Syllabus Unit 5 – Statistics Percentiles Concept of Percentile Rank

Concept of Percentile Rank

Master Percentile Rank in CBSE Class 11 Applied Mathematics. Learn formulas, calculations, and step-by-step solutions for exam success.

Definition of Percentile Rank

The percentile rank of a score is the percentage of scores in its frequency distribution that are equal to or lower than it. It provides a relative position of an individual within a group of size N. If a student scores at the 80th percentile, it means they performed better than or equal to 80% of the total group.

PR = (B/N) × 100
Example: Solved Example: Calculate Percentile Rank
Show Step-by-Step Solution

Step 1: Identify B (number of scores below or equal to x) = 15, N (total scores) = 20.
Step 2: Apply formula PR = (15/20) × 100.
Step 3: Calculate 0.75 × 100 = 75.
Answer: The percentile rank is 75.

Properties of Percentile Rank

Percentile rank ranges from 0 to 100. It is a non-parametric measure, meaning it does not assume a normal distribution of data. It is highly useful for comparing performance across different tests or groups of varying sizes.

0 ≤ PR ≤ 100
Example: Solved Example: Verify Range
Show Step-by-Step Solution

Step 1: Given N = 50, score x is the lowest, so B = 1.
Step 2: PR = (1/50) × 100 = 2.
Step 3: Given score x is the highest, so B = 50.
Answer: PR = (50/50) × 100 = 100. The range is [2, 100].

Calculation for Grouped Data

For grouped data, the percentile rank is calculated by identifying the cumulative frequency up to the upper boundary of the class interval containing the score. We use linear interpolation to estimate the position within the interval.

PR = [cfₗ + ((x - L)/i) × f] / N × 100
Example: Solved Example: Grouped Data Rank
Show Step-by-Step Solution

Step 1: Given N = 100, cfₗ = 40, L = 20, i = 10, f = 20, x = 25.
Step 2: PR = [40 + ((25 - 20)/10) × 20] / 100 × 100.
Step 3: PR = [40 + 0.5 × 20] = 50.
Answer: The percentile rank is 50.