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\%.

Read the full concept explanation →

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: The box spans the interquartile range (IQR = Q3 - Q1), the line inside is the median, and the whiskers extend to the data's extremes (or to a fence for outliers).

Common stuck point: Whiskers don't always go to min/max—they may stop at 1.5 \times \text{IQR} from the box.

Sense of Study hint: Find the five-number summary first: min, Q1, median, Q3, max. Draw the box from Q1 to Q3, mark the median inside, then add whiskers.

Worked Examples

Example 1

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

Solution

  1. 1
    Order the data (already ordered). Find quartiles: Q_1 = 8 (median of lower half \{3,7,8,10,12\}... median is 8), Q_2 = 13 (average of 12 and 14), Q_3 = 20 (median of upper half \{14,18,20,25,100\}... median is 20)
  2. 2
    Calculate IQR: IQR = Q_3 - Q_1 = 20 - 8 = 12
  3. 3
    Find fences: Lower fence = Q_1 - 1.5 \times IQR = 8 - 18 = -10; Upper fence = Q_3 + 1.5 \times IQR = 20 + 18 = 38
  4. 4
    Identify outliers: any value below -10 or above 38. The value 100 exceeds 38, so it is an outlier
  5. 5
    Draw box plot: whiskers extend to 3 (min non-outlier) and 25 (max non-outlier); box from Q_1=8 to Q_3=20; line at median Q_2=13; plot 100 as a separate dot

Answer

100 is an outlier (exceeds upper fence of 38). Whiskers: [3, 25]. Box: [8, 20]. Median: 13.
The 1.5×IQR rule identifies potential outliers by establishing fences beyond which data is unusual. Box plots provide a five-number summary (min, Q1, median, Q3, max) while flagging extreme values as separate dots.

Example 2

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

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, Q_1=25, median=40, Q_3=55, maximum=70. Calculate the IQR and determine the fence values for outlier detection.

Example 2

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

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

medianquartiles