Split the ordered data in half, then in half again: Order the data.
The median splits it in half.
The lower quartile Q₁ is the median of the lower half, and the upper quartile Q₃ is the median of the upper half.
Q₁, the median and Q₃ split the data into four quarters; the box spans Q₁–Q₃ (the IQR), whiskers to the min and max.
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IB-style question — quartiles of a list
Find the median, Q₁ and Q₃ of 12, 15, 15, 18, 20, 22, 27.
Step by step
- n = 7; median is the 4th value.
- Lower half 12, 15, 15 → Q₁; upper half 20, 22, 27 → Q₃.
Final answer
Median = 18, Q₁ = 15, Q₃ = 22.
Odd n: don't include the median in the halves: With an odd number of values, leave the median out of both halves before finding the quartiles.
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Two measures of spread: The range = max − min uses the extremes.
The interquartile range IQR = Q₃ − Q₁ measures the spread of the middle 50% and ignores outliers.
IB-style question — range and IQR
For the data 12, 15, 15, 18, 20, 22, 27 (Q₁ = 15, Q₃ = 22), find the range and the IQR.
Step by step
- Range = max − min.
- IQR = Q₃ − Q₁.
Final answer
Range = 15; IQR = 7.
IQR ignores outliers: Because the IQR uses only the quartiles, one extreme value barely changes it — unlike the range.
IB-style question — is it an outlier?
A data set has lower quartile Q1 = 12 and upper quartile Q3 = 20.
Show that the value 35 is an outlier.
Step by step
- Find the IQR, then the upper fence Q3 + 1.5 × IQR.
- Compare 35 with the fence.
Final answer
35 > 32 (the upper fence), so it is an outlier.
Q1 = 12, Q3 = 20 → IQR = 8. Anything beyond Q3 + 1.5·IQR = 32 is an outlier, so 35 qualifies.
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Even n: average the two middle values: With an even number of values, the median is the mean of the two middle values, and each quartile is the median of its half.
On Paper 2, entering the list and running 1-Var Stats gives Q₁, the median and Q₃ directly.
IB-style question — even-sized list
Find the median, Q₁, Q₃ and IQR of 5, 8, 8, 10, 12, 15.
Step by step
- n = 6; median = mean of the 3rd and 4th values.
- Lower half 5, 8, 8 → Q₁ = 8; upper half 10, 12, 15 → Q₃ = 12.
Final answer
Median = 9, Q₁ = 8, Q₃ = 12, IQR = 4.
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