Statistics How To

Interquartile Mean (IQM) / Midmean

Descriptive Statistics > Interquartile Mean

You may want to read this article first: What is the trimmed mean?

What is the Interquartile Mean?

The interquartile mean (IQM) is the mean of the middle 50 percent of data in a data set. Unlike the “regular” arithmetic mean, it is resistant to outliers.

interquartile mean

The IQM is the mean of the interquartile range (IQR).


How to Find the Interquartile Mean

The calculation is different depending on if your data is divisible by 4 or not.

Data is Divisible by Four

Example question: Find the IQM for the following data set:
5 6 17 30 44 55 56 8 9 11 13 15 1 3 16 65

Step 1: Sort the data from smallest to largest:
1 3 5 6 8 9 11 13 15 16 17 30 44 55 56 65

Step 2: Discard the bottom 25% and top 25% of numbers. In other words, split the data set into quarters and remove the top and bottom quarters:
1 3 5 6 | 8 9 11 13 | 15 16 17 30 | 44 55 56 65

Step 3: Find the mean of the remaining numbers:
(8 + 9 + 11 + 13 + 15 + 16 + 17 + 30 ) / 8 = 14.875

That’s it!

Data is NOT Divisible by Four

Example question: Find the IQM for the following data set:
6 17 30 44 55 56 8 9 11 13 15 1 3 16 65

Step 1: Sort the data from smallest to largest:
1 3 6 8 9 11 13 15 16 17 30 44 55 56 65

Step 2: Divide the number of items in the set by four. The set has 15 items, so 15/4 = 3.75.

Step 3: Remove the whole number (Step 2) from the bottom and the top of the set. For this example, the whole number is 3 (from 3.75):
1 3 6 8 9 11 13 15 16 17 30 44 55 56 65
which leaves:
8 9 11 13 15 16 17 30 44

Step 4: Figure out how many items are in the interquartile range. The IQR is the middle two quarters, so there would be 3.75 * 2 = 7.5 numbers.

Step 5: Place parentheses around the middle set of numbers using the whole number from Step 4. In this example, the whole number is 7:
8 (9 11 13 15 16 17 30) 44

Step 6: Take the fractional part from Step 4 (.5 in this case) and divide it by two (because there are two numbers on the outside of the parentheses):
.5/2 = .25
This means that the numbers 8 and 44 will each contribute 25% to the IQM.

Step 7: Multiply the two “outside” numbers (8 and 44 in this case) by the fraction in Step 6:
8 * .25 = 2
44 * .25 = 11

Step 8: Replace the two outside numbers by the fractional numbers (Step 7) and find the mean. When dividing by “n”, use the number of items in the IQR from Step 4 (7.5 in this case), not the actual number count (9 in this example):
82 (9 11 13 15 16 17 30) 4411 =
(2 + 9 + 11 + 13 + 15 + 16 + 17 + 30 + 11)/7.5 = 16.53.

That’s it!

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Interquartile Mean (IQM) / Midmean was last modified: October 12th, 2017 by Andale