D. Data · Data, Statistics and Mathematical Modelling

Quartiles, the Five-Number Summary and the Interquartile Range

D1.2 A1About 70 minutes

Learning goals

I can find the quartiles and five-number summary of a data set with an odd or even number of values.
I can calculate the interquartile range and explain what it tells me.
I can use the 1.5 × IQR rule to decide whether a value is an outlier.
I can explain what a percentile means.

Words to know

quartiles
Three values (Q1, Q2, Q3) that split ordered data into four groups with about the same number of values.

lower quartile (Q1)
The median of the lower half of the data; about 25% of the data is below it.

upper quartile (Q3)
The median of the upper half of the data; about 75% of the data is below it.

five-number summary
Minimum, Q1, median, Q3 and maximum.

interquartile range (IQR)
Q3 − Q1: the spread of the middle 50% of the data.

fence
A cut-off for outliers: Q1 − 1.5 × IQR (lower) or Q3 + 1.5 × IQR (upper).

percentile
A value with about a given percent of the data at or below it.

resistant measure
A statistic that is not changed much by outliers, like the median or IQR.

What does "75th percentile" mean?

When a baby visits the doctor, the doctor may say the baby is "in the 75th percentile for height". That means about 75% of babies the same age are shorter or the same height, and about 25% are taller.

That idea of splitting a data set into parts is what quartiles do. Quartiles split ordered data into four groups with about the same number of values. They tell you where the middle half of the data is, which gives a measure of spread that outliers cannot fool.

Quartiles and the five-number summary

The lower quartile (Q1) has about 25% of the data below it.
The median (Q2) has about 50% of the data below it.
The upper quartile (Q3) has about 75% of the data below it.

Together with the least value (minimum) and greatest value (maximum), these make the five-number summary:

minimum, Q1, median, Q3, maximum

How to find quartiles (the method used in this course)

Put the data in order from least to greatest.
Find the median. It splits the data into a lower half and an upper half.
If there is an odd number of values, the median itself is not included in either half.
Q1 is the median of the lower half. Q3 is the median of the upper half.

Different tools, slightly different quartiles

Statisticians use several slightly different rules for quartiles. Graphing calculators, Desmos and spreadsheets do not all use the same rule, so a tool may give a Q1 or Q3 that is a little different from your hand calculation, especially for small data sets. Use the method above when you calculate by hand, and say which tool you used when you use technology.

Worked example

Example 1: An even number of values

Twelve maple trees produced these amounts of sap (litres) in one week:

38, 45, 41, 52, 36, 48, 44, 50, 39, 47, 43, 55

Find the five-number summary.

Order: 36, 38, 39, 41, 43, 44, 45, 47, 48, 50, 52, 55 (n = 12).
Median: the middle two values are the 6th and 7th: 44 and 45. Median = (44 + 45) ÷ 2 = 44.5.
Lower half (first 6 values): 36, 38, 39, 41, 43, 44. Its median is (39 + 41) ÷ 2 = 40, so Q1 = 40.
Upper half (last 6 values): 45, 47, 48, 50, 52, 55. Its median is (48 + 50) ÷ 2 = 49, so Q3 = 49.
Minimum = 36 and maximum = 55.

Five-number summary: 36, 40, 44.5, 49, 55.

The 12 ordered values split into four groups of 3. Q1 is halfway between 39 and 41, the median is halfway between 44 and 45, and Q3 is halfway between 48 and 50.

Worked example

Example 2: An odd number of values

Find the five-number summary of: 14, 9, 20, 16, 11, 22, 15, 18, 12.

Order: 9, 11, 12, 14, 15, 16, 18, 20, 22 (n = 9).
Median: the 5th value, 15.
Leave out the median. Lower half: 9, 11, 12, 14. Q1 = (11 + 12) ÷ 2 = 11.5.
Upper half: 16, 18, 20, 22. Q3 = (18 + 20) ÷ 2 = 19.
Minimum = 9, maximum = 22.

Five-number summary: 9, 11.5, 15, 19, 22.

The interquartile range (IQR)

The interquartile range is the spread of the middle 50% of the data:

IQR = Q3 − Q1

For the maple sap: IQR = 49 − 40 = 9 L. The middle half of the trees produced amounts within a 9 L span.

The range (55 − 36 = 19 L) depends on the two most extreme trees. The IQR ignores the top 25% and bottom 25%, so one outlier cannot change it much. That makes the IQR a more resistant measure of spread.

Using the IQR to test for outliers

A common rule: a value is an outlier if it is

less than Q1 − 1.5 × IQR (the lower fence), or
greater than Q3 + 1.5 × IQR (the upper fence).

Worked example

Example 3: Is it an outlier?

Ten students recorded their commute to school in minutes: 12, 15, 18, 20, 22, 25, 25, 28, 30, 65. Use the 1.5 × IQR rule to decide if any value is an outlier.

Order (already done). n = 10. Median = (22 + 25) ÷ 2 = 23.5.
Lower half: 12, 15, 18, 20, 22, so Q1 = 18. Upper half: 25, 25, 28, 30, 65, so Q3 = 28.
IQR = 28 − 18 = 10. Then 1.5 × IQR = 15.
Lower fence = 18 − 15 = 3. Upper fence = 28 + 15 = 43.
No value is less than 3. The value 65 is greater than 43, so it is an outlier.

65 minutes is an outlier. (Maybe that student lives far away or takes two buses. An outlier is not necessarily a mistake, but you should check it.)

The fences are 1.5 IQR beyond the quartiles. Every value between the fences is normal for this data; 65 lies beyond the upper fence, so it is an outlier.

Just for fun

Why did Q1 and Q3 never argue about the outlier?

They agreed it was beyond their fences, so it wasn't their problem.

Percentiles

The kth percentile is a value with about k% of the data at or below it. So Q1 is about the 25th percentile, the median is the 50th percentile, and Q3 is about the 75th percentile. Percentiles are used for growth charts, standardized tests and fitness tests. Being in the 90th percentile does not mean you got 90%; it means you did as well as or better than about 90% of the group.

Hands-on activity

Flyer prices

You need: a grocery or store flyer (paper or online), paper and a pencil.

Choose one section of the flyer (for example, produce or snacks). Write down the prices of 16 items.
Order the prices. Find the five-number summary and the IQR.
Use the 1.5 × IQR rule to check for outliers. Is any item unusually cheap or expensive? Can you explain why?
Write one sentence that describes the middle 50% of prices, for example: "Half of the snack prices were between $2.99 and $4.49."
Bonus: enter the prices in a spreadsheet and use =QUARTILE(A1:A16,1) and =QUARTILE(A1:A16,3). Do you get exactly the same Q1 and Q3? (You may not: see the tip about different methods.)

What to look for: ordered list, Q1 and Q3 found as medians of the halves, IQR = Q3 − Q1, fences calculated correctly.

Your turn

Find the five-number summary: 4, 7, 8, 10, 12, 13, 15.

Find Q1, the median and Q3: 22, 25, 25, 27, 30, 31, 33, 35.

Find the IQR of the data in the last question.

Test marks: 61, 72, 75, 78, 80, 82, 85, 88, 90, 93, 95. Find the five-number summary and IQR.

Is 61 an outlier in the test-mark data? Use the 1.5 × IQR rule.

Is 30 an outlier in: 5, 8, 9, 10, 11, 13, 14, 16, 30?

A student scored at the 60th percentile on a reading test. Explain what that means.

More practice: warm-up

What percent of the data lies between Q1 and Q3?

Q1 = 12 and Q3 = 30. Find the IQR.

Which quartile is the same as the median?

List the five numbers in a five-number summary, in order.

Find the median of the lower half 3, 6, 8, 11.

More practice: on your own

Find the five-number summary: 14, 18, 21, 23, 24, 26, 27, 29, 31, 34, 36, 40, 42, 70.

For the data in the last question, find the fences and any outliers.

For the same data, compare the range and the IQR. Which describes the spread of a typical value better?

Find the five-number summary: 3, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 16, 19, 24.

A data set has Q1 = 40 and Q3 = 58. What is the smallest whole-number value that would count as a high outlier?

Challenge

Create a data set of 8 whole numbers with minimum 10, Q1 = 15, median 20, Q3 = 30 and maximum 50.

Two data sets have the same median, 50. Set A has IQR 4 and Set B has IQR 30. Describe the difference in words, using a context (for example, wait times in minutes at two clinics).

A data set of 11 values has Q3 = 40. Two new values, both greater than the maximum, are added. Explain why Q3 must stay the same or increase.

Check your understanding

Quiz

What is Q1 for 2, 4, 5, 7, 9, 10, 12, 15?

Q1 = 18 and Q3 = 33. What is the IQR?

Why is the IQR often better than the range for describing spread?

Q1 = 20 and Q3 = 28. Using the 1.5 × IQR rule, which value is an outlier?

About what percent of data values are greater than Q3?

A runner's time is at the 20th percentile for a race (lower times are faster). What does this mean?

For parents and teachers

Big idea: D1.2 names quartile values directly. Quartiles are new in Grade 9 (earlier grades used mean, median, mode and range). This lesson builds the skills needed for box plots in the next lesson.

How to help: The most common mistake is not ordering the data. The second is including the median in the halves when n is odd. Have your teen say the rule out loud: "Odd: leave the median out. Even: split straight down the middle."

Other difficulties:

Confusing percent with percentile.
Thinking an outlier must be an error. Outliers should be checked, not automatically deleted.
Getting slightly different quartiles from a calculator or spreadsheet. This is normal; different tools use different rules.