\chapter{Statistics --- continued}  
  
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: 
  
\begin{center}  
\begin{tabular}[c]{|*{5}{>{$}c<{$}|}} \hline  
\multicolumn{1}{|c|}{variable} & \multicolumn{1}{c|}{frequencies} &  
  &  \multicolumn{1}{c|}{difference} & \\  
x_i &  f_i &x_i \times f_i & x_i-\bar{x} & \parent{x_i-\bar{x}}\times f_i \\ \hline  
25 & 10 & 250 & -16 & -160 \\  
35 & 35 & 1225 & -6 & -210 \\  
45 & 40 & 1800 & 4 & 160 \\  
55 & 15 & 825 & 14 & 210 \\  
sum &100 &4100 &  &0 \\ \hline  
\end{tabular}  
\end{center}  
  
The third column of the table enables us to calculate the mean $\dps \bar{x}$ for 
$$\bar{x}=\dfrac{4100}{100}=41. $$ 
  
In the fourth column we can find the difference between each value of the variable and $\dps \bar{x}$ aka the 
\nuvo{deviation} \pron{£bd£Iv£i£h£q£sn} 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 
$$ 
m = \parent{\parent{f_1x_1} +\parent{f_2x_2}+\cdots \parent{f_5x_5}}\times{1}{N} - 
\parent{f_1+f_2+\cdots+f_5}\times\bar{x}\times\dfrac{1}{N}$$ 
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 \pron{£v£Pr£Q£hb£il£t£i} 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} \pron{£hv£Pr£Qns}. 
  
\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} \pron{£hst£and£ed £bd£Iv£i£h£q£sn} 
which is the square root of the variance and denoted by §s.%  
\index{$\sigma$}\index{deviation!standard} 
  
  
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