The arithmetic mean is the most commonly calculated measure of central tendency, and a large number of statistical techniques in inferential statistics use the arithmetic mean. It represents the average score in a distribution of scores.

The arithmetic mean is calculated according to the formula

Formula for arithmetic mean

where is

n - the number of subjects in the sample, and
ΣXi - the sum of all scores in the sample.

Based on the formula, it can be seen that the arithmetic mean is the quotient of the sum of all scores and their number. The arithmetic mean is calculated in two steps. In the first step, all scores are added up, and in the second, the sum of all scores is divided by their number.

Example: Calculating the arithmetic mean

Let's say we have the following data on the heights of girls in the 52 grade of the "X" elementary school (in cm): 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140. We get the average height of girls in that grade by adding 140+141+138+140+122+160+154+132+148 +135+140=1550, which we then divide by the number of girls, i.e. by 11.

The answer is: The average height of girls in the 52 grade of the "X" elementary school is 140.91 cm.

The arithmetic mean is calculated on data that are at the interval or ratio level of measurement. It is also possible to calculate it on data at the nominal or ordinal level of measurement, but this does not make much sense. For example: if we code the nominal variable gender with 1 - men and 2 - women, the arithmetic mean would be 1.5 and would have no informative value.

The value of the arithmetic mean is affected by extreme values. Because of the way the arithmetic mean is calculated, an extremely high score or scores will increase the value of the arithmetic mean and conversely, an extremely low score or scores will decrease the value of the arithmetic mean. Therefore, the arithmetic mean is not the best measure of central tendency for skewed distributions of scores.

The arithmetic mean value is within the range of results, i.e. between the values ​​of the lowest and highest achieved results in the sample. It is possible for the arithmetic mean to have the value of the lowest or highest result in the distribution, but this would imply that all respondents achieved the minimum or maximum result. In that case, it would be a constant, because all respondents achieved the same result.

The sample mean is denoted by X and is a statistic. The population mean is denoted by μ and is a parameter.

The value of the arithmetic mean is expressed in the same unit of measurement in which the distribution is expressed.