Percentiles Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Percentiles.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Percentiles are values that divide a ranked distribution into 100 equal parts. The nth percentile is the value below which n\% of the data falls, telling you where a specific observation stands relative to the entire dataset.

Being in the 90th percentile means you scored better than 90% of people. It's not about your raw score - it's about your position relative to everyone else.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Percentiles show relative position in a distribution. The nth percentile means you scored higher than n% of the group — it says nothing about your raw score.

Common stuck point: Students confuse their percentile rank with their percentage score. Scoring at the 90th percentile does not mean answering 90% of questions correctly.

Sense of Study hint: When you need to find a percentile, first sort all values from smallest to largest. Then use the formula L = \frac{n}{100} \times N where n is the desired percentile and N is the number of data points. Finally, if L is a whole number, average the Lth and (L+1)th values; otherwise round up to find the position.

Worked Examples

Example 1

easy
A student scored in the 85th percentile on a standardised test with 200 test-takers. What does this mean and how many students scored lower?

Solution

  1. 1
    Step 1: The 85th percentile means the student scored higher than 85% of all test-takers.
  2. 2
    Step 2: Number who scored lower: 200 \times 0.85 = 170 students.
  3. 3
    Step 3: This means 200 - 170 = 30 students scored the same or higher.

Answer

The student scored higher than 170 out of 200 test-takers (85%). 30 students scored the same or higher.
A percentile indicates the percentage of values in a distribution that fall below a given value. Being in the 85th percentile does not mean the student scored 85% on the test — it means they outperformed 85% of the other test-takers.

Example 2

medium
Given the sorted data set of 20 values: 12, 15, 18, 20, 22, 25, 27, 30, 32, 35, 38, 40, 42, 45, 48, 50, 55, 60, 65, 70. Find the value at the 40th percentile.

Example 3

medium
In a class of 40 students, a student scored higher than 32 others. What percentile is the student at?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A baby's weight is at the 25th percentile for her age group. Her parents are worried she is underweight. Is their concern justified? Explain what the 25th percentile means in this context.

Example 2

hard
In a normally distributed data set with mean 500 and standard deviation 100, approximately what value corresponds to the 84th percentile? (Hint: use the empirical rule — 84% is one standard deviation above the mean.)
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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