Box-and-Whisker Plot — Definition, Examples & Table

Box-and-Whisker Plot

A visual display of the five number summary. The box-and-whisker plot is a simplified boxplot taught to beginners . It does not show outliers. The whiskers extending all the way to the minimum and maximum values regardless of how far out they may be.

Key Formula

\text{Five-Number Summary: } \{\,\text{Min},\; Q_1,\; M,\; Q_3,\; \text{Max}\,\}

Where:

$\text{Min}$ = The smallest value in the data set
$Q_1$ = First quartile — the median of the lower half of the data
$M$ = Median — the middle value of the entire data set
$Q_3$ = Third quartile — the median of the upper half of the data
$\text{Max}$ = The largest value in the data set
$\text{IQR} = Q_3 - Q_1$ = Interquartile range — the width of the box, measuring the spread of the middle 50% of the data

Worked Example

Problem: Draw a box-and-whisker plot for the data set: 4, 7, 10, 12, 15, 18, 20, 22, 25.

Step 1: Order the data from least to greatest. The data is already sorted.

4,\; 7,\; 10,\; 12,\; 15,\; 18,\; 20,\; 22,\; 25

Step 2: Find the minimum and maximum values.

\text{Min} = 4, \quad \text{Max} = 25

Step 3: Find the median. There are 9 values, so the median is the 5th value.

M = 15

Step 4: Find Q₁. The lower half is {4, 7, 10, 12}. The median of these four values is the average of the 2nd and 3rd values.

Q_1 = \frac{7 + 10}{2} = 8.5

Step 5: Find Q₃. The upper half is {18, 20, 22, 25}. The median of these four values is the average of the 2nd and 3rd values.

Q_3 = \frac{20 + 22}{2} = 21

Answer: The five-number summary is {4, 8.5, 15, 21, 25}. Draw a number line, then place a box from 8.5 to 21 with a vertical line at 15 for the median. Extend a whisker left from the box to 4 and a whisker right from the box to 25.

Another Example

This example focuses on reading and interpreting a box-and-whisker plot rather than constructing one from raw data, showing students how to analyze spread and skewness.

Problem: The five-number summary for test scores in a class is {52, 68, 76, 85, 98}. Use the box-and-whisker plot to describe the distribution of scores.

Step 1: Identify the five-number summary values directly.

\text{Min} = 52,\; Q_1 = 68,\; M = 76,\; Q_3 = 85,\; \text{Max} = 98

Step 2: Calculate the IQR to measure the spread of the middle 50% of the data.

\text{IQR} = Q_3 - Q_1 = 85 - 68 = 17

Step 3: Analyze the whisker lengths. The left whisker spans from 52 to 68 (length 16), and the right whisker spans from 85 to 98 (length 13). The left whisker is slightly longer, indicating the lower scores are a bit more spread out.

\text{Left whisker} = 68 - 52 = 16, \quad \text{Right whisker} = 98 - 85 = 13

Step 4: Look at the box itself. The median (76) is closer to Q₁ (68) than to Q₃ (85), so the data within the box is slightly skewed to the right.

76 - 68 = 8 \quad \text{vs.} \quad 85 - 76 = 9

Answer: The box extends from 68 to 85 with a median line at 76. Whiskers reach to 52 and 98. The overall range is 46, the IQR is 17, and the distribution is roughly symmetric with a very slight right skew inside the box.

Frequently Asked Questions

What is the difference between a box-and-whisker plot and a boxplot?

A box-and-whisker plot always extends its whiskers to the minimum and maximum of the data, regardless of how extreme those values are. A standard boxplot (sometimes called a modified boxplot) caps the whiskers at 1.5 × IQR beyond Q₁ and Q₃, and displays any values beyond that range as individual dots called outliers. The box-and-whisker plot is the simplified version commonly taught first.

How do you find the quartiles for a box-and-whisker plot?

First, sort the data and find the median, which splits the data into a lower half and an upper half. Q₁ is the median of the lower half, and Q₃ is the median of the upper half. If the overall data set has an odd number of values, exclude the median itself when forming the two halves.

What does the length of the box tell you?

The length of the box equals the interquartile range (IQR = Q₃ − Q₁). It represents the spread of the middle 50% of your data. A wider box means more variability among the central data values, while a narrow box means those values are tightly clustered.

Box-and-Whisker Plot vs. Boxplot (Modified)

	Box-and-Whisker Plot	Boxplot (Modified)
Whisker endpoints	Always extend to the minimum and maximum values	Extend at most 1.5 × IQR beyond Q₁ and Q₃
Outlier display	Does not identify or display outliers	Shows outliers as individual dots beyond the whiskers
Typical use	Introductory statistics courses	Standard practice in statistics and data science
Sensitivity to extreme values	Whiskers can be very long if extreme values exist, potentially misleading	Extreme values are separated out, giving a clearer picture of the main distribution

Why It Matters

Box-and-whisker plots appear throughout middle school and high school math courses, standardized tests (SAT, ACT), and AP Statistics. They give you a quick way to compare the center, spread, and shape of different data sets side by side — for example, comparing test scores between two classes. Understanding them is also the foundation for reading modified boxplots, which are standard in college-level statistics and real-world data analysis.

Common Mistakes

Mistake: Forgetting to sort the data before finding quartiles and the median.

Correction: Always arrange the data in ascending order first. The five-number summary is defined in terms of ordered data, so skipping this step produces incorrect quartile values.

Mistake: Including the overall median in both halves when computing Q₁ and Q₃ for an odd-sized data set.

Correction: When the data set has an odd number of values, exclude the median from both the lower and upper halves before finding Q₁ and Q₃. Including it inflates or distorts the quartile values.

Related Terms

Five Number Summary — The five values displayed by the plot
Boxplot — Modified version that shows outliers separately
Outlier — Extreme values hidden in box-and-whisker plots
First Quartile — Left edge of the box (Q₁)
Median of a Set of Numbers — Vertical line inside the box
Third Quartile — Right edge of the box (Q₃)
Stemplot — Another way to display data distribution