Box Plots and Distribution: Reading the Shape of Data

A box plot compresses an entire data set into five landmarks: minimum, lower quartile, median, upper quartile and maximum. That compression is powerful because it lets us compare centre, spread and possible skew quickly without listing every observation.

This guide develops box plots as a way to read distributions, not merely draw rectangles. It connects quartiles to position, explains the interquartile range, and shows how two groups can be compared fairly using both centre and spread.

The five-number summary

• minimum
• lower quartile, Q1
• median, Q2
• upper quartile, Q3
• maximum

The median splits the ordered data into two halves. The lower and upper quartiles then mark central positions within those halves.

Interquartile range

IQR = Q3 − Q1

The IQR measures the spread of the middle 50% of the data. Unlike the full range, it is less sensitive to extreme minimum or maximum values.

Worked example

Suppose a box plot has minimum 12, Q1 = 18, median = 22, Q3 = 30 and maximum 45.

Range = 45 − 12 = 33
IQR = 30 − 18 = 12

The middle half of the observations lies between 18 and 30, while the full set spans 12 to 45.

What the box itself tells you

The box runs from Q1 to Q3, so it contains the middle 50% of the ordered data. The median line shows the central position. Longer sections of the box or whiskers indicate that values are more spread out across that part of the distribution.

Comparing two groups

Suppose Group A has median 65 and IQR 8, while Group B has median 70 and IQR 20.

Group B has the higher median, so its typical central performance is higher by that measure. Group A has the smaller IQR, so its middle 50% is more tightly clustered.

A strong comparison says both things. “Group B is better” throws away the spread information and may overstate what the box plots actually show.

Reading possible skew

If the median is close to Q1 and the upper half of the box is much longer, or if one whisker is substantially longer than the other, the distribution may be skewed in that direction.

A box plot gives evidence about shape but does not show every detail. Different data sets can share the same five-number summary, so avoid claiming more than the graph supports.

Quartiles are positional summaries

Quartiles divide ordered observations by position. They do not say that the numerical distances between Q1, median and Q3 must be equal. Equal proportions of observations can occupy unequal numerical intervals.

Range versus IQR

Range uses the two extremes. IQR uses the middle half. If one group contains an extreme outlier, its range may grow dramatically while its IQR changes little.

This makes IQR useful when we want a measure of typical spread that is more resistant to extremes.

Common errors

Calling the width of the box the range. The width of the box represents IQR, not full range.

Assuming each geometric segment contains the same numerical width. Quartiles divide observations by position, not the number line into equal distances.

Comparing medians but ignoring spread. Two groups can have similar centre but very different consistency.

Claiming exact frequencies from a box plot without the sample size. The plot shows proportions and positions, not necessarily raw counts.

Over-interpreting shape. A box plot is a summary; it can suggest skew but does not reveal every gap, cluster or repeated value.

A reliable comparison protocol

1. Compare medians for centre.
2. Compare IQRs for the spread of the middle half.
3. Compare ranges if the extremes matter.
4. Inspect relative whisker and box lengths for possible skew.
5. State conclusions in the context of the data rather than describing only the geometry.

Practice

1. A box plot has Q1 = 14 and Q3 = 26. Find the IQR.
2. Its minimum is 8 and maximum is 35. Find the range.
3. Group X has median 72 and IQR 6. Group Y has median 75 and IQR 18. Compare the two groups.
4. Why is IQR less sensitive to a single extreme maximum than range?

Answers

1. 12. 2. 27. 3. Y has the higher median, while X has the smaller IQR and therefore a more tightly clustered middle 50%. 4. IQR depends on Q1 and Q3 rather than directly on the maximum.

Connected routes

Use Mean, Median and Mode for centre and Range and Spread for variability. Use Reading Histograms to compare a different representation of distributions. Return to the Mathematics Learning Hub for the wider Secondary Mathematics estate.