9.4Box Plots & Five-Number Summary
Construct and interpret box plots using the five-number summary. Analyze quartiles, IQR, outliers, and compare data distributions.
Why This Matters
Box plots give you a compact visual summary of an entire data set — including outliers. They're used in medical research, business analytics, and AP Statistics to compare groups and identify unusual values quickly.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do box plots and the five-number summary help us understand and compare data distributions?
Lesson Overview
A box plot (also called a box-and-whisker plot) summarizes an entire data set using just five numbers: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum — the five-number summary. The box spans from Q1 to Q3, capturing the middle 50% of the data. The whiskers extend to the minimum and maximum (or to the fences when outliers are present). The interquartile range (IQR = Q3 − Q1) measures the spread of the middle half and is resistant to outliers. Box plots are especially useful for comparing two or more data sets side by side and for quickly identifying shape, center, and spread.
Five-Number Summary
Min, Q1, Median, Q3, Max
IQR
IQR = Q3 − Q1
Lower Fence
Q1 − 1.5 × IQR
Upper Fence
Q3 + 1.5 × IQR
How to Identify Outliers
- Find Q1 and Q3 from the ordered data.
- Calculate IQR = Q3 − Q1.
- Lower Fence = Q1 − 1.5(IQR). Upper Fence = Q3 + 1.5(IQR).
- Any value below the Lower Fence or above the Upper Fence is an outlier.
- In a modified box plot, outliers are plotted as separate points; whiskers extend only to the last non-outlier value.
Finding Quartiles — Two Methods
Method 1 — Exclude median (Algebra 1 / SAT)
Split the ordered data into lower and upper halves, excluding the median when n is odd. Q1 = median of lower half. Q3 = median of upper half.
Method 2 — Include median (some calculators)
Split the ordered data into lower and upper halves, including the median in both halves when n is odd. Either method is acceptable — be consistent.
Anatomy of a Box Plot
Five-Number Summary — Example
Data (ordered): 12, 15, 18, 20, 22, 25, 28, 30, 35
| Statistic | Value | How Found |
|---|---|---|
| Minimum | 12 | Smallest value |
| Q1 | 16.5 | Lower half {12,15,18,20} → (15+18)/2 = 16.5 |
| Median | 22 | 9 values → 5th value = 22 |
| Q3 | 29 | Upper half {25,28,30,35} → (28+30)/2 = 29 |
| Maximum | 35 | Largest value |
Box plot:
Modified Box Plot — With Outlier
Data: 5, 8, 10, 12, 14, 15, 16, 18, 20, 45 → Q1=10, Q3=18, IQR=8. Upper fence = 18+12 = 30. Value 45 > 30 → outlier.
Right whisker stops at 20 (last non-outlier). Outlier 45 plotted as a separate circle.
Comparing Two Box Plots
Class A
Class B
Class B: higher median (80 vs. 65). Class A: greater spread (IQR = 20 vs. 18). Class B performed better overall; Class A has more variability.
Box Plot Shapes
Symmetric
Median centered. Whiskers equal.
Right-Skewed
Median near Q1. Long right whisker.
Left-Skewed
Median near Q3. Long left whisker.
Worked Examples
Find the five-number summary for: 8, 3, 15, 7, 12, 20, 5, 10, 18, 6
Order the data: 3, 5, 6, 7, 8, 10, 12, 15, 18, 20.
Minimum = 3. Maximum = 20.
Median: 10 values → average of 5th and 6th = (8 + 10) / 2 = 9.
Q1: lower half = {3, 5, 6, 7, 8} → median = 6.
Q3: upper half = {10, 12, 15, 18, 20} → median = 15.
Construct a box plot for: Min = 10, Q1 = 25, Median = 40, Q3 = 55, Max = 80.
Draw a number line from 5 to 85.
Mark the five values: 10, 25, 40, 55, 80.
Draw a box from Q1 (25) to Q3 (55).
Draw a vertical median line at 40 inside the box.
Draw left whisker from 25 to 10; right whisker from 55 to 80.
Add caps at 10 and 80. Label the graph.
Find Q1, Q3, and IQR for: 4, 7, 9, 11, 14, 17, 20, 23, 26
n = 9 (odd). Median = 5th value = 14.
Lower half (exclude median): {4, 7, 9, 11}. Q1 = (7 + 9) / 2 = 8.
Upper half (exclude median): {17, 20, 23, 26}. Q3 = (20 + 23) / 2 = 21.5.
IQR = Q3 − Q1 = 21.5 − 8 = 13.5.
Identify any outliers in: 5, 8, 10, 12, 14, 15, 16, 18, 20, 45
n = 10. Median = (14 + 15) / 2 = 14.5.
Lower half: {5, 8, 10, 12, 14} → Q1 = 10.
Upper half: {15, 16, 18, 20, 45} → Q3 = 18.
IQR = 18 − 10 = 8.
Lower Fence = 10 − 12 = −2. Upper Fence = 18 + 12 = 30.
Values 5–20 are all within [−2, 30]. Value 45 > 30 → outlier.
Compare using box plots: Set A — Min=20, Q1=35, Median=50, Q3=65, Max=80. Set B — Min=30, Q1=55, Median=70, Q3=78, Max=90.
Center: Set A median = 50; Set B median = 70. Set B has a higher center.
Spread: Set A IQR = 65 − 35 = 30; Set B IQR = 78 − 55 = 23. Set A has greater spread.
Range: both = 60. Equal overall range.
Shape: Set A — median centered → roughly symmetric. Set B — median near Q3 → left-skewed.
Guided Practice
Guided Practice Video: Box Plots and Five-Number Summary
Review how to find the five-number summary and construct box plots — including identifying outliers and comparing distributions — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the five-number summary for: 14, 22, 8, 31, 17, 25, 11, 19, 28, 5
Hint: Order the data first. Then find Min, Max, Median (average of 5th and 6th), Q1 (median of lower half), Q3 (median of upper half).
Calculate the IQR for: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
Hint: n = 11. Median = 6th value. Q1 = median of lower 5 values. Q3 = median of upper 5 values. IQR = Q3 − Q1.
Determine whether 52 is an outlier in: 10, 14, 16, 18, 20, 22, 24, 26, 52
Hint: Find Q1, Q3, and IQR. Upper fence = Q3 + 1.5(IQR). Is 52 above the upper fence?
A box plot has Min=5, Q1=20, Median=35, Q3=50, Max=65. Find the IQR, range, and describe the shape.
Hint: IQR = Q3 − Q1. Range = Max − Min. Where is the median relative to Q1 and Q3? Compare whisker lengths.
Two box plots: Group X has Median=45, IQR=20. Group Y has Median=60, IQR=10. Which group has a higher typical value? Which has more variability?
Hint: Median = typical value. IQR = spread of middle 50%. Compare both.
Key Vocabulary
Box Plot
Displays the five-number summary using a box (Q1 to Q3) and whiskers to Min/Max. Also called a box-and-whisker plot.
Five-Number Summary
Min, Q1, Median, Q3, Max — the five values that summarize a data set.
First Quartile (Q1)
Median of the lower half of the data. 25% of values fall at or below Q1. Left edge of the box.
Third Quartile (Q3)
Median of the upper half of the data. 75% of values fall at or below Q3. Right edge of the box.
Interquartile Range (IQR)
IQR = Q3 − Q1. Spread of the middle 50% of data. Resistant to outliers.
Outlier
A value more than 1.5 × IQR below Q1 or above Q3. Plotted as a separate point in a modified box plot.
Modified Box Plot
Whiskers extend only to the last non-outlier value. Outliers are plotted as individual dots.
Symmetric Box Plot
Median centered in the box; both whiskers roughly equal length. Mean ≈ Median.
Right-Skewed Box Plot
Median near Q1; long right whisker. Mean > Median.
Left-Skewed Box Plot
Median near Q3; long left whisker. Mean < Median.
Interactive Practice — 5 Questions
The five-number summary is: Min=10, Q1=25, Median=40, Q3=55, Max=70. What is the IQR?
Q1=20, Q3=40, IQR=20. What is the upper fence?
A box plot has the median line closer to Q3 and a long left whisker. This distribution is:
The box in a box plot spans from Q1=30 to Q3=50. Approximately what percentage of the data is inside the box?
Which measure of spread is MOST resistant to outliers?
Independent Practice
Independent Practice
Find the five-number summary for: 22, 35, 14, 41, 28, 19, 33, 47, 26, 38. Show all steps.
Calculate the IQR for: 8, 12, 16, 20, 24, 28, 32, 36. Then find the lower and upper fences.
Determine whether 75 is an outlier in: 20, 25, 30, 35, 40, 45, 50, 75. Show all fence calculations.
A box plot has Min=5, Q1=10, Median=12, Q3=20, Max=60. Describe the shape. Is the mean greater than or less than the median? Explain.
Two box plots: Set A — Min=10, Q1=20, Median=30, Q3=40, Max=50. Set B — Min=15, Q1=30, Median=45, Q3=55, Max=65. Compare the center and spread of both sets using median and IQR.
Common Mistakes
Forgetting to order the data before finding the five-number summary.
Always sort data from least to greatest first. Quartiles and the median are meaningless on unsorted data.
Confusing Q1 and Q3 — labeling the left edge of the box as Q3.
Q1 is the lower quartile (left edge). Q3 is the upper quartile (right edge). The box spans Q1 to Q3.
Extending whiskers to outliers in a modified box plot.
In a modified box plot, whiskers extend only to the last non-outlier value. Outliers are plotted as separate points.
Confusing range with IQR — using them interchangeably.
Range = Max − Min (affected by outliers). IQR = Q3 − Q1 (resistant to outliers). They measure different things.
Math Tips
Always order the data first before finding any part of the five-number summary.
The box contains the middle 50% of the data — from Q1 to Q3. Width of box = IQR.
IQR is resistant to outliers — unlike range, it is not affected by extreme values.
Symmetric box plot: median centered in box, whiskers roughly equal. Right-skewed: median near Q1, long right whisker. Left-skewed: median near Q3, long left whisker.
Each section of a box plot (left whisker, left box half, right box half, right whisker) contains approximately 25% of the data.
For skewed distributions, use median as the measure of center — the mean is pulled toward the tail.