Box Plot Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Box Plot.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A box plot displays the five-number summary (minimum, Q1, median, Q3, maximum) of a data set using a box and whiskers.

A summary of spread and center in one picture. Box shows the middle 50%50\%.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A box plot pictures min, Q1, median, Q3, and max as a box with whiskers, showing center and spread at a glance.

Common stuck point: The procedure for box plot is the easy part; the trap is reading box width as a count of data. Asking "Am I summarizing or comparing distributions using min, Q1, median, Q3, and max?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I summarizing or comparing distributions using min, Q1, median, Q3, and max?

Worked Examples

Example 1

medium
For the data set {3,7,8,10,12,14,18,20,25,100}\{3, 7, 8, 10, 12, 14, 18, 20, 25, 100\}, construct a box plot and identify any outliers using the 1.5×IQR1.5 \times IQR rule.

Answer

100 is an outlier (exceeds upper fence of 38). Whiskers: [3, 25]. Box: [8, 20]. Median: 13.

First step

1
Order the data (already ordered). Find quartiles: Q1=8Q_1 = 8 (median of lower half {3,7,8,10,12}\{3,7,8,10,12\}... median is 8), Q2=13Q_2 = 13 (average of 12 and 14), Q3=20Q_3 = 20 (median of upper half {14,18,20,25,100}\{14,18,20,25,100\}... median is 20)

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Example 2

medium
Two box plots compare exam scores for two classes. Class A: median=72, Q1=65Q_1=65, Q3=80Q_3=80. Class B: median=78, Q1=70Q_1=70, Q3=85Q_3=85. Compare center and spread for both classes.

Example 3

medium
Data {2,4,5,7,8,10,12,15}\{2,4,5,7,8,10,12,15\}. Find the five-number summary.

Example 4

hard
Data {6,7,8,9,10,11,12,13,14,15}\{6,7,8,9,10,11,12,13,14,15\}. Find the five-number summary.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A box plot shows: minimum=10, Q1=25Q_1=25, median=40, Q3=55Q_3=55, maximum=70. Calculate the IQR and determine the fence values for outlier detection.

Example 2

hard
A data set has Q1=20Q_1 = 20, Q3=35Q_3 = 35, and a suspected outlier at value 60. Determine whether 60 is truly an outlier using the 1.5×IQR1.5 \times IQR rule, and explain how removing it would affect the box plot.

Example 3

easy
A box plot has five-number summary min=2, Q1=5, median=8, Q3=11, max=15. What is the median?

Example 4

easy
For min=2, Q1=5, median=8, Q3=11, max=15, find the IQR.

Example 5

easy
With min=2 and max=15, what is the range shown by a box plot?

Example 6

easy
What percent of data lies inside the box of a box plot?

Example 7

easy
In a box plot, the median line sits much closer to Q1 than Q3. What does this suggest?

Example 8

easy
What percent of data lies above the median in a box plot?

Example 9

easy
What percent of data lies between Q1 and the median?

Example 10

easy
A box plot's whiskers can extend at most to what, in the standard 1.5×IQR rule?

Example 11

medium
Data: 3,5,7,9,11,13,15. Find the five-number summary.

Example 12

medium
With Q1=10, Q3=22, what is the upper fence (Q3+1.5×IQRQ3+1.5\times\text{IQR})?

Example 13

medium
Q1=10, Q3=22. A data value of 4 — is it an outlier (lower fence rule)?

Example 14

medium
Two box plots: A has IQR 4, B has IQR 12, same medians. Which data is more spread?

Example 15

medium
Data: 4,8,8,12,16,20,24,28. Find Q1 and Q3 (8 values).

Example 16

medium
A box plot shows median 50, Q1 40, Q3 80. Is it skewed, and which way?

Example 17

medium
Min 5, Q1 12, median 15, Q3 18, max 40. Which side likely has an outlier?

Example 18

medium
If two data sets have identical box plots, must their histograms match?

Example 19

medium
Min 0, Q1 10, median 20, Q3 30, max 100. What is the IQR?

Example 20

challenge
Data: 2,4,6,8,10,12,14,16,18,20. Find the IQR.

Example 21

challenge
A symmetric box plot has IQR 8 and median 20. Estimate Q1, Q3 and the 1.5×IQR fences.

Example 22

challenge
Box plot A (median 30, IQR 4) vs B (median 30, IQR 20). Both symmetric. Compare standard deviations qualitatively.

Example 23

easy
A box plot has min 44, Q1 99, median 1414, Q3 1919, max 2424. What is the range?

Example 24

easy
In a box plot, between which two summary values do the middle 50%50\% of data lie?

Example 25

easy
Data {4,7,9,12,15}\{4,7,9,12,15\}. What is the median?

Example 26

easy
In a box plot, the median line is exactly in the middle of the box. What does this suggest about the middle 50%50\%?

Example 27

easy
For a box plot, does the box width equal the range of the data?

Example 28

easy
What percent of the data lies above Q3 in a box plot?

Example 29

medium
Data 1,3,4,6,8,9,121,3,4,6,8,9,12. Find Q1, median, and Q3.

Example 30

medium
A box plot has Q1 =15=15, Q3 =25=25. Compute the lower and upper fences (1.5×1.5\timesIQR).

Example 31

medium
With Q1 =8=8 and Q3 =20=20, is a value of 4242 an outlier?

Example 32

medium
Two box plots: A has IQR 55, B has IQR 55. A has range 3030, B has range 1212. Which has heavier tails?

Example 33

medium
In a box plot the median is much closer to Q3 than to Q1. Which way is the data skewed?

Example 34

medium
A box plot displays Q1 =30=30, median =42=42, Q3 =50=50, with a far-low whisker tip at 55. Where does skew likely lie?

Example 35

medium
Q1 =12=12, Q3 =22=22. Find the lower fence.

Example 36

medium
Two box plots overlap completely (identical five-number summaries). Must the data sets have the same standard deviation?

Example 37

hard
Data: 5,7,8,10,11,12,14,18,305,7,8,10,11,12,14,18,30. Determine whether 3030 is an outlier.

Example 38

hard
A box plot has min 33, Q1 1010, median 1414, Q3 2020, max 4848. What does the long upper whisker plus a far max suggest?

Example 39

hard
Comparing two box plots with the same median: A's box is much wider than B's. What does it say about consistency?

Example 40

hard
Box plot A has Q1 =20=20, Q3 =30=30. Plot B has Q1 =10=10, Q3 =40=40. Both medians equal 2525. Which is more variable?

Example 41

challenge
A box plot is drawn from n=200n=200 values. Approximately how many values lie between Q1 and Q3?

Example 42

challenge
Box plot summary: min 00, Q1 1010, median 2525, Q3 3535, max 100100. Estimate the location of any outliers using the 1.5×1.5\timesIQR rule.

Example 43

challenge
Two products have test-score box plots: A symmetric, Q1 7070, Q3 9090; B left-skewed with median near Q3, Q1 5050, Q3 8585. Which has higher typical scores and lower spread?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

medianquartiles