Arithmetic Mean

The most widely used measure of central tendency is the arithmetic mean. Most commonly, mean refers to the arithmetic mean. The arithmetic mean is defined as all the scores for a variable added together and then divided by the number of observations. Therefore, the formula to compute the arithmetic mean is as follows:

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where

• X represents the data points,
• Σ is the summation of all the Xs,
• n is the number of data points or observations, and
• X¯ is the computed mean.

For example, take the data presented in Table 1.

The sum of the observations (ΣX) is 1 + 5 + 7 + 2 + 10 + 4 + 6 + 5 + 4 + 6 = 50. Then, we divide this value by n, which in this example is 10 because we have 10 observations. Thus, 50/10 = 5. The arithmetic mean for this set of observations is 5.

The SPSS statistical software package provides several ways to compute a mean for a variable. The Mean command can be found under Descriptives, then Frequencies, Explore, and finally Means. Furthermore, the mean can be added to output for more advanced calculations, such as multiple regression. The output for the Descriptives mean is presented in Figure 1.

As seen in the output, the variable “data points” has a total of 10 observations (seen under the column headed N), the lowest value in the data set is 1, the highest value is 10, the mean is 5, and the standard deviation is 2.539.

Table 1 Data for Arithmetic Mean Computation
Observation	Data Points
1	1
2	5
3	7
4	2
5	10
6	4
7	6
8	5
9	4
10	6

There are two major issues you should be aware of when using the arithmetic mean. The first is that the arithmetic mean can be influenced by outliers, or data values that are outside the range of the majority of the data points. Outliers can pull the mean toward themselves. For example, if the data set in Figure 1 included a data point (which would be observation 11) of 40, the mean would be 8.2. Thus, when the data set is extremely skewed, it can be more meaningful to use other measures of central tendency (e.g., the median or the mode).

The second issue is that the arithmetic mean is difficult to interpret when the variable of interest is nominal with two levels (e.g., gender) and not meaningful when there are more than two levels or groups for a given variable (e.g., ethnicity). The mean has been found to be consistent across time. With repeated measures of the same variable, the arithmetic mean tends not to change radically (as long as there are no extreme outliers in the data set). Furthermore, the arithmetic mean is the most commonly used measure of central tendency in more advanced statistical formulas.

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