Measures of Center

When we use the word “average” in mathematics and statistics, we want to be more specific. There are three types of averages used in statistics. Each one describes a “typical” data value. If the data is numerical, then the average will be somewhere between the least and the greatest data value.

Mean

The mean is the sum of the data values divided by the number of data values. We use $LaTeX: \overline{x}$ to represent the mean.

$LaTeX: \overline{x}$ $LaTeX: =\frac{x_1+x_2+\cdots+x_n}{n}$

The mean of the numbers {1, 4, 10} is $LaTeX: \frac{1+4+10}{3}=5$

Median

The median is another measure of center for numeric data which is especially useful if there are a few unusually low or unusually high data values (outliers).

When the data is placed in numeric order, the median is the central value. If there are an even number of data values, then the median is the arithmetic mean of the middle two values.

The median for {1, 4, 10} is 4.

The median for {1, 4, 5, 10} is 4.5.

Mode

The mode is the most commonly occurring value. Unlike the mean and the median, which can only be used for numerical data, the mode can represent a typical value for categorical data.

There may be no mode, one mode, or multiple modes.

The data set {1, 4, 10} has no mode.

The mode for {1, 4, 10, 10} is 10.

If the data is categorical, then a mode can be determined. The mode for

{yes, no, no, no, yes, yes, yes, no, yes, yes, yes} is “yes”.

Relating the mean and the median

The median is called a resistant measure because is unaffected by outliers. The mean, on the other hand, is sensitive to outliers. Consider these examples:

Data Set 1: {10, 20, 30, 40, 50} median is 30, mean is also 30.

Data Set 2: {10, 20, 30, 40, 500} median is still 30, mean is 120. In this data set, the mean does not describe a “typical, central” value as well as the median does.

When there are unusually low or high values possible (such as home prices), the median is preferred to the mean to describe a typical value.

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