Key takeaway: The interquartile range (IQR) is the third quartile minus the first quartile, giving the spread of the middle half of a data set and ignoring extreme values.
Grade 66.SP.B.5cStatistics and Data
Definition of Interquartile Range
The interquartile range, or IQR, measures how spread out the middle 50% of the data is. It is the length of the box in a box plot.
To compute it, order the data, find Q1 and Q3, and subtract. Because it looks only at the center of the data, a single unusually high or low value does not affect it. That makes it a more stable measure of spread than the range when outliers are present. A small IQR means the middle values are tightly packed; a large one means they are scattered. It is paired with the median as a summary.
Formula and notation
With the first and third quartiles:
\[\text{IQR} = Q_3 – Q_1\]
For example, \(46 – 28.5 = 17.5\). The IQR uses the same units as the data.
Key facts
- The IQR covers the central half of the values, so extremes cannot change it.
- The interquartile range is always less than or equal to the range.
- Its natural companion for center is the median.
- An IQR of zero means at least half of the values are identical.
Interquartile Range example
A food truck owner recorded daily sales (in hundreds of dollars) for 12 days, sorted: 22, 25, 27, 30, 31, 34, 38, 40, 45, 47, 52, 58. Find the IQR.
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The median is between the 6th and 7th values: \(\frac{34 + 38}{2} = 36\).
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Lower half: 22, 25, 27, 30, 31, 34; its middle is \(\frac{27 + 30}{2} = 28.5\), so \(Q_1 = 28.5\).
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Upper half: 38, 40, 45, 47, 52, 58; its middle is \(\frac{45 + 47}{2} = 46\), so \(Q_3 = 46\).
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Subtract: \(46 – 28.5 = 17.5\).
The IQR is 17.5, or \$1,750 in daily sales.
Common mistake
Watch out
Pablo reports an IQR of \(58 – 22 = 36\) for the same sales data.
Why it is wrong
That subtracts the maximum and minimum, which gives the range. The IQR subtracts the quartiles, \(Q_3 – Q_1 = 17.5\), so it describes only the middle half of the data.
Related terms and lessons
- Grade 6 · Mean, Median, Range, IQR and Mean Absolute DeviationLessonWorksheetQuiz
- Algebra 1 · Statistics: Center, Spread and Data DisplaysLessonWorksheetQuiz
Browse the glossary: A–Z list · Grade 6 vocabulary · Statistics and Data