Sort a class by exam score and split them into four equal groups. The three cut points — Q1, Q2, Q3 — tell you more about the spread than the average ever could.
Quartiles divide sorted data into four equal parts. Q1 (25th percentile) cuts off the bottom quarter, Q2 is the median, Q3 (75th percentile) cuts off the top quarter.
Box plots, the interquartile range, outlier detection, salary bands, percentile ranks — quartiles are the backbone of robust descriptive statistics.
Finding quartiles
- Sort the data.
- Q2 = median of the whole set.
- Q1 = median of the lower half.
- Q3 = median of the upper half.
- IQR = Q3 − Q1 — the spread of the middle 50%.
Data: 3, 7, 8, 5, 12, 14, 21, 13, 18. Find Q1, Q2, Q3.
Why use the IQR instead of the range for spread?
The range (max − min) is destroyed by a single outlier. The IQR ignores the extreme 25% on each end, so it's a robust measure of how spread the 'typical' data is.
Different software uses slightly different quartile rules (whether to include the median in the halves, interpolation methods). For a school dataset it rarely matters; for published stats, state your method.
Outlier rule of thumb: anything below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is flagged as a potential outlier — exactly what box-plot whiskers mark.
- Quartiles split sorted data into four equal parts (Q1, Q2=median, Q3).
- IQR = Q3 − Q1 — robust spread of the middle 50%.
- Drive box plots and the 1.5×IQR outlier rule.