Range and Interquartile Range (IQR)

1. Range

One measure of variation is range. The range is one of the simplest ways to calculate variation. It is calculated by simply subtracting the minimum value from the maximum value.

formula to know

Range

EXAMPLE

This chart shows the height of the Chicago Bulls basketball team for a particular year:

Height of Chicago Bulls Players
Omer Asik	84
Carlos Boozer	81
Ronnie Brewer	79
Jimmy Butler	79
Luol Deng	81
Taj Gibson	81
Richard Hamilton	79
Mike James	74
Kyle Korver	79
John Lucas III	71
Joakim Noah	83
Derrick Rose	75
Brian Scalabrine	81
Marquis Teague	74
C.J. Watson	74

It's easy to see from the list that the minimum value is 71 and the maximum value is 84. The range is actually the easiest measure of spread or variation to find: Take the maximum value (in this case, the tallest person) and subtract the minimum value (in this case, the shortest person).

Therefore, 84 minus 71 equals 13 inches. This means that every individual on the team falls within a 13-inch range.

term to know

Range

The difference between the largest and smallest number in a data set.

2. Interquartile Range (IQR)

Range and interquartile range are similar ideas in that they're both measured by subtraction, so they're not particularly difficult to calculate. While they are calculated similarly, they do measure different measures of variation.

The interquartile range, also abbreviated IQR, is another measure of spread, but it's median-based. To review, the median is the middle number of an ordered data set.

The IQR represents the range in which the middle 50% of the data points lie. Finding the interquartile range takes a few steps.

step by step

Step 1: First, order the data set from smallest to largest and then work your way in towards the center until you find the median.

71, 74, 74, 74, 75, 79, 79, 79, 79, 81, 81, 81, 81, 83, 84

The median, in this case, is 79.

Step 2: Once you identify the two data sets on either side of the median—a low half and a high half—you can find the middle number of each of those halves. These three numbers together are called the quartiles.

• The first quartile, or Q1, is where 1/4 of the data falls at or below this point.
• The median, which can also be called the second quartile (Q2), is where half the data falls at or below this point.
• The third quartile, or Q3, is where 3/4 of the data falls at or below this point.

With this data set:

71, 74, 74, 74, 75, 79, 79, 79, 79, 81, 81, 81, 81, 83, 84,

• Q1 = 74
• Q2 (median) = 79
• Q3 = 81

Step 3: Finally, to calculate the interquartile range, find the difference between the third and first quartiles.

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Interquartile Range

Going back to our example, the IQR is calculated as Q3 minus Q1, or 81 minus 74, which equals 7 inches. This means that the middle half of the data set falls within a 7-inch range, whereas the entire data set falls within a 13-inch range.
Visually, the IQR is the box on a box plot, shown below:

The range gives the entire spread of the data set lowest to highest, whereas the IQR gives the range of the middle 50%.

hint

Using IQR has several advantages:

• Outliers strongly affect range calculations, but IQR is unaffected because only the middle 50% of the data is included.
• For skewed data, the mean (and therefore the standard deviation) can be misleading, as the mean can be pulled in the direction of the skew, not accurately representing the center of the data. IQR focuses on the middle 50% of the data, providing a better sense of the spread and variability of the data without being distorted by skewness.

term to know

Interquartile Range

The difference between the third and first quartiles. It represents the range in which the middle 50% of the data points lie.