Let’s say that some college professor wants to see how was the score of his students. What would he do? Obviously, it’ll be too complicated to read the scores from all the students. Instead of this, he would probably find the average score.
The arithmetic mean is just an another name for the mean or the average. It is preferably to call it ‘arithmetic mean’ instead of simply ‘mean’ because in math there are several means; for example, there are geometric mean and harmonic mean.
Definition
Let $X: S \rightarrow X(S) = \{x_{1}, x_{2}, \cdots, x_{k}\} \subseteq \mathbf{R}$ be the numeric variable given on finite population with $N$ elements. Furthermore, let $\{(x_{i}, f_{i}), i = 1, \cdots, k\}$ be its frequency distribution.
Definition: Arithmetic mean $\mu$ of numeric variable $X$ is defined by
$$\mu = \frac{\sum_{i = 1}^{k}f_{i}x_{i}}{N}.$$
If numeric variable is given with statistical sequence $y_{1}, \cdots, y_{N}$, then its arithmetic mean is obviously
$$\mu (y_{1}, \cdots, y_{N}) = \frac{\sum_{i = 1}^{N} y_{i}}{N}.$$
Therefore, finding an arithmetic mean simply means that we have to sum all the numbers up and then divide that sum by the number of elements in our set.
If our data is population, then the mean is called population mean and it is denoted by the letter $\mu$. If the data is a sample, it’s called a sample mean and it is denoted by $\bar{x}$.
Examples
Example 1: The manufacture of some detergent during one decade was (in tons): $$105, 100, 110, 112, 108, 100, 104, 115, 96, 120.$$ The average manufacture is $$\mu = \frac{1}{10} \cdot (105 + 100 + 110 + 112 + 108 + 100 + 104 + 115 + 96 + 120) = 107.$$
Example 2: The distribution of number of days by number of absent employees of some company is given:
The arithmetic mean of numeric variable which joins to every employee the number of days of absence is:
$$\mu = \frac{4 \cdot 0 + 10 \cdot 1 + \cdots + 3 \cdot 9 + 2 \cdot 10 }{4 + 10 + \cdots + 2} =\frac{416}{110} = 3.78182.$$
Example 3: The distance (in kilometres) from place of residence to work location of employees of some company is given:
The average distance is $$\mu = \frac{\sum_{i = 1}^{5}f_{i}x_{i}}{\sum_{i = 1}^{5} f_{i}} = \frac{1 \cdot 10 + 2 \cdot 15 + \cdots + 1 \cdot 40}{1 + 2 + \cdots + 1} = \frac{270}{12} = 22.5 \ km.$$
We have seen examples of calculating an arithmetic mean for ungrouped observations. If we want to calculate the arithmetic mean of grouped data, we need to calculate the class mark.
Precisely, if we have data grouped in classes, then the whole class is identified with class mark, i.e. $x_{i}f_{i}$ represents the product of frequency $f_{i}$ of the class $\left[L_{1i}, L_{2i}\right>$ and class mark $x_{i} = \frac{L_{1i}+L_{2i}}{2}$.
This arithmetic mean is called the weighted arithmetic mean.
Example 4: Find the average traffic data for shops if you have the following informations:
Solution:
$$x_{i} = \frac{L_{1i} + L_{2i}}{2}$$
$$x_{1} = \frac{30 + 40}{2} = 35$$
$$x_{2} = \frac{40 + 50}{2} = 45$$
$$\vdots$$
$$x_{7} = \frac{110 + 150}{2} = 130$$
$$\mu = \frac{\sum_{i = 1}^{7}x_{i}f_{i}}{50} = \frac{3585}{50} = 71.7 \ thousands \ of \ units$$
Geometric interpretation of the arithmetic mean
The first step in geometric construction of the arithmetic mean of numbers $x$ and $y$ is to draw a semicircle with diameter of length $x + y$.
Next step is to draw a perpendicular to the diameter from the midpoint of the line segment of length $x + y$.
Finally, the arithmetic mean of $x$ and $y$ is the length of the perpendicular from the semicircle to the diameter.
