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R73 • Lesson 103 of 105

Box Plots & Interquartile Range

Five-number summary, box-and-whisker plots, and measuring spread with IQR

Reserve & Extensions • K-12

Prerequisites: H47

Key Concepts

  • box plots
  • IQR
  • quartiles
  • five-number summary
  • outliers

Box Plots and Interquartile Range

A box plot (also called a box-and-whisker plot) gives you a visual snapshot of how data is spread out. In one compact diagram, you can see the center, spread, skewness, and outliers of a data set -- making it one of the most useful tools in statistics.

The Five-Number Summary

Every box plot is built from five key values:

StatisticWhat It Is
MinimumSmallest value (excluding outliers)
Q1 (First Quartile)Median of the lower half (25th percentile)
Median (Q2)Middle value (50th percentile)
Q3 (Third Quartile)Median of the upper half (75th percentile)
MaximumLargest value (excluding outliers)

Worked Example 1: Finding the Five-Number Summary

Data (already sorted): 2, 4, 5, 7, 8, 10, 11, 13, 15, 18, 20

There are 11 values.

  1. Minimum: 2
  2. Median (Q2): The 6th value = 10
  3. Q1: Median of the lower half (2, 4, 5, 7, 8) = 5
  4. Q3: Median of the upper half (11, 13, 15, 18, 20) = 15
  5. Maximum: 20

Five-number summary: 2, 5, 10, 15, 20

Interquartile Range (IQR)

IQR = Q3 − Q1

The IQR measures the spread of the middle 50% of the data. It is not affected by extreme values, making it a more robust measure of spread than the range.

From Example 1: IQR = 15 − 5 = 10.

Identifying Outliers: The 1.5 × IQR Rule

A data point is an outlier if it falls:

Below Q1 − 1.5 × IQR    or    Above Q3 + 1.5 × IQR

Worked Example 2: Detecting Outliers

Data: 3, 5, 7, 8, 9, 10, 11, 12, 14, 35

Q1 = 7, Q3 = 12, IQR = 5.

Lower fence: 7 − 1.5(5) = 7 − 7.5 = −0.5

Upper fence: 12 + 1.5(5) = 12 + 7.5 = 19.5

Any value below −0.5 or above 19.5 is an outlier. The value 35 is an outlier.

On the box plot, the right whisker extends to 14 (the largest non-outlier), and 35 is plotted as a separate dot.

Drawing a Box Plot

  1. Draw a number line covering the data range.
  2. Draw a box from Q1 to Q3.
  3. Draw a vertical line inside the box at the median.
  4. Extend whiskers from Q1 to the minimum non-outlier and from Q3 to the maximum non-outlier.
  5. Plot any outliers as individual dots beyond the whiskers.

Comparing Distributions

Worked Example 3: Comparing Two Classes

Class A scores: Min=55, Q1=68, Median=75, Q3=84, Max=95

Class B scores: Min=40, Q1=72, Median=80, Q3=88, Max=98

Comparing:

  • Center: Class B has a higher median (80 vs 75).
  • Spread: Class A: IQR = 16. Class B: IQR = 16. Same middle spread.
  • Range: Class A: 40. Class B: 58. Class B has more total variability.
  • Skewness: In Class A, the median is closer to Q1, suggesting a slight right skew. In Class B, the long left whisker suggests left skew.

Common Mistake

The whiskers do NOT always extend to the minimum and maximum of the data. If there are outliers, the whiskers stop at the most extreme non-outlier values. Outliers are shown as separate points. Many students draw whiskers to every data point, which defeats the purpose of outlier detection.

Reading Box Plot Shape

If the median line is closer to Q1, the data is right-skewed (tail stretches right). If closer to Q3, it is left-skewed. If roughly centered in the box, the data is approximately symmetric. Longer whiskers also indicate skewness in that direction.

Practice Problems

1. Find the five-number summary: 12, 15, 18, 22, 25, 28, 30, 35, 40.

Solution

Min = 12. Q1 = (15+18)/2 = 16.5. Median = 25. Q3 = (30+35)/2 = 32.5. Max = 40. Five-number summary: 12, 16.5, 25, 32.5, 40.

2. For the data in Problem 1, find the IQR and determine if there are any outliers.

Solution

IQR = 32.5 − 16.5 = 16. Lower fence: 16.5 − 24 = −7.5. Upper fence: 32.5 + 24 = 56.5. All values are within these fences. No outliers.

3. Data: 1, 2, 3, 4, 5, 6, 7, 8, 9, 50. Find Q1, Q3, IQR, and identify outliers.

Solution

Q1 = 3, Q3 = 8, IQR = 5. Upper fence: 8 + 7.5 = 15.5. The value 50 exceeds 15.5, so 50 is an outlier.

4. Two box plots show: Group X has IQR = 8 and Group Y has IQR = 20, but both have the same median. What does this tell you?

Solution

Both groups have the same typical value (same center), but Group Y has much more variability in the middle 50% of its data. Group Y's scores are more spread out around the median.

5. A box plot shows the median very close to Q3 and a long left whisker. Describe the shape of the distribution.

Solution

The distribution is left-skewed (negatively skewed). Most data is concentrated at the higher end, with a long tail stretching toward lower values.

Summary

Overview