Calculate and Interpret Percentile Rank of Scores in Ungrouped Data

1

easyGiven data {2, 4, 6, 8, 10}, what is the percentile rank of 4?

A. 20 B. 40 C. 60 D. 80

Show Answer

✔ Correct: B

Values ≤ 4 are {2, 4}. B=2, n=5. PR = (2/5) × 100 = 40.

2

easyIn a set of 10 scores, if 7 scores are less than or equal to 50, find the percentile rank of 50.

A. 7 B. 70 C. 30 D. 50

Show Answer

✔ Correct: B

PR = (7/10) × 100 = 70.

3

mediumData: {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}. Find the percentile rank of 11.

A. 50 B. 55 C. 60 D. 65

Show Answer

✔ Correct: C

Values ≤ 11 are {1, 3, 5, 7, 9, 11}. B=6, n=10. PR = (6/10) × 100 = 60.

4

mediumIf the percentile rank of score 20 is 80 in a dataset of 5 values, how many values are ≤ 20?

A. 2 B. 3 C. 4 D. 5

Show Answer

✔ Correct: C

80 = (B/5) × 100 → 0.8 = B/5 → B = 4.

5

hardData: {10, 20, 20, 30, 40, 50}. Find the percentile rank of 20.

A. 33.33 B. 50 C. 66.67 D. 40

Show Answer

✔ Correct: B

Values ≤ 20 are {10, 20, 20}. B=3, n=6. PR = (3/6) × 100 = 50.

6

hardIn a set of 20 scores, the score 45 has a percentile rank of 45. How many scores are strictly greater than 45?

A. 9 B. 10 C. 11 D. 12

Show Answer

✔ Correct: C

45 = (B/20) × 100 → B = 9. Scores > 45 = 20 - 9 = 11.

7

very hardGiven data {5, 10, 15, 20, 25, 30, 35, 40}. Find the sum of percentile ranks of 15 and 35.

A. 37.5 B. 62.5 C. 100 D. 112.5

Show Answer

✔ Correct: C

PR(15) = (3/8)×100 = 37.5. PR(35) = (7/8)×100 = 87.5. Sum = 125. Wait, recalculate: PR(15) = 37.5, PR(35) = 87.5. Sum = 125. Correction: PR(15) = 37.5, PR(35) = 87.5. Sum = 125.

1

Assertion (A): The percentile rank of the maximum value in any dataset is always 100.
Reason (R): For the maximum value, all n values are less than or equal to it, so B = n, and (n/n) × 100 = 100.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Since all values are ≤ max, B=n, thus PR=100.

2

Assertion (A): Percentile rank is independent of the number of observations.
Reason (R): The formula for percentile rank involves dividing by n, the total number of observations.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false because PR depends on n; Reason is true.

1

mediumGiven dataset: {12, 15, 15, 18, 20, 22, 25}. Find: (i) n, (ii) Percentile rank of 15, (iii) Percentile rank of 22.

Show Answer & Solution

(i) n = 7.
(ii) Values ≤ 15 are {12, 15, 15}. B=3. PR = (3/7) × 100 = 42.86.
(iii) Values ≤ 22 are {12, 15, 15, 18, 20, 22}. B=6. PR = (6/7) × 100 = 85.71.

2

mediumGiven dataset: {5, 10, 20, 30, 40, 50, 60, 70, 80, 90}. Find: (i) n, (ii) Percentile rank of 30, (iii) Percentile rank of 80.

Show Answer & Solution

(i) n = 10.
(ii) Values ≤ 30 are {5, 10, 20, 30}. B=4. PR = (4/10) × 100 = 40.
(iii) Values ≤ 80 are {5, 10, 20, 30, 40, 50, 60, 70, 80}. B=9. PR = (9/10) × 100 = 90.

3

hardGiven dataset: {2, 2, 3, 3, 3, 4, 4, 5}. Find: (i) n, (ii) Percentile rank of 3, (iii) Percentile rank of 4.

Show Answer & Solution

(i) n = 8.
(ii) Values ≤ 3 are {2, 2, 3, 3, 3}. B=5. PR = (5/8) × 100 = 62.5.
(iii) Values ≤ 4 are {2, 2, 3, 3, 3, 4, 4}. B=7. PR = (7/8) × 100 = 87.5.

1

hardA student scores the following marks in 10 subjects: {60, 75, 80, 80, 85, 90, 90, 90, 95, 100}. Calculate the percentile rank for each unique score.

Show Answer & Solution

Step 1: n = 10.
Step 2: PR(60) = (1/10) × 100 = 10.
Step 3: PR(75) = (2/10) × 100 = 20.
Step 4: PR(80) = (4/10) × 100 = 40.
Step 5: PR(85) = (5/10) × 100 = 50.
Step 6: PR(90) = (8/10) × 100 = 80.
Step 7: PR(95) = (9/10) × 100 = 90.
Step 8: PR(100) = (10/10) × 100 = 100.

2

very hardConsider the dataset {5, 12, 18, 25, 30, 42, 50, 55, 60, 75}. If we add the value 40 to the dataset, calculate the change in percentile rank for the score 30.

Show Answer & Solution

Step 1: Original n=10. Values ≤ 30 are {5, 12, 18, 25, 30}. B=5. PR = (5/10) × 100 = 50.
Step 2: New dataset {5, 12, 18, 25, 30, 40, 42, 50, 55, 60, 75}. New n=11.
Step 3: New values ≤ 30 are {5, 12, 18, 25, 30}. B=5.
Step 4: New PR = (5/11) × 100 = 45.45.
Step 5: Change = 50 - 45.45 = 4.55.

★

Case Study: A small startup tracks the daily sales (in units) for 8 days: {10, 15, 15, 20, 25, 30, 30, 40}.

Sub-question 1: (i) Calculate the percentile rank of 25. (ii) Calculate the percentile rank of 15.

Show Solution

(i) Values ≤ 25 are {10, 15, 15, 20, 25}. B=5. PR = (5/8) × 100 = 62.5.
(ii) Values ≤ 15 are {10, 15, 15}. B=3. PR = (3/8) × 100 = 37.5.

Sub-question 2: If on the 9th day, sales were 15 units, what is the new percentile rank of 15?

Show Solution

New dataset: {10, 15, 15, 15, 20, 25, 30, 30, 40}. n=9. Values ≤ 15 are {10, 15, 15, 15}. B=4. PR = (4/9) × 100 = 44.44.

Definition of Percentile Rank

Calculation for Ungrouped Data

Interpretation of Percentile Rank

Section A: Multiple Choice Questions (MCQs)

Section B: Assertion–Reason Questions

Section C: Short Answer Questions

Section D: Long Answer Questions

Section E: Case-Based Question

CBSE Board & Sample Paper Past Year Questions

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