Five-Number Summary
The five-number summary describes a dataset with five values and no arithmetic beyond sorting, which is why it is the first summary taught before box plots appear.
The five numbers
- Minimum the smallest value.
- Q1 the lower quartile: the median of the lower half.
- Q2 the median of the whole dataset.
- Q3 the upper quartile: the median of the upper half.
- Maximum the largest value.
Worked example
For 12, 15, 15, 18, 20 the summary is Min 12, Q1 15, Median 15, Q3 18, Max 20, and the interquartile range is 18 − 15 = 3.
Worked example with negative numbers
For −10, −5, 0, 5, 10 the summary is Min −10, Q1 −5, Median 0, Q3 5, Max 10, and the interquartile range is 5 − (−5) = 10. Negative quartiles are written with their signs, and subtracting a negative quartile widens the range rather than narrowing it.
The interquartile range
The interquartile range covers the middle half of the data. It ignores the smallest quarter and the largest quarter, which is why it survives outliers that would distort the range, and why it is the basis for the 1.5 × IQR rule.
Spotting outliers
The fences sit at Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. For 1, 2, 3, 4, 100 the quartiles are 2 and 4, so the IQR is 2 and the upper fence is 4 + 3 = 7. The value 100 sits far above it, so it is flagged as a possible outlier and is worth checking before any mean is reported.
From five numbers to a box plot
A box plot draws the minimum and maximum as whiskers, the quartiles as the ends of the box, and the median as a line inside it. A long whisker on one side shows skew, and the position of the median inside the box shows which half the data crowds into. Every one of those values is on the calculator result, so the plot can be drawn straight from it.