# 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 $$\overline{x}$$ 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.