Simple Descriptive Statistics

Mean

The mean ($\overline{\mathrm{X}}$) is one of the most common statistical measures used to describe a sample population. Commonly referred to as the “average,” the [Page 1602]mean is calculated by finding the sum of all points in a data set and dividing it by the total number of observations. The formula for $\overline{\mathrm{X}}$ is as follows:

$$\overline{X}=\mathrm{\Sigma }\left(X_1+X_2...X_n\right)/N$$

where X represents each observation in the data set, and N is the total number of observations. Using this formula, the professor can take the sum of all 100 of the individual midterm exam scores (score student 1 + score student 2 + score student 3 + . . . score student 100) and divide by the total number of scores (N = 100). The resulting sample mean ($\overline{\mathrm{X}}$) provides useful information related to the average performance on the midterm exam by this group of students (i.e., one score that resides in the center of a distribution). The professor can also compare the means from semester to semester or professor to professor, assuming that the same or very comparable exam is being used, to identify basic trends. It is important to remember, however, that inferring why scores may vary or remain unchanged is beyond the capabilities of descriptive statistics.

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