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Interpreting
Box Plots

Learn to read box plots like a data detective: identify quartiles, calculate the IQR, and connect what you see to real-world meaning.

Pre-Algebra

What Is a Box Plot?

A box plot (also called a box-and-whisker plot) is a visual summary of a data set. It shows how data is spread out by splitting it into four equal groups, each containing about 25% of the values.

Anatomy of a Box Plot

10 20 30 40 50 \(Q_1\) = 20 Median = 30 \(Q_3\) = 40 Min = 10 Max = 50 IQR = 40 − 20 = 20

Why Do Box Plots Matter?

Quick Summary
Summarize hundreds of data points in one compact graphic
See the Spread
Instantly see how spread out or clustered data values are
Compare Groups
Place two box plots side by side to compare different groups

Box plots are used everywhere: comparing test scores between classes, temperatures across months, prices across stores, and much more.

The Five-Number Summary

1
Minimum: the smallest value in the data set
2
\(Q_1\) (First Quartile): the median of the lower half: 25% of data falls below this
3
Median (\(Q_2\)): the middle value: 50% of data falls below this
4
\(Q_3\) (Third Quartile): the median of the upper half: 75% of data falls below this
5
Maximum: the largest value in the data set

Quartiles Divide Data into Fourths

25% 25% 25% 25% Min \(Q_1\) Median \(Q_3\) Max IQR = middle 50% of data

The Interquartile Range (IQR)

Small IQR
Data is clustered together: values are consistent
Large IQR
Data is spread out: values vary a lot

The IQR is more useful than the total range because it ignores extreme values (outliers) and focuses on where most of the data lives.

Median vs. Mean: Key Distinction

Median Mean
What it measures Middle value of ordered data Average of all values
Shown on box plot? ✓ Yes (center line) ✕ No
Affected by outliers? Resistant Pulled by extremes
Best for Skewed data Symmetric data

How to Read a Box Plot

1
Identify the five-number summary: read Min, \(Q_1\), Median, \(Q_3\), and Max from the plot
2
Calculate the IQR: compute \(Q_3 - Q_1\) to find the spread of the middle 50%
3
Look at whisker lengths: longer whiskers mean data extends further from the box
4
Check for symmetry: is the median centered in the box, or shifted left/right?
5
Interpret in context: connect numbers to what the data actually represents

Example 1: Test Scores

A class of students took a math quiz (out of 100). Here is the box plot of their scores:

50 60 70 80 90 100 55 65 78 90 98

Example 1: Interpretation

1
The median score was 78, so about half the class scored above 78 and half below
2
The IQR is 25 points, meaning the middle 50% of scores were spread across a 25-point range
3
The right whisker is short (90 to 98), so the top scorers were clustered near each other
4
The left whisker is longer (55 to 65), so the lower scores were more spread out

Example 2: Comparing Two Classes

40 50 60 70 80 90 100 A 45 60 75 85 95 B 60 72 80 88 98

Comparing: What the Data Tells Us

Class A
Median: 75, IQR: 25. Wider spread: some students struggled while others excelled. More variability in performance.
Class B
Median: 80, IQR: 16. Higher center, tighter spread: most students performed consistently well.

Example 3: Skewed Data

A restaurant tracked wait times (minutes) for customers one evening:

0 10 20 30 40 50 2 5 8 18 45

Common Mistakes to Avoid

✕ Confusing IQR with Range
The range is Max − Min (entire spread). The IQR is \(Q_3 - Q_1\) (middle 50% only).
✕ Reading Mean from Box Plot
The line inside the box is the median, not the mean. Box plots do not show the mean.
✕ Wider Box = More Data
A wider box does not mean more data points. It means the middle 50% is more spread out.

Summary of Key Concepts

Concept What It Tells You How to Find It
Minimum Smallest value Left whisker endpoint
\(Q_1\) 25th percentile Left edge of box
Median 50th percentile Line inside box
\(Q_3\) 75th percentile Right edge of box
Maximum Largest value Right whisker endpoint
IQR Spread of middle 50% \(Q_3 - Q_1\)

Question 1 of 5

A box plot shows \(Q_1 = 12\) and \(Q_3 = 28\). What is the IQR?

Question 2 of 5

The line inside the box on a box plot represents the:

Question 3 of 5

A box plot for daily temperatures shows: Min = 58, \(Q_1 = 64\), Median = 72, \(Q_3 = 78\), Max = 95. About what percentage of days had temperatures between 64 and 78?

Question 4 of 5

Two box plots are compared. Class A has IQR = 30 and Class B has IQR = 12. Which statement is correct?

Question 5 of 5

A box plot has a very long right whisker and a short left whisker. The box is shifted toward the left. This data is most likely: