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Descriptive Statistics: Skewness and the Mean, Median, and Mode

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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Consider the following data set:

• 4 ; • 5 ; • 6 ; • 6 ; • 6 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 8 ; • 8 ; • 8 ; • 9 ; • 10

This data produces the histogram shown below. Each interval has width one and each value is located in the middle of an interval.

A histogram with a symmetrical data distribution, with a mean, median, and mode of 7.

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each 7 for these data. In a perfectly symmetrical distribution, the mean, the median, and the mode are the same.

The histogram for the data:

• 4 ; • 5 ; • 6 ; • 6 ; • 6 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 8

is not symmetrical. The right-hand side seems "chopped off" compared to the left side. The shape distribution is called skewed to the left because it is pulled out to the left.

A histogram that is skewed to the left. The mode is still 7, but the mean and median are less than 7.

The mean is 6.3, the median is 6.5, and the mode is 7. Notice that the mean is less than the median and they are both less than the mode. The mean and the median both reflect the skewing but the mean more so.

The histogram for the data:

• 6 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 7 ; • 8 ; • 8 ; • 8 ; • 9 ; • 10

is also not symmetrical. It is skewed to the right.

A histogram skewed to the right. The mode is still 7, but the mean and median are both greater than 7.

The mean is 7.7, the median is 7.5, and the mode is 7. Notice that the mean is the largest statistic, while the mode is the smallest. Again, the mean reflects the skewing the most.

To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is less than the mode. If the distribution of data is skewed to the right, the mode is less than the median, which is less than the mean.

Skewness and symmetry become important when we discuss probability distributions in later chapters.

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