Box Plots and Quartiles: Five-Number Summary and Outliers

A box plot summarises data with five numbers: minimum, lower quartile (Q1), median, upper quartile (Q3) and maximum. The box covers the middle half of the data, and the interquartile range IQR = Q3 − Q1 measures its spread. Values more than 1.5 × IQR beyond the box are outliers.

Finding the quartiles

1. Sort the data.
2. Find the median (Q2).
3. Q1 is the median of the lower half; Q3 is the median of the upper half. With an odd count, leave the overall median out of both halves.

Worked example

Data: $7, 15, 36, 39, 40, 41, 42, 43, 47, 49, 95$ (11 values, already sorted).

• Median: the 6th value, $41$.
• Lower half: $7, 15, 36, 39, 40$, so $Q_1 = 36$.
• Upper half: $42, 43, 47, 49, 95$, so $Q_3 = 47$.
• $\text{IQR} = 47 - 36 = 11$.

Outliers

Fences are $Q_1 - 1.5 \times \text{IQR} = 19.5$ and $Q_3 + 1.5 \times \text{IQR} = 63.5$.

So $7$, $15$ and $95$ are outliers. In the box plot, the whiskers stop at the most extreme values inside the fences (36 and 49), and outliers are drawn as dots.

Why use the IQR?

The range ($95 - 7 = 88$) is dragged around by a single extreme value. The IQR depends only on the middle half of the data, so it is much more stable.

Common mistakes

• Forgetting to sort first.
• Including the median in both halves for an odd count (this calculator's method leaves it out; some textbooks differ).
• Drawing whiskers all the way to outliers.

Practice questions

For $3, 5, 8, 9, 12$:

1. Find the median.
2. Find Q1 and Q3.
3. Find the IQR.

Answers: 1) 8 2) Q1 = 4, Q3 = 10.5 3) 6.5

The box plot calculator finds the five-number summary and draws the plot with outliers. For averages see the mean, median and mode calculator, and for spread read standard deviation explained.

Frequently asked questions

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