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: Picture It
Start with something you already know.
Four teams from one race
- Twenty runners finish a race.
- Their times are lined up from fastest to slowest.
- The coach cuts the line into four teams of five.
- The fastest five finished seconds apart.
- The slowest five straggled in over ten minutes.
The five-number summary marks the fastest time, the three cuts between teams, and the slowest time.
A box plot draws the four teams end to end, from cut to cut. A long section is a team spread out, like the stragglers, not a bigger team.
With an odd count or tied values the teams are only about equal, so each section holds about a quarter of the data.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Eight friends timed their walk to school, in minutes: 2, 3, 5, 6, 8, 9, 11, 14. Split the list into its lower four and its upper four. What is the median of the lower four, in minutes?
minutes
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Ana Summarize Eleven Mornings
Ana noted how many minutes her train was late on eleven mornings: 12, 3, 9, 30, 7, 11, 4, 14, 6, 10, 8. She wants the five-number summary and wants to know whether any morning is an outlier.
- Step 1
In order, the eleven times read 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 30. Every number in the summary is a position in this ordered list, so the ordering comes first.
Watch Priya Check Both Fences
Priya timed ten pizza deliveries, in minutes: 18, 25, 20, 22, 31, 19, 24, 21, 4, 23. Two times look odd to her, the 4 and the 31, and she wants the rule to decide.
- Step 1
In order, the ten times are 4, 18, 19, 20, 21, 22, 23, 24, 25, 31. With an even count there is no single middle value, so the median is the mean of the 5th and 6th: (21 + 22) ÷ 2 = 21.5 minutes.
Step 6: Your Turn
Practice makes it stick.
The Sandwich Menu
Problem 1 of 17
A cafe sells seven sandwiches, priced $6, $10, $7, $25, $8, $7 and $9. By the 1.5 IQR rule, above what price does a sandwich count as a high outlier, in dollars? Give a decimal if it is not whole.
dollars
The Phone Bills
Problem 2 of 17
Eight friends compare their monthly phone bills: $48, $30, $55, $42, $35, $60, $40 and $50. What is the IQR of the bills, in dollars?
dollars
Five Numbers and Two Fences
Order first, then split. For a fence, find the IQR, take one and a half of it, and measure from the quartile.
Problem 3 of 17
Seven values, in order: 4, 6, 9, 11, 13, 16, 20. What is Q3?
Problem 4 of 17
Nine values, in order: 2, 5, 7, 8, 10, 12, 13, 16, 21. What is Q1?
Problem 5 of 17
A data set has Q1 = 15 and Q3 = 21. What is the lower fence?
Problem 6 of 17
A data set has Q1 = 10 and Q3 = 14. The values above Q3 are 15, 17, 19, 21 and 23. How many of them are outliers?
values
Problem 7 of 17
A box plot is drawn from 40 values. About how many of the values lie between Q1 and the median?
values
Whose Step?
Each page has one line that slipped. Find it, then put it right.
Problem 8 of 17
Noor wanted Q3 for the values 6, 15, 8, 3, 9, 12, 20. One line is a slip. Which?
Noor's working: Q3 of 6, 15, 8, 3, 9, 12, 20
- 1data: 6, 15, 8, 3, 9, 12, 20
- 2median: the 4th value, 3
- 3upper half: 9, 12, 20
- AnswerQ3 = 12
Problem 9 of 17
What is Q3 for the values 6, 15, 8, 3, 9, 12, 20?
Problem 10 of 17
Leo has a summary: minimum 4, Q1 14, median 18, Q3 22, maximum 33. He tested whether 33 is an outlier. One line is a slip. Which?
Leo's working: minimum 4, Q1 14, median 18, Q3 22, maximum 33
- 1IQR = 22 − 14 = 8
- 21.5 × 8 = 12
- 3upper fence = 18 + 12 = 30
- Answer33 > 30: 33 is an outlier
Problem 11 of 17
Leo’s summary again: minimum 4, Q1 14, median 18, Q3 22, maximum 33. With the fence measured from the right place, is 33 an outlier?
Problem 12 of 17
Maya has a summary: minimum 2, Q1 10, median 14, Q3 18, maximum 40. She tested whether 40 is an outlier. One line is a slip. Which?
Maya's working: minimum 2, Q1 10, median 14, Q3 18, maximum 40
- 1spread = 40 − 2 = 38
- 21.5 × 38 = 57
- 3upper fence = 18 + 57 = 75
- Answer40 < 75: no outlier
Problem 13 of 17
Maya’s summary again: minimum 2, Q1 10, median 14, Q3 18, maximum 40. With the fence built on the IQR, where is the upper fence?
Pick the Move
Which measure answers the question? No working is needed here.
Problem 14 of 17
Two classes took the same quiz. Their box plots have the same median, but one box is twice as wide as the other. Which comparison shows which class had more varied scores?
Problem 15 of 17
Ten homes on a street have recently sold, and one of them is a mansion. Which number best reports how spread out the typical prices are?
Same Summary?
Judge what each summary keeps and what it forgets.
Problem 16 of 17
Select every data set with the same five-number summary as 1, 2, 3, 4, 5, 6, 7.
Problem 17 of 17
Data set A is 2, 4, 6, 8, 10, 12, 14. Data set B is 2, 4, 6, 8, 10, 12, 50. Select every measure that is the same for both.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A data set has Q1 = 10 and Q3 = 16. What is the smallest whole number that would count as a high outlier?
Stretch 2 of 2
Seven whole numbers, in order from smallest to largest, have the five-number summary 1, 3, 5, 7, 9. A number may appear more than once. How many different lists of seven could they be?
lists
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.
For the grown-up
The misconception this lesson targets is reading a wide section of a box plot as holding more data. Every section holds about a quarter of the values; width shows how thinly that quarter is spread, so a long whisker is a thin scatter, not a crowd. Two slips travel with it: taking quartiles from a list that was never put in order, and building the outlier fence on the range instead of the IQR, which lets an extreme value stretch its own fence and escape. If a student hesitates, ask "how many values are in that section?" before "how wide is it?". Worth knowing: textbooks and calculators split an odd-sized list in slightly different ways, so a quartile can differ by a little between them; this lesson always leaves the median out of both halves.