give 

Moan values of log-transformed data are encountered in analylieai chemistry, e.g. the particle si/e of the droplets formed by a nebulizer used for flame or inductively coupled plasma (ICPi spectroscopy, where log-transformed valites are plotted (Figs 40.4a and hi. The use of geometric means in sitch circumstantces serves to reduce the effects of outliers on the moan. 

Fig 40.4 (a) An approximately log-transformed distribution particle size of droplets In flame spectroscopy. (b)The results in (a) plotted against the logarithm of particle size. 

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