Quartile Deviation – The Intuition First
You already know the range — the gap between the smallest and largest value. That tells you the total spread, but it’s fragile: one extreme outlier can make the range huge even if most data is tightly packed. Quartile Deviation solves that by ignoring the extremes entirely.
Think of a class of 30 students. Their marks in a test: most are between 40 and 70, but one student scored 98 and another scored 12. The range is 86, which is misleading — it suggests huge variation, but the middle 80% of the class is actually quite similar. Quartile Deviation looks only at the middle 50% of the data, cutting off the bottom 25% and the top 25%. That gives you a stable, realistic measure of spread.
The Precise Definition
First, you need quartiles:
- Q1 (First Quartile): the value below which 25% of the data lies.
- Q3 (Third Quartile): the value below which 75% of the data lies.
- Q2 (Second Quartile): the median — 50% below, 50% above.
The Interquartile Range (IQR) is simply:
IQR=Q3−Q1
This is the range of the middle 50% of the data. The Quartile Deviation (QD) — also called the semi-interquartile range — is half of that:
QD=2Q3−Q1
Quartile Deviation = 2Q3−Q1. It measures the average spread of the middle half of the data, in units of the original variable.
Why Halve It?
The IQR itself is a range — it tells you the total width of the middle 50%. But when we talk about "deviation" or "average spread", we usually want a typical distance from the center. Halving the IQR gives you something like the average distance of the middle 50% from the median. It’s analogous to why we sometimes use the mean deviation — we want a single number that represents how far, on average, values lie from the middle.
A Worked Example
Consider the sorted marks:
12, 15, 20, 22, 25, 30, 35, 40, 42, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 98
There are 20 values.
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Q1 position = 420+1=5.25 → between the 5th (25) and 6th (30) values.
Q1=25+0.25(30−25)=26.25
-
Q3 position = 3×420+1=15.75 → between the 15th (70) and 16th (75) values.
Q3=70+0.75(75−70)=73.75
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IQR = 73.75−26.25=47.5
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Quartile Deviation = 247.5=23.75
So the middle 50% of marks are spread, on average, about 23.75 marks away from the median.
For grouped data, use the cumulative frequency to locate Q1 and Q3 classes, then apply the interpolation formula:
Qk=L+f4kN−cf×h
where L is the lower boundary of the quartile class, cf is cumulative frequency before it, f is its frequency, h is class width, and k=1,3.
When to Use Quartile Deviation …