Wiki Page, Math 13 - Perez Sandbox : Module 5 - Mean and Median (1a of 2)

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2026-01-17T21:15:50-08:00

Making the Concept of Center More Precise

When we describe the distribution of a quantitative variable, we describe the overall pattern (shape, center, and spread) in the data and deviations from the pattern (outliers).

In our previous discussion of patterns in quantitative data, we identified a typical value in the distribution. We used this single value of the variable to represent the entire group. This is an informal way to think about the center of the distribution. Now we focus on describing the center of a distribution more precisely.

We develop two different measurements for identifying the center of a distribution: the mean and the median. Each measure has special properties.

Mean

The mean is the average. It is written as and pronounced “x-bar.” To calculate the mean, we add the data values and divide by the number of data points.

We can write this as the following formula.

In this formula, the symbol means sum (add up the values). The represents the data values. The letter represents the number of data values.

EXAMPLE Calculating the mean

Let’s find the mean of a set of three quiz scores: 70, 85, 82. In this situation, because there are 3 quiz scores. We add the values, 70 + 85 + 82 to get 237, then divide by 3 to get a mean of 79.

We could write this calculation using the formula:

Some Properties of the Mean

The mean is a single number that represents the distribution.

Suppose Beth’s homework scores are 70, 80, 80, 80, 85, 86, 90, 90, 95.

The mean of her scores is

When we examine the mean relative to the distribution of scores, the mean may, or may not, do a good job representing the distribution. In Beth's situation, the mean appears to be a pretty good measure of how Beth performed overall.

There are times when a mean does not represent a distribution well. For example, if Beth scored an 82 on 8 assignments and a 100 on one assignment, her mean would still be 84. But in this case, the mean is not the best representation of the distribution since all but one of the scores is an 82.

The mean is a fair share measurement.

Some people call the mean the fair share measurement. If the points are evenly distributed across the 9 assignments, the score on each assignment would be 84. In other words, if Beth earned the same number of points (756) on the 9 assignments, but earned the same score on each assignment, that score is the mean.

Notice, however, that Beth did not actually make an 84 on any assignment. The mean does not give us information about any individual homework score or about how the homework scores vary in the distribution. It only gives us a sense of her performance by averaging the values across all the assignments as if the distribution had no variability at all.

The mean is the balancing point of the distribution.

The mean is also referred to as the balancing point of a distribution. If we measure the distance between each data point and the mean, the distances are balanced on each side of the mean.

For example, a homework score of 95 is 11 points above the mean, as shown.

A homework score of 80 is 4 points below the mean. In the table, we calculate the sum of the distances above and below the mean. Notice that the sum of the distances above and below the mean are equal. In this way, the mean is a balancing point for the distribution.

We can also view the distances below the mean as negative and the distances above the mean as positive. When we add these “signed” distances together, we get 0.

The mean is the only measure of center with this special property.

This short movie on the mean as a balancing point shows more examples. (4:45minutes).

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