Probability and Statistics
What is the arithmetic mean?
The arithmetic mean is the most common and intuitive form of average. As a special case within the broader family of power means it expresses the representative value of a data set by dividing the total sum of all observations by their number. Since it is grounded on additive aggregation, the arithmetic mean is ideal for describing quantities that combine linearly (for example, raw measurements or values that do not evolve proportionally or exponentially). In essence, it identifies the equilibrium point of the distribution, the value around which the data tend to balance.
In general form, the arithmetic mean is expressed as:
\[M_{a} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}\]where $x_{1} , x_{2} , \ldots , x_{n}$ are the observed values and $n$ is the total number of elements.
- The arithmetic mean can be applied to any set of real numbers, including negative and zero values.
- Because it is sensitive to extreme values, the arithmetic mean can be distorted by outliers, making other means, like the median or geometric mean, more appropriate in some cases.
- The arithmetic mean is always greater than or equal to the geometric mean.
Example 1
Let’s consider the following data set of five numerical values and let’s calculate the arithmetic mean:
| $\mathbf{x} \mathbf{ᵢ}$ | Values |
|---|---|
| x₁ | 7.2 |
| x₂ | 4.8 |
| x₃ | 9.1 |
| x₄ | 5.5 |
| x₅ | 6.4 |
In this case, $n = 5$. Substituting the values, we get:
\[M_{a} = \frac{7.2 + 4.8 + 9.1 + 5.5 + 6.4}{5} = \frac{33.0}{5} = 6.6\]Hence, the arithmetic mean of the series is approximately: \(M_{a} = 6.6\)
Weighted arithmetic mean
In some cases, not all data points contribute equally to the overall result. The weighted arithmetic mean extends the idea of the simple arithmetic mean by assigning a weight $w_{i}$ to each observation $x_{i}$, reflecting its relative importance or frequency within the dataset. It is defined as:
\[M_{a w} = \frac{\sum_{i = 1}^{n} w_{i} x_{i}}{\sum_{i = 1}^{n} w_{i}}\]where $x_{i}$ are the observed values and $w_{i} > 0$ are their associated weights.
- The weighted arithmetic mean generalizes the simple arithmetic mean by introducing importance factors $w_{i}$.
- It ensures that larger or more relevant observations have a stronger influence on the final result.
- When all weights are equal, the weighted arithmetic mean reduces to the standard arithmetic mean.
Example 2
Let’s consider a business case where a company wants to calculate the weighted arithmetic mean of its monthly sales. Each month has a different number of working days, which serve as the weights for the calculation.
| Month | $x_{i}$ = daily sales in $ | $w_{i}$ = working days |
|---|---|---|
| January | 420 | 20 |
| February | 380 | 22 |
| March | 460 | 18 |
| April | 400 | 21 |
| May | 440 | 19 |
By applying the formula of the weighted arithmetic mean, we obtain:
\[M_{a w} & = \frac{( 420 \times 20 ) + ( 380 \times 22 ) + ( 460 \times 18 ) + ( 400 \times 21 ) + ( 440 \times 19 )}{20 + 22 + 18 + 21 + 19} \\ & = \frac{8400 + 8360 + 8280 + 8400 + 8360}{100} \\ & = \frac{41800}{100} \\ & = 418\]Hence, the weighted arithmetic mean of the company’s sales is $M_{a w} = 418$ $ per day.