TL;DR:
A box plot shows the minimum, first quartile, median, third quartile and maximum.
The box is the middle half of the data. Its width is the interquartile range,
IQR = Q3 − Q1.A common rule flags values more than
1.5 × IQRbeyond the box as outliers.
A teacher posts the results of a test as a single picture: a box with whiskers. At a glance you can see where the middle student sits, how spread out the class is, and whether anyone is far from the pack. Learning to build and read that picture takes five numbers.
The five numbers
Minimum and maximum: the smallest and largest values.
Median (second quartile, Q2): the middle value when sorted.
First quartile (Q1): the median of the lower half.
Third quartile (Q3): the median of the upper half.
Together they cut the sorted data into four parts with about a quarter of the values in each. The box runs from Q1 to Q3 with a line at the median, and the whiskers reach toward the extremes.
Worked example: a five-number summary of test scores
Eleven scores, sorted: 30, 58, 61, 64, 67, 70, 72, 75, 78, 83, 96.
Median: with 11 values, the middle is the 6th, which is 70.
Lower half (the 5 values below the median): 30, 58, 61, 64, 67. Its median is the 3rd, so
Q1 = 61.Upper half (the 5 values above the median): 72, 75, 78, 83, 96. Its median is the 3rd, so
Q3 = 78.Minimum 30, maximum 96.
Interquartile range:
IQR = Q3 − Q1 = 78 − 61 = 17.
The five-number summary is 30, 61, 70, 78, 96. The middle half of the class scored between 61 and 78, a spread of 17 points. The full range is 96 − 30 = 66.
The outlier rule
A common rule of thumb marks a value as an outlier if it lies more than 1.5 × IQR below Q1 or above Q3. Here 1.5 × 17 = 25.5.
Lower fence:
61 − 25.5 = 35.5. The score 30 is below it, so it is an outlier.Upper fence:
78 + 25.5 = 103.5. The top score 96 is inside it, so it is not an outlier.
On the plot, the lower whisker stops at 58, the smallest value that is not an outlier, and the 30 is drawn as a separate dot. The upper whisker reaches 96.
Reading the shape
Look at where the median sits inside the box and how long the whiskers are. Here the median 70 is closer to Q3 (8 away) than to Q1 (9 away), nearly centred, so the middle half is fairly symmetric. The lower whisker is longer, and an outlier sits below, so one unusually low score pulls the left tail out.
A box plot hides how many values sit in each part and does not show a mode or a mean. It is best for comparing groups side by side. Two classes with the same median but different box widths differ in spread even though their middles match.
Try one yourself: an even number of scores
Take ten values: 4, 6, 7, 9, 10, 12, 13, 15, 18, 20. With 10 values the median is the average of the 5th and 6th: (10 + 12)/2 = 11. The lower half is 4, 6, 7, 9, 10, whose median is 7, so Q1 = 7. The upper half is 12, 13, 15, 18, 20, whose median is 15, so Q3 = 15. Then IQR = 15 − 7 = 8.
The fences are 7 − 1.5 × 8 = 7 − 12 = −5 and 15 + 12 = 27. Every value is inside, so there are no outliers, and the whiskers run to 4 and 20. Check the counts: each half has 5 values, and 5 + 5 = 10.
Comparing two groups
The real power of a box plot comes from putting two side by side. Suppose class X has the five-number summary 40, 55, 65, 75, 90 and class Y has 50, 60, 65, 70, 80 (invented values). Both medians are 65. Class X has IQR = 75 − 55 = 20 and class Y has IQR = 70 − 60 = 10. Class X is twice as spread out in the middle half. A reader who looked only at the medians would see two identical classes.
A note on quartile methods
Textbooks and calculators use slightly different rules for quartiles, especially whether the median is included in each half when n is odd. Here we left it out. Another method might give slightly different Q1 and Q3. Use the rule your course teaches, and write down which you used.
How to check your answer
Order. Min ≤ Q1 ≤ median ≤ Q3 ≤ max.
Counts. Each half should have the same number of values.
Re-sort. A common error is finding the median of unsorted numbers.
Fences. Compute both fences, even if one side has no outliers.
Common mistakes
Forgetting to sort first.
Including the median in both halves when n is odd under a method that excludes it.
Drawing the whisker to an outlier instead of to the last ordinary value.
Reading the length of a box as the number of data points. It shows spread, not count.
Confusing the range with the IQR.
Where you will use it next
Spread is measured differently in standard deviation by hand, which uses every value rather than just five. Questions about what a pattern in data means come next in correlation is not causation.
Frequently asked questions
Does half of the data lie inside the box?
About half, yes. Exactly half when the data splits evenly, but ties and odd counts can shift it slightly.
Is the 1.5 rule a law?
No. It is a convention for flagging unusual values. An outlier by the rule is a prompt to investigate, not proof of an error.
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