Percentile & Percentile Rank

Percentiles and Percentile Ranks

  • Definition: Percentiles describe the relative standing of an individual within a known group based on a cumulative percentage distribution.

  • Percentile: A specific point on the measurement scale indicating the percentage of cases that fall below it in the distribution.

  • Percentile Rank: The percentage of individuals whose scores are at or below a given score in the distribution.

Describing and Interpreting Percentile & Percentile Rank

Example

“If there is a test, and let's say the 90th Percentile on a test was 175”

  • If person A scored 175 on the test → Person A score was better than 90% of the test takers

  • If person A’s score is in the 90th percentile, 90% of a comparison group achieved scores at or below person A’s score

  • This means that only 10% of the individuals scored higher than Person A, indicating a strong performance relative to peers.

Cumulative Frequency and Cumulative Percentage

  • Cumulative Frequency (cf): The running total of frequencies up to a certain point.

  • Cumulative Percentage (c%): The cumulative frequency expressed as a percentage of the total.

Example Calculations

  1. 95th Percentile Calculation

    • For the 95th percentile, the score (X) is identified as 4.5.

  2. Percentile Rank Calculation

    • For X = 3.5, the percentile rank is 70%.

Interpolation Method

  • We can calculate the percentile rank for non-exact scores using interpolation.

  • Example Distribution:

    • Scores: 10, 9, 8, 7, 6, 5, 2, 8, 4, 6, 4, 1

    • Cumulative Percentages: 100, 92, 60, 44, 20, 4

  • To find the percentile for X = 7, determine real limits:

    • Located between 6.5 (20%) and 7.5 (44%).

  • Step-by-Step Interpolation:

    1. Interval Width: Score width = 1 point; Percentage width = 24 points.

    2. Position of the Score: X = 7 is 0.5 points from the top of the interval.

    3. Calculate Percentage: Halfway (12 points) down from 44% results in 32%.

Finding the 50th Percentile

  • Using a second distribution to find the 50th percentile:

  • Score intervals: 20-24, 15-19, 10-14, 5-9, 0-4.

  • Associated percentage values:

    • 10% at 4.5 and 60% at 9.5.

    • 50% lies between these two percentages.

  • Interpolation Steps:

  1. Interval Width: 5 points for scores; 50 points for percentage.

  2. Locate 50%: 10 points from the top of the percentage interval, which is 1/5 of the total.

  3. Calculate Score: 1 point down from 9.5 gives 8.5 as the 50th percentile.