### Chapter 1. Descriptive Statistics: Measures of Variability

### Range, Interquartile Range, and the Five-Number Summary

The most basic measure of variability is the *range*.

Range

**Definition**

The **range **is the difference between the highest and the lowest score of a distribution.

**Formula**

\[\text{range} = X_{max} - X_{min} \]

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Since the range measures variability by looking at the end-points of a distribution, it is extremely sensitive to the presence of outliers in the dataset. An alternative measure of variability that much less sensitive to outliers is the *interquartile range*.

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Interquartile range

**Definition**

The **interquartile range** (IQR) is the difference between the first quartile and the third quartile of a distribution.

**Formula**

\[\text{IQR} = Q_3 - Q_1\]

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Calculating Quartiles with Statistical Software

To calculate *quartiles* in Excel, make use of the following function:

QUARTILE.INC(array, quart)

array: The array or cell range of numeric values for which you want to calculate the sample variance.quart:Indicates which quartile to return.

To calculate quartiles in R, make use of the following function:

quantile(x)

x: The numeric vector whose quartiles you wish to calculate.

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Remember that *quartiles* are measures of location which divide a distribution into four equal parts, similar to how the median divides a distribution into two equal parts. The interquartile range thus measures how spread out the middle 50% of the data is. This means that the interquartile range is completely unaffected by the values of the smallest and largest 25% of the scores.

Often, the range and interquartile range are combined with the median to form a so-called *five-number summary.*

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Five-number summary

**Definition**

The **five-number summary** is made up out of the following five points of a distribution:

- Minimum
- First quartile
- Median
- Third quartile
- Maximum

To plot the five-number summary, construct a **boxplot**.

**#\phantom{0000000000}#Boxplot**

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