Arithmetical mean

The arithmetical mean is the average value of the samples. It is independent of the width of the intervals. It is symbolized as $\overset{―}{x}$ and it is only used for quantitative variables. We find it by adding up all the values and dividing by the total number of data.

The general formula for $N$ elements is: $\overset{―}{x} = \frac{x_{1} + x_{2} + x_{3} + \dots + x_{n}}{n}$

Example

In a basketball match, we have the following points for the players of a team: $0, 2, 4, 5, 8, 8, 10, 15, 38$ Calculate the mean of points of the team.

Applying the formula $\overset{―}{x} = \frac{0 + 2 + 4 + 5 + 8 + 9 + 10 + 15 + 38}{9} = \frac{90}{9} = 10$

Calculation of the mean for grouped information

The average in the case of $N$ data grouped in $n$ intervals is given by the formula $\overset{―}{x} = \frac{x_{1} \cdot f_{1} + x_{2} \cdot f_{2} + x_{3} \cdot f_{3} + \dots + x_{n} \cdot f_{n}}{f_{1} + f_{2} + f_{3} + \dots + f_{n}}$

where $f_{i}$ represents the times that the value $x_{i}$ is repeated. The grouping can also be done by intervals, using then the intermediate value of the interval to calculate the mean.

Example

The height in $c m$ of the players of a basketball team is in the following table. Calculate the mean.

Interval	$x_{i}$	$f_{i}$	$x_{i} \cdot f_{i}$
$[160, 170)$	$165$	$1$	$165$
$[170, 180)$	$175$	$2$	$350$
$[180, 190)$	$185$	$4$	$740$
$[190, 200)$	$195$	$3$	$585$
$[200, 210)$	$205$	$2$	$410$
		$12$	$2250$

We calculate the mean for grouped data: $\overset{―}{x} = \frac{165 \cdot 1 + 175 \cdot 2 + 185 \cdot 4 + 195 \cdot 3 + 205 \cdot 2}{1 + 2 + 4 + 3 + 2} =$ $= \frac{2250}{12} = 187.5$

If there is an interval with a non determinated width it is not possible to calculate the mean:

$[160, 170)$	$165$	$1$	$16$
$[170, 180)$	$175$	$2$	$350$
$[180, 190)$	$185$	$4$	$740$
$[190, 200)$	$195$	$3$	$585$
$[200,)$	$2$
		$12$	$2250$

It is also important to mention that the arithmetical mean is very sensitive to extreme punctuations.

Example

In a basketball match, we have the following points for the players of a team: $0, 1, 3, 4, 5, 6, 7, 8, 47$ Calculate the mean of points of the team.

$\overset{―}{x} = \frac{0 + 1 + 3 + 4 + 5 + 6 + 7 + 8 + 47}{9} = \frac{81}{9} = 9$

In this case the mean does not illustrate well the information, since all the values except one are below the mean.