D. Statistical Analysis · One-Variable Data AnalysisQuartiles, Percentiles and Percentile Rank
D1.2About 70 minutes
Learning goals- I can find the quartiles and the five-number summary of a data set.
- I can find the kth percentile of a data set using a position rule, and explain that tools may use slightly different rules.
- I can calculate the percentile rank of a value, including when there are ties.
- I can estimate percentiles and percentile ranks from a cumulative percent graph, and interpret them in context.
Words to know
quartiles
The three values Q1, Q2 (the median) and Q3 that split ordered data into four parts with about the same number of values.
kth percentile (Pk)
A value with about k% of the data at or below it. Q1 ≈ P25, the median = P50, Q3 ≈ P75.
percentile rank
The percent of the data at or below a given value (counting half of any tied values): PR = (B + 0.5E) ÷ n × 100.
decile
One of the nine values that split ordered data into ten equal parts: D1 = P10, D2 = P20, ..., D9 = P90.
five-number summary
Minimum, Q1, median, Q3, maximum.
"Your child is in the 90th percentile"
Growth charts at the doctor's office, standardized test reports and university admissions all use percentiles. A Grade 12 student who scores in the 90th percentile on a national math contest did as well as or better than about 90% of the students who wrote it. That says nothing directly about the student's mark out of 100; it describes the student's position in the group.
The curriculum asks you to "determine the positions of individual data points within a one-variable data set using quartiles, percentiles, and z-scores". This lesson covers quartiles and percentiles. Lesson 10 covers z-scores.
Quartiles: the method used in this course
Remember from Grade 9:
- Order the data and find the median.
- The median splits the data into a lower half and an upper half. If n is odd, leave the median out of both halves.
- Q1 is the median of the lower half and Q3 is the median of the upper half.
The five-number summary is minimum, Q1, median, Q3, maximum, and the IQR = Q3 − Q1.
Worked exampleExample 1: Five-number summary
Twenty Grade 12 students' test marks, in order: 52, 58, 61, 64, 65, 67, 70, 71, 72, 74, 75, 76, 78, 79, 81, 83, 85, 88, 91, 96. Find the five-number summary and the IQR.
- n = 20 is even, so the median is the mean of the 10th and 11th values: (74 + 75) ÷ 2 = 74.5.
- Lower half: the first 10 values, 52 to 74. Its median is the mean of its 5th and 6th values: (65 + 67) ÷ 2 = 66 = Q1.
- Upper half: the last 10 values, 75 to 96. Its median is (81 + 83) ÷ 2 = 82 = Q3.
- IQR = 82 − 66 = 16.
Five-number summary 52, 66, 74.5, 82, 96; IQR = 16 marks.
Percentiles
The kth percentile, Pk, is a value with about k% of the data at or below it. Q1 is about P25, the median is P50 and Q3 is about P75. Deciles are P10, P20, ..., P90.
A position rule for Pk (n values in order):
- Compute L = (k ÷ 100) × n.
- If L is a whole number, Pk is the mean of the Lth and (L + 1)th values.
- If L is not a whole number, round L up and take that value.
For the 20 marks: P35 has L = 0.35 × 20 = 7, a whole number, so P35 = (70 + 71) ÷ 2 = 70.5. P42 has L = 8.4, so round up to the 9th value: P42 = 72.
There is no single official rule. Spreadsheets have several percentile and quartile functions (PERCENTILE.INC, PERCENTILE.EXC, QUARTILE.INC, QUARTILE.EXC), and each gives slightly different answers for small data sets. They agree closely for large data sets. When you report a percentile, say which method or tool you used.
Worked exampleExample 2: Finding percentiles
For the 20 marks in Example 1, find P90 and the 3rd decile D3, and interpret P90.
- P90: L = 0.90 × 20 = 18, a whole number. P90 = mean of the 18th and 19th values = (88 + 91) ÷ 2 = 89.5.
- D3 = P30: L = 0.30 × 20 = 6, a whole number. D3 = mean of the 6th and 7th values = (67 + 70) ÷ 2 = 68.5.
- Interpret: about 90% of the class scored 89.5 or less, so a mark above 89.5 puts a student in the top 10%.
Percentile rank: going the other way
A percentile answers "what value has k% below it?". A percentile rank answers the reverse: "what percent of the data is at or below this value?"
PR = (B + 0.5E) ÷ n × 100
where B is the number of values below x and E is the number equal to x. Counting half of the tied values places x in the middle of its ties. Round to a whole number; percentile ranks are usually reported between 1 and 99.
Worked exampleExample 3: Percentile rank
(a) In the 20 marks, find the percentile rank of 78. (b) In the data 5, 6, 6, 6, 7, 8, 8, 9, 10, 10, find the percentile ranks of 6 and of 8.
- (a) Values below 78: 52 up to 76, which is B = 12 values. E = 1. PR = (12 + 0.5) ÷ 20 × 100 = 62.5, so about the 63rd percentile.
- (b) For 6: B = 1 (just the 5), E = 3. PR = (1 + 1.5) ÷ 10 × 100 = 25.
- For 8: B = 5 (5, 6, 6, 6, 7), E = 2. PR = (5 + 1) ÷ 10 × 100 = 60.
(a) A mark of 78 is at about the 63rd percentile. (b) 6 is at the 25th percentile and 8 is at the 60th.
Large data sets: read percentiles from a graph
For big data sets (or when you only have a grouped table), plot a cumulative percent graph (an ogive with the vertical axis in percent, as in lesson 3).
- Percentile Pk: go across from k% to the curve, then down.
- Percentile rank of x: go up from x to the curve, then across.
You can also interpolate in the table, exactly as you estimated the median in lesson 5.
Worked exampleExample 4: Estimating from a cumulative table
For the 40 trout, the cumulative frequencies at 18, 22, 26, 30, 34, 38 and 42 cm are 0, 2, 8, 19, 31, 37, 40. Estimate (a) P90 and (b) the percentile rank of a 32 cm trout.
- (a) 90% of 40 = 36. Between 34 cm (31 fish) and 38 cm (37 fish). Need 5 of the 6 fish in that class: 34 + (5/6)(4) ≈ 37.3 cm.
- (b) 32 cm is halfway through the class [30, 34), where the cumulative frequency rises from 19 to 31. Halfway: 19 + 6 = 25 fish. 25 ÷ 40 = 62.5%, about the 63rd percentile.
(a) P90 ≈ 37.3 cm; (b) a 32 cm trout is at about the 63rd percentile.
Percentile is not percent
A mark of 62% can be at the 90th percentile on a very hard test, and a mark of 90% can be at the 40th percentile on an easy one. A percentile always compares you with a group, so always ask: which group?
Just for funA teen tells his parents he finished in the 99th percentile of his chess club. Proud parents ask how many members the club has. "Two hundred," he says. "I finished second." (Check: B = 198 below him and E = 1, so PR = 198.5 ÷ 200 × 100 ≈ 99.)
Hands-on activityWhere do you stand? Percentiles in the family
You need: a tape measure, paper and a pencil, and at least 8 people (family members, friends, or teammates; ask permission first and keep the names private).
- Measure each person's arm span (fingertip to fingertip, arms stretched out) to the nearest centimetre. If fewer people are available, measure the hand span (thumb tip to little-finger tip, hand stretched) of each person's two hands and treat each hand as one value.
- Order the data and find the five-number summary.
- Find P20 and P80 with the position rule.
- Find the percentile rank of your own measurement.
- Ask: would your percentile rank be different if the group were all adults? All Grade 1 students? Why?
What to look for: correct ordering, the median of each half for the quartiles, the whole-number check in the position rule, and the formula (B + 0.5E) ÷ n × 100. The last question should mention that a percentile rank depends on the comparison group.
Your turn
1Data (n = 15): 12, 15, 18, 20, 21, 23, 25, 26, 28, 30, 31, 34, 37, 40, 45. Find the five-number summary.
Median = 8th value = 26. Lower half (first 7): 12, 15, 18, 20, 21, 23, 25 → Q1 = 20. Upper half (last 7): 28, 30, 31, 34, 37, 40, 45 → Q3 = 34. Summary: 12, 20, 26, 34, 45.
2For the same data, find P20.
L = 0.20 × 15 = 3, whole. P20 = mean of the 3rd and 4th values = (18 + 20) ÷ 2 = 19.
3For the same data, find P70.
L = 0.70 × 15 = 10.5, not whole, so round up to 11. P70 = 11th value = 31.
4For the same data, find the percentile rank of 30.
B = 9, E = 1. PR = 9.5 ÷ 15 × 100 ≈ 63.3, so about the 63rd percentile.
5A student's mark is at the 85th percentile. Explain what that means.
About 85% of the students in the comparison group scored the same or lower, and about 15% scored higher.
6Which percentile is the 7th decile?
More practice: warm-up
1Which percentile is the median?
2Q3 is approximately which percentile?
3In 40 values, what is L for P25?
4If 30 of 50 values are below x and none equal x, what is the percentile rank of x?
5Find the IQR when Q1 = 18 and Q3 = 31.
More practice: on your own
1For the data 3, 7, 8, 10, 12, 15, 16, 19, 21 (n = 9), find the five-number summary and the IQR.
Median = 5th = 12. Lower half 3, 7, 8, 10 → Q1 = 7.5. Upper half 15, 16, 19, 21 → Q3 = 17.5. Summary 3, 7.5, 12, 17.5, 21; IQR = 10.
2For the same data, find P25 with the position rule and compare it with Q1. Why are they different?
L = 0.25 × 9 = 2.25 → round up to 3 → P25 = 8. Q1 = 7.5. The two rules differ slightly for small data sets; both are reasonable estimates of the 25th percentile.
3In a class of 32, Leila's mark is tied with 2 other students, and 21 students scored lower. Find her percentile rank.
B = 21, E = 3. PR = (21 + 1.5) ÷ 32 × 100 ≈ 70.3, so about the 70th percentile.
4A national test had 125 000 writers. Jonah's percentile rank is 92. About how many writers scored the same as or lower than him?
0.92 × 125 000 = about 115 000 writers.
5Use the trout cumulative table (0, 2, 8, 19, 31, 37, 40 at 18, 22, ..., 42 cm) to estimate the first decile D1.
10% of 40 = 4, between 22 cm (2) and 26 cm (8). Need 2 of 6: 22 + (2/6)(4) ≈ 23.3 cm.
Challenge
1Show that in a data set with no ties, the percentile rank of the median (n odd) is exactly 50.
With n odd and no ties, the median has B = (n − 1) ÷ 2 values below it and E = 1. PR = ((n − 1)/2 + 0.5) ÷ n × 100 = (n/2) ÷ n × 100 = 50. ∎
2A university program says it admits students "above the 80th percentile" of applicants. There are 2 400 applicants. About how many are admitted? If 30 applicants are tied exactly at P80, how could the program decide fairly?
About 20% of 2 400 = 480 applicants. Ties at the cut-off need a second criterion announced in advance (for example, the mark in a required course, or an interview), because the percentile alone cannot separate them.
3For a data set of 1 000 values, explain why different spreadsheet percentile functions give almost the same P90, but for 10 values they can differ a lot.
For 1 000 values, P90 is near position 900, and neighbouring values (899th, 900th, 901st) are usually very close, so small differences in the position rule barely matter. For 10 values, the rules may choose the 9th value, the 10th, or something in between, and those can be far apart.
Quiz
1. A student is at the 70th percentile on a test. Which statement is correct?
2. In 24 ordered values, the position rule for P75 gives
3. In 50 values, 34 are below x and 2 are equal to x. What is the percentile rank of x?
4. The five-number summary of 9, 10, 12, 14, 15, 17, 20 (n = 7) is
5. Why should you state the method or tool when you report a percentile?
Big idea (D1.2): quartiles, percentiles and percentile ranks describe the position of a value within a group, not its size. Percentile rank = (B + 0.5E) ÷ n × 100.
How to help: Look at a real percentile together: a child's growth chart, a standardized test report, or a fitness test. Ask, "Who is the comparison group?" and "What would change if the group changed?"
Common difficulties:
- Confusing a percentile with a percent mark.
- Forgetting to order the data first.
- In the position rule, forgetting to average when L is a whole number, or rounding L down instead of up.
- Leaving out tied values (E) in the percentile rank formula.
- Expecting a spreadsheet to match a hand calculation exactly for small data sets.