Range and Interquartile Range
Simple Explanation
The range is the simplest measure of spread: the highest value minus the lowest. The interquartile range (IQR) is more robust: it measures the spread of just the middle 50% of the data, Q₃ − Q₁, ignoring extreme values at either end.
Why Do We Need It?
A single extreme outlier can massively distort the range, but barely affects the IQR — knowing both, and when each is more appropriate, is essential for honestly describing a dataset's spread.
Formula
Range and Interquartile Range
Range = maximum − minimum. IQR = Q₃ − Q₁
Two simple measures of how spread out a dataset is. The range uses only the two extreme values; the interquartile range (IQR) uses the middle 50% of the data, making it far less sensitive to outliers.
- Q₁
- — the first quartile — the value 25% of the way through the ordered data
- Q₃
- — the third quartile — the value 75% of the way through the ordered data
When to use it: Whenever you need a quick measure of spread — the range for a rough idea, the IQR for a measure resistant to extreme outliers.
Worked Example
Find the range and IQR of a dataset
Find the range and IQR of: 4, 7, 9, 12, 15, 18, 21, 25, 30.
Why Does This Work?
The range only looks at the two most extreme values, so a single unusually high or low value dominates it entirely — the IQR instead brackets the "typical" middle half of the data, so a handful of extreme outliers barely move Q₁ or Q₃ at all.
Real-Life Example
Comparing household income spread
A city's household incomes include a small number of extremely high earners alongside a large middle-income group.
The range would be dominated by the highest earner, but the IQR gives a much more honest picture of the spread among typical households.
Practice
Find the range of: 12, 5, 19, 8, 23, 3.
MediumCommon mistake
Computing the IQR using the overall median as one of the boundaries instead of properly splitting the data into lower and upper halves first — always find Q₁ and Q₃ from each half separately.
Quick Review
- Range = max − min. Simple, but sensitive to outliers.
- IQR = Q₃ − Q₁. More robust, describing the middle 50% of the data.
- Q₁ = median of the lower half; Q₃ = median of the upper half.