The arithmetical mean is the average value of the samples. It is independent of the width of the intervals. It is symbolized as $$\overline{x}$$ 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: $$$\displaystyle \overline{x}=\frac{x_1+x_2+x_3+\ldots+x_n}{n}$$$
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 $$$\displaystyle \overline{x}=\frac{0+2+4+5+8+9+10+15+38}{9}=\frac{90}{9}=10$$$
Calculation of the mean for grouped information
The average in the case of $$N$$ data grouped in $$n$$ intervals is given by the formula $$$\displaystyle \overline{x}=\frac{x_1\cdot f_1+x_2\cdot f_2+x_3\cdot f_3+\ldots+x_n\cdot f_n}{f_1+f_2+f_3+\ldots+f_n}$$$
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.
The height in $$cm$$ 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: $$$\displaystyle \overline{x}=\frac{165 \cdot 1+175 \cdot 2+185\cdot 4+195\cdot 3+205\cdot 2}{1+2+4+3+2}=$$$ $$$=\frac{2250}{12}=187.5$$$
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.
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.
$$$\displaystyle \overline{x}=\frac{0+1+3+4+5+6+7+8+47}{9}=\frac{81}{9}=9$$$
In this case the mean does not illustrate well the information, since all the values except one are below the mean.