The linear mean, commonly known as the arithmetic mean, is the most basic and frequently used measure of central tendency in statistics. Its purpose is to distill a set of numerical observations into a single, representative value that indicates the center of the data distribution. The term “linear mean” is sometimes used to distinguish this standard average from specialized averages, such as the geometric mean or the harmonic mean, because it assumes a straight relationship in the data’s composition.
How to Calculate the Arithmetic Mean
Calculating the arithmetic mean is a straightforward, two-step process requiring only the values within the data set and the total count of those values. The procedure begins by finding the sum of all the numbers in the data set. For example, if a set contains the numbers 10, 15, 20, and 25, the sum is 70.
The second step involves dividing this total sum by the number of observations. In the example, there are four numbers, so 70 is divided by 4, resulting in an arithmetic mean of 17.5. This process is mathematically represented by the formula: Mean = (Sum of all observations) / (Total number of observations).
Statisticians use compact notation for this calculation, where the Greek letter sigma ($\Sigma$) symbolizes “the sum of,” and the result is denoted by $\bar{x}$ (read as “x-bar”). The formula states that to find the mean ($\bar{x}$), you sum all the individual data points ($x_i$) and divide that result by the total number of points ($n$). This method ensures that every value in the data set contributes equally to the final average.
Where the Mean is Used
The arithmetic mean is widely applied across numerous fields because it offers an effective way to summarize large amounts of data. In education, the mean is used to calculate a student’s average test score or grade point average, representing overall performance across multiple assignments. Businesses use the mean to track average daily sales, compare average revenue generated by different product lines, or calculate the average time customers spend on a website.
In finance, investors use the arithmetic mean to determine the average return of a stock or investment portfolio over a specific period. Scientists and researchers apply the mean to experimental data to find a midpoint result, which is used for testing hypotheses and drawing conclusions. Furthermore, the mean is utilized in economics to compute per capita income, which is the average income of a nation’s population.
While the mean is a powerful summary statistic, it is susceptible to extreme values, known as outliers. A single number that is significantly larger or smaller than the rest of the data set can disproportionately shift the mean away from the majority of the data points. For example, including a single, highly paid executive when calculating the average salary in a small company can dramatically inflate the mean, making it a less accurate representation of the typical employee’s salary.