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\chapter*{Statistics -- 2}

To measure how closely a set of data cluster round their mean, we could try to determine the
`average difference' between the mean and all the data. Let us assume that the data we are
considering are given by the following table:

\medskip

\begin{tabular}[c]{|c|c|c|c|c|} \hline
 variable & frequencies &  &  difference & \\
$x_i$ &  $f_i$ &$x_i \times f_i$ & $x_i-\bar{x}$ & $\parent{x_i-\bar{x}}\times f_i$ \\ \hline

sum &  &  &  &\\ \hline
\end{tabular}

\medskip

The third column of the table enables us to calculate the mean $\dps \bar{x}$ for
$\bar{x}=\dfrac{}{}= $. In the fourth column we can find the difference between each value of the
variable and $\dps \bar{x}$ aka the \nuvo{deviation} from the mean. The fifth and last one enables
us to determine what we were looking for. It is $0$. The fact is general, what we have computed is
(using symbols)
$$ m = \parent{f_1\parent{x_1 -\bar{x}} +f_2\parent{x_2 -\bar{x}}+\cdots f_5\parent{x_5 -\bar{x}}}
\times\dfrac{1}{N} $$
where $N$ is the sum of the frequencies that is the total number of individuals in the population. 
We then obtain
$$ m = \parent{\parent{f_1x_1} +\parent{f_2x_2}+\cdots \parent{f_5x_5}}\times\dfrac{1}{N} -
\parent{f_1+f_2+\cdots+f_5}\times\bar{x}\times\dfrac{1}{N}$$
and, because of the definition of the mean and that of $N$,
$$ m = \bar{x} - \dfrac{N}{N}\bar{x} = 0$$.

So this first attempt is fruitless. To avoid cancelling of differences we have to annihilate the
signs \idest, `to make all the differences positive'. It is possible to take the absolute value of
the differences, what we obtain then, as the `mean of the absolute (value of the) deviations from
the mean' is known as the \nuvo{mean deviation}. But if this measure of variability has been widely
used in the past for it is easy to calculate, it is little used now because it is very difficult
to develop any mathematical theory for the mean deviation and because nowadays there is no premium
on easy computations  ---~computers can do that for us quickly and accurately enough.

The other well known way of getting rid of the sign of a real number is just to square it. The `mean
of the squared deviation from the mean' is called the \nuvo{variance}.

\begin{Def}
  The variance of the statistics \accol{\couple{x_1}{f_1}\cdots\couple{x_n}{f_n}} is given by
$$ V = \dfrac{1}{N}\sum_{i=1}^{n}f_i\parent{x_i-\bar{x}}^2 = 
\dfrac{1}{N}\parent{f_1\parent{x_1-\bar{x}}^2 +\cdots f_n\parent{x_n-\bar{x}}^2}.$$
\end{Def}

\begin{prop}
  $$ V =\dfrac{1}{N}\sum_{i=1}^{n}f_i\parent{x_i}^2 - \parent{\bar{x}}^2.$$

The variance is the mean of the squares minus the square of the mean.
\end{prop}

As is clear from its definition, the variance is measured in units of $x^2$ rather than $x$. That is
sometimes a nuisance as well as making it harder to understand what the size of the variance is
telling us, and to compare variability in two sets of data. We therefore define a measure of
variability which is closely related to the variance but is expressed in the same units as the
data: the \nuvo{standard deviation} which is the square root of the variance and denoted by §s.%
\index{$\sigma$}\index{deviation!standard}

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