# Measures of Dispersion: Definition, Equations & Examples

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• 0:04 Dispersion Definition
• 1:23 Dispersion Measures
• 6:18 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

The dispersion of a data set is the amount of variability seen in that data set. This lesson will review the three most common measures of dispersion, defining and giving examples of each.

## Dispersion Definition

Pretend that you want to sell your house. You narrow your search to two companies: SCT Housing and WCT Housing. Both companies advertise that sellers receive, on average, 90% of their asking price. Does it matter which company you choose?

The real question is, does the mean (average), describe the data accurately enough to make an informed decision? No, it doesn't. The mean is not a reliable predictor; it only describes the data set as a whole and doesn't tell what's happening within the set.

Let's add some data to the example to illustrate the point. Assume the following shows the percent of asking price received on the previous nine sales for each company:

• SCT: 88, 92, 91, 89, 89, 91, 91, 89, and 90
• WCT: 71, 100, 100, 83, 100, 95, 86, and 90

How can you make an informed decision about which company will offer you the greatest benefit for the least risk? You must analyze each set of the dispersion, which is the amount of variation within a data set. Only when you do that, will you be able to truly compare these two companies.

## Dispersion Measures

Data sets with strong central tendencies are sets in which items are tightly grouped around the mean. Weak central tendency in data indicates that individual items aren't grouped with any significance, which makes predictions based on this data less reliable than those based on data sets with strong central tendencies.

For example, if your mail is always delivered between 8:02 a.m. and 8:08 a.m., you can reliably predict when the mail will come. However, if your mail delivery can range from 8:00 a.m. to 5:30 p.m., you are no longer able to pinpoint a delivery time and planning for special deliveries becomes much more tricky.

In this lesson, we will review three measures of dispersion:

• Range, the distance between the lowest and highest values in the set
• Interquartile range
• Standard deviation

Let's explore these measures of dispersion by applying them to our opening scenario.

### 1. Range

To find the range of any data set, you need to first put the values in order from lowest to highest. Then you simply subtract the lowest from the highest. Before continuing, go back and find the range of each of the previous data sets. So:

• SCT Housing: 88, 89, 89, 89, 90, 91, 91, 91, 92 = 92 - 88 = 4. Range is 4.
• WCT Housing: 71, 83, 85, 86, 90, 95, 100, 100, 100 = 100 - 71 = 29. Range is 29.

A small range indicates a strong central tendency. Remember that a strong central tendency tells us that all the data is grouped tightly around the mean.

The range identifies how varied a data set is but does not account for outliers, or pieces of data that fall far outside the remainder of the data set (like the 71 in the WCT set). Outliers can skew measures of central tendency artificially.

### 2. Interquartile Range

To account for possible outliers use the interquartile range (IQR). This is a measure of the range within only the middle 50% of the data set.

To find the IQR, you separate the data set into quartiles (or four equal parts) by first putting the data set into numerical order (as we did with the range). Then find the median (meaning the middle) of the set. This is identified as Q2, or the beginning of the second quartile. After finding the median of the whole set, identify the median of each half of the set.

The median of the lower half is Q1 and the median of the upper half is Q3. These two points mark the top and bottom of the middle 50% of the data set. The range is the difference between Q1 and Q3.

See if you can identify the IQR of each set before moving on.

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