Measures of Spread: Range and Interquartile Range (IQR)

Learning Objectives

Define measures of spread and why they are important.

Calculate the range of a data set.

Calculate the five-number summary (minimum, Q1, median, Q3, maximum).

Calculate and interpret the interquartile range (IQR).

Describing Data Beyond the Center

In a previous lesson, we learned about measures of central tendency (mean, median, mode), which describe the 'typical' value in a data set. But that's only half the story. Consider these two data sets:

Set A: {10, 20, 50, 80, 90}

Set B: {48, 49, 50, 51, 52}

Both sets have the same mean (50) and the same median (50). But are they the same? Absolutely not! Set A is very spread out, while Set B is tightly clustered together. We need measures of spread (or variability) to describe this difference.

Range: The Simplest Spread

The range is the simplest measure of spread. It's the difference between the maximum and minimum values in a data set.

Range = Maximum Value - Minimum Value

For Set A: Range = 90 - 10 = 80

For Set B: Range = 52 - 48 = 4

The range gives us a quick idea of how spread out the data is, but it can be misleading because it is only affected by the two most extreme values (outliers).

The Five-Number Summary and IQR

A more robust way to understand the spread of data is to look at how it's divided into quarters, or quartiles. This starts with the five-number summary.

1.Minimum: The smallest value in the data set.

2.Maximum: The largest value in the data set.

3.Median (Q2): The middle value of the entire data set.

4.First Quartile (Q1): The median of the lower half of the data set.

5.Third Quartile (Q3): The median of the upper half of the data set.

Example: Let's find the five-number summary for the data set {2, 3, 5, 6, 8, 10, 11}.

Step 1: Order the data. (It's already ordered).

Step 2: Find the Min and Max. Min = 2, Max = 11.

Step 3: Find the Median (Q2). The middle value is 6.

Step 4: Find Q1. Look at the lower half of the data (the numbers to the left of the median): {2, 3, 5}. The median of this half is 3. So, Q1 = 3.

Step 5: Find Q3. Look at the upper half of the data: {8, 10, 11}. The median of this half is 10. So, Q3 = 10.

Five-Number Summary: {2, 3, 6, 10, 11}

Interquartile Range (IQR): The Middle 50%

The interquartile range (IQR) is the range of the middle 50% of the data. It is a very useful measure of spread because it is not affected by outliers.

How to find it: Subtract the first quartile from the third quartile.

IQR = Q3 - Q1

For our example data set: IQR = 10 - 3 = 7.

This tells us that the middle half of our data is spread over a range of 7 units.

Key Terms

**Measures of Spread: Statistics that describe how similar or varied the set of observed values are for a particular variable.
**Range: The difference between the maximum and minimum values in a data set.
**Quartiles: The values that divide a list of numbers into four equal parts.
**Median (Q2): The middle quartile, which divides the data set in half.
**First Quartile (Q1): The median of the lower half of a data set.
**Third Quartile (Q3): The median of the upper half of a data set.
**Interquartile Range (IQR): A measure of statistical dispersion, being equal to the difference between the upper and lower quartiles (IQR = Q3 - Q1).