18. Measures of Dispersion

Suppose two stocks have the same mean expected return (10%). However, one stock always has a 10% return, while the other stock is very unpredictable. Which one would you want? Probably the first one, right? This section defines some basic measures of dispersion, also known as spread or variability, and how they help interpret data.

Range

The simplest measure of dispersion is range, which is defined as the span of values that a variable takes. You can find the range by determining the difference between the maximum and minimum values, as follows:

Range = Maximum Value Minimum Value

The range is useful because it is easy to calculate and even easier to understand.

Comparing the range and the mean indicates how significant the dispersion is. However, the range does not measure the dispersion of the majority of the observations; it focuses only on the highest and lowest values. Therefore, it is very sensitive to outliers.

Consider the following two 10-item data sets:

Data Set A

12, 12, 12, 13, 13, 14, 22, 23, 23, 24

Data Set B

124, 126, 127, 129, 129, 130, 132, 132, 133, 136

Both data sets have a range of , but they certainly are not dispersed the same.

Quartiles

The range shows the minimum and maximum values, and the median shows the middle value. If you combine the range and the median, the picture of a variables distribution starts to come into focus. This process can be taken a step further by calculating quartiles and plotting the data.

Consider this data set:

87	120	87	84	83	105	80	89	107	107
98	98	84	83	82	79	94	76	86	87

If put in order and split into quartiles, the data set becomes

76	79	80	82	83
83	84	84	86	87
87	87	89	94	98
98	105	107	107	120

	1st quartile =
	2nd quartile (median) =
	3rd quartile =

The minimum, 1st quartile, median, 3rd quartile, and maximum are shown in the graph above. This drawing is called a , and it shows the data to be considerably skewed to the right.