**Introduction to Measures of Dispersion for Numerical Data**

*When describing a distribution, it is important to have some measure of where the middle is. With a little thought, we can see that, if we have a measure of the middle, we still need more information to describe the distribution fully. For example, are all of the population values the same or are they spread out over a range of values? If they are spread out, how should we measure the spread? Measuring the spread, or dispersion, of population or sample values is the focus of this lesson.*

**Population Measures of Dispersion**

Two distributions can have the same middle but still look very different. As an illustration, consider the two populations presented graphically in the dotplots in Figure 6.1.

For each of the populations, the mean, median, and mode are all 15.Yet, they are different; one is more spread out than the other. The measures of dispersion are used to quantify the spread of the distribution. Range, interquartile range, mean absolute deviation, and standard deviation are four such measures that will be discussed in this lesson.

**Range**

The *range* is the difference in the largest and smallest population values; it is the total spread in the population. In Figure 6.1, *Y* assumes values from 13 to 17, giving a range of 17 – 13 = 4. The range of *X* is 20 – 10 = 10. Because the range of *X* is larger than that of *Y*, more than twice as large in this example, *X* is more spread out; that is, *X* has a larger dispersion than *Y*.

**Example**

Find the range of heights of the high school orchestra members that were first discussed in Lesson 4.

**Solution**

The tallest orchestra member is 83.8 inches tall, and the shortest is 53.5 inches tall. The range of the heights is 83.8 – 53.5 = 30.3 inches, more than 2.5 feet!

**Interquartile Range**

Although the range in the heights of orchestra members is large (30.3 inches), most of the orchestra members are much closer in height than this indicates. The range is a crude measure of the dispersion of the population distribution. For example, consider the dotplot for the population distribution of *Z* in Figure 6.2.

Notice the range of *Z* is 4 as is the range of *Y* in Figure 6.1. Both *Y* and *Z* have means, medians, and modes of 15. Yet, the distribution of *Z* appears to be more spread out than that of *Y*. We need another measure to capture this dispersion. We will explore two such measures, beginning with the interquartile range.

The range is greatly affected by exceptionally small or large values in a population. Instead of looking at the range of all population values, the *interquartile range* measures the spread in the middle half of the data. To find the interquartile range, we must first find the quartiles. The *quartiles* are the values that divide the population into fourths, just as the median divided the population in half. The first quartile separates the bottom 25% of the data from the top 75&% of the data. The second quartile separates the bottom 50% of the data from the top 50% of the data. But, wait. That is exactly what the median does! The *second quartile* and the *median* are different names for the same quantity. The third quartile separates the bottom 75% of the data from the top 25% of the data.

The first and third quartiles are found by separating the lower half of the population values from the upper half of the population values. If there is an odd number of population values, the median is excluded from both halves. The first quartile, denoted by *Q*_{1}, is the median of the bottom half of the population. The third quartile, *Q*_{3}, is the median of the top half of the population values. The *interquartile range* (IQR) is then:

*IQR* = *Q*_{3} – *Q*_{1}

The *IQR* is the range or spread of the middle half of the population values.

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