28
Feb 16

What is a mean value - all means in one place




What is a mean value - all means in one place

In introductory Stats texts, various means are scattered all over the place, and there is no indication of links between them. This is what we address here.

The population mean of a discrete random variable is the starting point. Such a variable, by definition, is a table values+probabilities, see this post, and its mean is EX=\sum_{i=1}^nX_ip_i. If that random variable is uniformly distributed, in the same post we explain that EX=\bar{X}, so the sample mean is a special case of a population mean.

The next point is the link between the grouped data formula and sample mean. Recall the procedure for finding absolute frequencies. Let Y_1,...,Y_n be the values in the sample (it is convenient to assume that they are arranged in an ascending order). Equal values are joined in groups. Let X_1,...,X_m denote the distinct values in the sample and n_1,...,n_m their absolute frequencies. Their total is, clearly, n. The sample mean is
\bar{Y}=(Y_1+...+Y_n)/n
(sorting out Y's into groups with equal values)
=\left(\overbrace {X_1+...+X_1}^{n_1{\rm{\ times}}}+...+\overbrace{X_m+...+X_m}^{n_m{\rm{\ times}}}\right)/n
=(n_1X_1 + ... + n_mX_m)/n,

which is the grouped data formula. We have shown that the grouped data formula obtains as a special case of the sample mean when equal values are joined into groups.

Next, denoting r_i=n_i/n the relative frequencies, we get

(n_1X_1 + ... + n_mX_m)/n=

(dividing through by n)

=r_1X_1+...+r_mX_m.

If we accept the relative frequencies as probabilities, then this becomes the population mean. Thus, with this convention, the grouped data formula and population mean are the same.

Finally, the mean of a continuous random variable X which has a density p_X is defined by EX=\int_{-\infty}^\infty tp_X(t)dt. in Section 6.3 of my book it is shown that the mean of a continuous random variable is a limit of grouped means.

Conclusion

Properties of means apply equally to all mean types.

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