Percentile
In short: Percentile ranks a value by the share of the group that does not beat it: being at the 75th mark means three quarters of all observations are equal to it or smaller.
AP StatisticsStatisticsAP Statistics Unit 1
What is Percentile?
A percentile reports standing, not score. If your result sits at the 75th spot, then 75 percent of the group did as well as you or worse, and the other 25 percent did better.
To place a particular value \(x\), count how many observations are less than or equal to \(x\), divide by the total, and convert to a percent:
\[\text{rank} = \frac{\text{number of values} \le x}{n} \times 100\]
This differs from a percent correct: scoring 60 percent on a hard test could still land at the 95th spot.
How to spot it
- A report says a child is in the 40th percentile for height, or a test result is given as a rank out of 100.
- The question asks how a value compares to the rest of the group rather than what the raw value is.
- A normal curve is shaded up to a point and the area to the left is requested.
- You are asked what score is needed to place in the top 10 percent.
The method
- Order the data, or use the cumulative relative frequency graph.
- Count the observations at or below the value of interest.
- Divide by the total number of observations and multiply by 100.
- Say what it means in context: about that many percent of the group scored at or below this value.
Percentile: solved example
Problem
In a class of 60 students, 45 earned 78 or lower on a final. Where does a 78 stand, and what does that tell the student who earned it?
Solution
Count at or below 78: 45 out of 60. \(\frac{45}{60} \times 100 = 75\).
A score of 78 sits at the 75th percentile: about 75 percent of the class scored 78 or lower, while about 25 percent scored higher.
Try it
Question 1
Your score is at the 40th mark among 250 test takers. About how many scored at or below it?
Show the answer
\(0.40 \times 250 = 100\) students scored at or below it.
Question 2
Is landing at the 80th spot the same as getting 80 percent of the questions right?
Show the answer
No. The rank compares you with other test takers; percent correct measures your own answers against the total possible.
Percentile vs. Quartiles
Quartiles are three particular percentiles: \(Q_1\) is the 25th, the median is the 50th and \(Q_3\) is the 75th. A percentile can be taken at any position from 1 to 99, so the idea is more general and more precise than quartiles.
Keep going
Lessons
Worksheets and quizzes
- z-Scores and the Normal Distribution – Worksheet
- z-Scores and the Normal Distribution – Quiz
- Five-Number Summary, Boxplots and Comparing Distributions – Worksheet
- Five-Number Summary, Boxplots and Comparing Distributions – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics