Box plots: how to draw, read and compare them

Data (already in order): 3, 5, 7, 8, 9, 11, 12, 14, 15, 18, 20. There are 11 values.

• Median: the $\tfrac{11 + 1}{2} = 6$th value, 11.
• Lower quartile: the median of the lower half (3, 5, 7, 8, 9), which is 7.
• Upper quartile: the median of the upper half (12, 14, 15, 18, 20), which is 15.
• Minimum 3, maximum 20.

So the IQR is $15 - 7 = 8$ and the range is $20 - 3 = 17$. Check the median with our mean, median and mode calculator.

1. Draw a scale that covers the minimum and maximum.
2. Draw a box from the lower quartile (7) to the upper quartile (15).
3. Draw a vertical line inside the box at the median (11).
4. Draw whiskers from the box out to the minimum (3) and maximum (20).

Use a ruler and keep the scale even: a box plot is a scale drawing.

For grouped data, read the median and quartiles from a cumulative frequency graph, and take the minimum and maximum from the question or the ends of the data. Box plots are often drawn directly underneath the graph using the same scale.

An outlier is a value far from the rest. A common rule: a value is an outlier if it's more than $1.5 \times \text{IQR}$ below the lower quartile or above the upper quartile.

In the example, $1.5 \times 8 = 12$, so the limits are $7 - 12 = -5$ and $15 + 12 = 27$. Every value lies between them, so there are no outliers. Outliers are sometimes plotted as crosses beyond the whiskers. Check which rule your exam board uses.

Make two comparisons, each in context:

1. Average: compare the medians. "The median score for class A (62) was higher than class B (55), so class A did better on average."
2. Spread: compare the IQRs (or ranges). "Class A's IQR (12) was smaller than class B's (20), so class A's scores were more consistent."

Don't just say "higher" or "bigger": explain what it means for the situation.

• Forgetting to put the data in order first.
• Reading the median as the middle of the box (it can be anywhere inside it).
• Comparing only the maximums, or only one measure.
• Including the median in the halves when finding quartiles from an odd-sized list (methods vary; follow your teacher's).

1. Data: 2, 4, 4, 6, 7, 9, 10. Find the median, LQ and UQ.
2. A box plot has LQ 24 and UQ 38. Find the IQR.
3. Using question 2, would a value of 62 be an outlier?

Answers: 1. Median 6, LQ 4, UQ 9. 2. 14. 3. $38 + 1.5 \times 14 = 59$, and 62 > 59, so yes.

More practice in our box plot questions.