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 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 asks how a value or feature behaves inside the full distribution.
Common stuck point:Students often know a procedure related to percentiles but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?
Worked Examples
Example 1
medium
On a test, Maria's percentile rank is 92 while Ron's is 80. If 500 students took the test, about how many more students scored below Maria than below Ron?
Answer
60
First step
1
Maria above 92%: 0.92ร500=460.
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Example 2
hard
Joe's height is at the 70th percentile for boys his age. His twin Jim is at the 90th. If the population mean is 60 inches with SD 3 inches (normal), estimate the height gap.
Example 3
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?
Example 4
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 5
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
easy
If you scored in the 80th percentile, what percent of test-takers did you score above?A score at the 80th percentile is above 80% of all scores (shaded region).
Example 2
easy
The median of a data set is the same as which percentile?What percentile is the median?
Example 3
easy
The first quartile Q1โ corresponds to which percentile?Q1 falls at which percentile?
Example 4
easy
The third quartile Q3โ corresponds to which percentile?Q3 falls at which percentile?
Example 5
easy
Being in the 99th percentile for height means you are taller than what percent of people?At the 99th percentile, 99% of the group falls below (shaded).
Example 6
easy
Does a higher percentile rank mean a higher relative standing in the data?
Example 7
easy
In a class of 100 students, how many scored below the 30th percentile (approximately)?
Example 8
easy
Can two different raw scores both be in the 90th percentile if they come from two different tests?
Example 9
medium
In the ranked list 2,5,7,8,11,14 (6 values), the value 8 has 3 of 6 values at or below partial-count... using rank below: 3 of 6 values are below 8. Estimate its percentile.
Example 10
medium
A data set has 200 values. The 75th percentile separates how many values below it?
Example 11
medium
To find the position of the 25th percentile in n=20 ordered values, use 100Pโรn. What position index does it give?
Example 12
medium
A student is in the 60th percentile on a test of 50 students. Roughly how many students did they outscore?
Example 13
medium
The IQR is built from which two percentiles, and what does it measure?The IQR spans from the 25th percentile (Q1) to the 75th percentile (Q3), covering the middle 50% of the data.
Example 14
medium
Why is a percentile rank often more informative than a raw score on a standardized test?
Example 15
medium
If a value sits at the 50th percentile in a right-skewed distribution, is it the mean or the median?
Example 16
medium
On a baby growth chart, a child is at the 10th percentile for weight. Interpret this.A child at the 10th percentile weighs more than 10% of peers (shaded region) and less than 90%.
Example 17
medium
In a data set of 40 values, the 90th percentile separates how many values below it?
Example 18
challenge
In an ordered data set of n=8 values 3,5,6,8,11,13,17,20, find the 25th percentile using the locator L=10025โร8 and the convention 'if L is a whole number, average positions L and L+1'.Find Q1 (the 25th percentile) for the data set {3, 5, 6, 8, 11, 13, 17, 20}.
Example 19
challenge
Two students both score 700. In test A (700 is the 95th percentile) and test B (700 is the 60th percentile), which performance is relatively stronger and why?Test A: a score of 700 falls at the 95th percentile โ above 95% of test-takers (shaded).
Example 20
challenge
Explain why a value can be at the 90th percentile yet have a negative z-score is impossible, while at the 40th percentile a negative z-score is possible (assume symmetric distribution).The 40th percentile (z โ โ0.25) is below the mean (z=0), so its z-score is negative.
Example 21
easy
A student's score is in the 92nd percentile. What percent of test-takers did the student score above (or equal to)?
Example 22
easy
In a data set of 400 values, how many values are below the 60th percentile (approximately)?
Example 23
easy
A baby's weight is in the 5th percentile. Should the pediatrician be concerned about being below most peers?
Example 24
easy
You are told to find the 100th percentile. Which value does this correspond to?
Example 25
easy
In a sorted list of 20 numbers, the 50th percentile is in approximately which position(s)?
Example 26
medium
In a sorted data set of 80 values, what is the position index of the 30th percentile using P/100รn?
Example 27
medium
A SAT score in the 85th percentile means the student outperformed approximately what percent of test-takers?
Example 28
medium
A standardized test gives Carla a score of 1200 at the 78th percentile and Don a score of 1320 at the 88th percentile. Whose score is higher in percentile rank?
Example 29
medium
Find the 40th percentile of the sorted data {2,4,6,8,10,12,14,16,18,20} using position =P/100โ n.
Example 30
medium
Out of 250 students, how many scored at or above the 90th percentile?
Example 31
medium
For a normal distribution with mean 100 and SD 15, the 50th percentile is what value?
Example 32
hard
A test has scores normally distributed with mean ฮผ=70 and SD ฯ=10. The 84th percentile is approximately what score?
Example 33
hard
Sorted data: {3,7,8,12,15,18,22,25,30,35} (n=10). Find the 70th percentile using linear interpolation: position =P(n+1)/100.
Example 34
hard
A normal distribution with ฮผ=500 and ฯ=100 has the 16th percentile at approximately what value?
Example 35
hard
Two students, A and B, took different tests. A scored 70 (90th percentile). B scored 95 (60th percentile). Which student performed better relative to their group?
Example 36
hard
A data set of 1000 values has the 95th percentile equal to 200. Roughly how many values exceed 200?
Example 37
challenge
For a uniform distribution on [0,100], what value is the 73rd percentile?
Example 38
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 39
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 โ 84% is one standard deviation above the mean.)