Use the stated median-of-halves convention consistently, then separate claims about the middle half from claims about the whole dataset.
- Calculate quartiles from ordered even- and odd-sized datasets.
- Read median, IQR and range from a five-number summary.
- Compare centre and the middle-half spread without overgeneralising.
The idea
A box plot summarises an ordered dataset using its minimum, lower quartile Q1, median, upper quartile Q3 and maximum. The box spans Q1 to Q3, so its width is the interquartile range, Q3 − Q1. In this lesson, quartiles are the medians of the lower and upper halves; for an odd-sized dataset, exclude the overall median from both halves.
Before you begin: Order data, find a median and subtract values on a consistent numerical scale.
Worked examples
01 Start with the method
Find Q1, the median, Q3 and the interquartile range of this dataset.
The data are already ordered. Average the fourth and fifth values to find the median.
The lower half is 2, 4, 6, 8 and the upper half is 10, 12, 14, 16. Find each half’s median.
Subtract the lower quartile from the upper quartile. The middle half extends from 5 to 13.
The minimum and maximum describe the full range, which is different from the IQR.
A mistake to avoid
The box does not run from the minimum to the maximum. Its ends are Q1 and Q3. Quartile conventions can differ between software packages; use the convention specified in the question.
02Find quartiles with an odd number of values
Find the median, Q1, Q3 and IQR. Exclude the overall median when splitting the data.
The fourth observation is the overall median. Exclude it from both halves under the stated convention.
Find the middle observation in each half.
Subtract the quartiles to find the IQR.
03Compare the middle half and full spread
Each summary lists minimum, Q1, median, Q3, maximum. Compare the centre, IQR and range.
The third value is the median. Both groups have the same median, 12.
Subtract Q1 from Q3 for each group. Group B’s middle half has less spread.
Now compare extremes. Group B has a larger full range, so a smaller IQR does not imply a smaller range.
Practice questions with answers
Work these out first, then open each solution to compare your method. These questions are untimed.
Question 1
A box plot has Q1 = 11 and Q3 = 19. Find its interquartile range.
Show solution 1
Subtract the lower quartile from the upper quartile. Neither the median nor the whisker endpoints are needed.
Question 2
Find the range and IQR from this five-number summary.
Show solution 2
The extremes determine range. The second and fourth entries are Q1 and Q3, which determine IQR.
Question 3
Use median-of-halves to find Q1, Q3 and IQR.
Show solution 3
The two halves are 2, 4, 6 and 8, 10, 12. Each has three values, so its median is its middle observation.
Question 4
Exclude the overall median from each half. Find the IQR.
Show solution 4
The median is 4. The halves are 1, 2 and 5, 7, whose medians are 1.5 and 6.
Question 5
A box has Q1 = 15 and Q3 = 27. State the interval it spans and its IQR.
Show solution 5
The box spans the lower to upper quartile. It represents the middle approximately 50% of observations, subject to repeated values and the quartile convention.
Common questions
Why might software give different quartiles?
There are several quartile conventions, especially for small datasets. This lesson uses medians of halves and excludes the overall median for odd-sized data. Follow the convention required in your class or question.
Does a wider box contain more observations?
Not necessarily. Each box represents roughly the middle half of its dataset. A wider box means that middle half is spread across a larger numerical interval, not that the sample is bigger.
Are whiskers always the minimum and maximum?
In a simple five-number-summary box plot, they are. In a modified box plot, outliers may be plotted separately and whiskers extend to the most extreme non-outliers. Check which kind of plot is being used.