The five-number summary is the minimum, lower quartile, median, upper quartile and maximum. Each of the four sections holds about a quarter of the data. The IQR is Q3 − Q1. A value more than 1.5 IQRs beyond a quartile is an outlier.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding the quartiles
Order the data. The median splits it; Q1 is the median of the lower half, Q3 of the upper half. 3, 5, 7, 8, 10, 12, 15, 19: median 9, Q1 = 6, Q3 = 13.5, IQR = 7.5.
Width means density
A wide section means the values there are spread thin. A narrow one means they are packed tightly. The counts are equal either way: a quarter of the data, not a quarter of the width.
The interquartile range
Q3 − Q1 measures the spread of the middle half and ignores the extremes entirely. The range, max − min, is thrown by one outlier; the IQR is not. That is why the outlier rule is built on it.
The lower fence
Q1 − 1.5 × IQR. With Q1 = 6 and IQR = 7.5: 6 − 11.25 = −5.25. Nothing in the list is below that, so no low outlier. Check both fences every time.
Comparing two box plots
Two classes with the same median but one box twice as wide: the same typical score, but one class far more varied. Compare medians for center, boxes for spread, whiskers for range.
What a box plot hides
Two very different distributions can share a five-number summary. A box plot cannot show two peaks. Never report a box plot alone for an unfamiliar data set; a dot plot beside it can.
Step 2: Try It Yourself
Tap and try it out.
Data: 12, 7, 15, 9, 22, 11, 14, 8, 10, 41. Find the five-number summary and test for outliers
- 17, 8, 9, 10, 11, 12, 14, 15, 22, 41order the ten values
Step 0 of 5
- Minimum3
- Lower quartile5
- Median8
- Upper quartile13
- Maximum19
- Interquartile range8
Each of the four sections holds a quarter of the values, however wide it looks. A narrow box means the middle half of the data is packed close together.
Step 3: In Real Life
Salaries at a company
Minimum, quartiles, median, maximum: five numbers describe every salary. A box plot draws them, and the IQR rule flags the CEO as an outlier rather than letting one number distort the picture.
Step 4: Watch an Example
One step at a time.
Watch Sana Test for an Outlier
A data set has lower quartile 5, upper quartile 13, and one value at 30.
- Step 1
The IQR is 13 − 5 = 8. She multiplies by 1.5, giving a fence distance of 12.
Step 5: Your Turn
Practice makes it stick.
The Spread
Problem 1 of 2
Lower quartile 5, upper quartile 13. What is the IQR?
The Fence
Problem 2 of 2
IQR 8 and upper quartile 13. What is the upper outlier fence?
Five Numbers
1 of 8
Q1 = 10, Q3 = 22. What is the IQR?
2 of 8
Data 3, 5, 7, 8, 9, 13, 19. What is the median?
3 of 8
Same data. What is Q1?
4 of 8
Same data. What is Q3?
5 of 8
Q1 = 4, IQR = 6. What is the lower outlier fence?
6 of 8
What fraction of the data lies in the box, as a decimal?
7 of 8
Match each part of the summary with what it reports.
Tap a card on the left to start.
8 of 8
A box section is very wide. Does it hold more points or the same?
Step 6: Quick Check
Show what you know.
Question 1 of 2
Q1 = 12, Q3 = 20. What is the IQR?
Question 2 of 2
What does a wide section of a box plot mean?
What You Learned
- The five-number summary is minimum, Q1, median, Q3 and maximum.
- The IQR measures the middle half and ignores extremes.
- A value beyond 1.5 IQRs past a quartile counts as an outlier; check both fences.
- A box plot cannot show two peaks, so pair it with a dot plot for unfamiliar data.