Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Dr. Roger Day (day@math.ilstu.edu)

### The Shape of a Distribution

The distribution shown at the conclusion of the last section, described as a bell-shaped or mound-shaped curve or a normal distribution, is just one example of a shape that a distribution can take on. The normal distribution is an example of a symmetric distribution, one whose left and right sides are mirror images of each other. Many distributions are asymmetric, meaning their left and right portions are not mirror images. The concept of skewness helps describe asymmetric distributions.

The distribution above is skewed to the right, or positively skewed. Although the "mound" of values occurs in the left portion of the distribution, it is the tail of the distribution, extending to the right and containing extremely large values, that determines the skewness of the distribution. As we have previously pointed out, those large extremes pull the mean of the distribution toward that tail, while the median of the distribution remains more firmly anchored in the center of the distribution. You will recall that we termed the median a resistant measure of location because of this characteristic.

Likewise, the distribution below is skewed to the left. Extreme values in the lower portion of the distribution pull the mean to the left, while the median resists the impact of the extreme values. Because the mean is less than the median, we also say that this distribution is negatively skewed. What can we say about the relative location of the mean and median in a symmetric distribution?

We can express a numeric degree of skewness in a distribution by using the following relationship:

For the set of 21 exam scores we have used in several previous examples, we get

This tells us there is some degree of positive skew in the data. (Note that the population standard deviation was used here, because the data set contained scores for the entire population of statistics students in that particular class.) Now consider the skewness in each of these sets of data, salaries for two companies. Here, assume these are a random sample of each companies salaries, so we use the sample standard deviation.

Examine the salaries for each company and use them to help justify the skewness calculated here.

Another characteristic of distributions is called kurtosis. This applies primarily to normal distributions and describes the degree of peakedness of a distribution. Here are examples of the three types of kurtosis typically identified in bell-shaped curves.

Leptokurtic distributions are characterized by high and narrow peaks, mesokurtic distributions are moderately peaked, any platykurtic distributions are flat-topped distributions.