What Does a Percentile Tell You?
The previous tutorial, Variance and Its Relationship to Standard Deviation, described how data values vary around their mean. Percentiles answer a different kind of question: where does a value stand in the ordered data? They describe a position in a group, not a distance from the mean.
Suppose a student’s score is reported at the 85th percentile among students who took an assessment. The basic interpretation is that about 85% of the scores in the reference group were at or below that score, and about 15% were above it. The reference group matters: a percentile compares a score with a particular group, such as students who took the same assessment.
A percentile is not the same as a percent correct. A score at the 85th percentile does not mean that 85% of the questions were answered correctly. It means the score’s position is high relative to the scores in the reference group. A student could answer 70% of the questions correctly and still be at a high percentile if many other scores were lower; the reverse could also happen.
Percentile statements are often approximate. Ties can affect how many observations are at or below a particular score, and different assessment or software conventions may locate percentile values in different ways. Therefore, do not turn a reported percentile into an exact number of people unless the data and the stated convention justify that count.
Calculating a Percentile Rank from Data
A percentile rank describes the position of a particular observed value in a data set. In this tutorial, use the following clear convention: count the observations at or below the value, divide by the total number of observations, and convert the proportion to a percentage. This convention includes ties with the value.
The calculation uses the observed data, so the resulting percentage can be exact for that data set under this convention. For example, if 9 of 12 observations are at or below a value, its percentile rank is exactly \(9/12 \times 100\%=75\%\) in that data set. That does not mean the value will be at the 75th percentile in every other group or in a larger population.
Pay particular attention to the phrase at or below. It includes observations equal to the value. The number strictly below a score can be smaller, especially when several observations are tied. Some references use a different rule for handling ties, such as counting only observations strictly below or splitting the tied observations. If a question supplies a convention, follow it and state it. If it does not, name the convention you use so the calculation is clear.
Worked Examples: Interpreting and Finding Percentile Ranks
Worked Example: Interpreting an 85th-Percentile Score
A fictional reading assessment reports that a student’s score of 76 is at the 85th percentile among students in the same reference group. Explain the report. The group includes 40 students, but individual scores and ties are not provided.
Interpret the percentile. The report says that about 85% of scores in the reference group were at or below 76. About 15% were above 76, since \(100\%-85\%=15\%\). This describes the score’s relative position in the group; it does not say that the student answered 85% of the questions correctly.
Relate the percentages to the group size cautiously. Multiplying gives \(0.85(40)=34\) and \(0.15(40)=6\). Thus, the percentile statement corresponds roughly to 34 students at or below the score and roughly 6 above it. These are approximate counts implied by the reported percentile, not established exact counts. The assessment’s percentile convention and any scores tied at 76 could affect the actual counts.
Check the relationship between the two proportions. The percentages add to \(85\%+15\%=100\%\), and the rough counts add to \(34+6=40\), the group total. This confirms the approximate split, but it does not turn the approximate counts into exact observed counts.
Answer in context. The student’s score of 76 is higher than or equal to about 85% of the scores in the specified reading-assessment group, and about 15% of the scores were above it. The report does not establish exact numbers of students in either group from the information given.
Worked Example: Percentile Rank in a List of Sensor Readings
A fictional sample of 12 temperature readings, in degrees Celsius, is already ordered:
Find the percentile rank of 18 using the at-or-below convention. Then compare it with the percentage of readings strictly below 18.
Count readings at or below 18. The values at or below 18 are \(12,14,15,15,16,18,18\), so there are 7 such readings out of 12.
Calculate the percentile rank.
Check the count and percentage. The ordered list has five readings below 18 and two equal to 18, giving \(5+2=7\) at or below 18. Also, \(0.58333\ldots \times 12=7\), confirming that the rounded percentile rank corresponds to the count used.
Compare with strictly below. Five of the 12 readings are strictly below 18, so \(5/12 \times 100\%=41.666\ldots\%\), or about 41.7%. The percentile rank under the stated convention is about 58.3%, not 41.7%, because both readings equal to 18 are included.
Interpret in context. In this sample of temperature readings, 18 degrees Celsius is at the 58.3rd percentile under the at-or-below convention: 58.3% of the readings were 18 degrees or lower. The percentage is based on this sample, not a claim about all temperatures in a larger population.
Worked Example: Percentile Rank from a Frequency Table
A fictional class records quiz scores. The frequency table summarizes 25 students. Find the percentile rank of a score of 80 using the at-or-below convention, and state how many students scored below, equal to, and above 80.
| Quiz score | Number of students |
|---|---|
| 60 | 2 |
| 65 | 4 |
| 70 | 5 |
| 75 | 6 |
| 80 | 5 |
| 85 | 3 |
Check the total number of students. Add the frequencies: \(2+4+5+6+5+3=25\). This agrees with the stated sample size.
Count students at or below 80. Add the frequencies for scores of 60, 65, 70, 75, and 80:
Calculate and check the percentile rank.
The calculation can also be checked by noting that \(25-22=3\) students scored above 80, so \(22/25=1-3/25=1-0.12=0.88\), or 88%.
Separate the three groups. There are \(2+4+5+6=17\) students below 80, 5 students equal to 80, and 3 students above 80. These counts total \(17+5+3=25\). Their percentages are \(17/25=68\%\) below, \(5/25=20\%\) equal, and \(3/25=12\%\) above. The at-or-below percentage is \(68\%+20\%=88\%\), matching the percentile-rank calculation.
Interpret in context. A score of 80 has a percentile rank of 88% in this class under the at-or-below convention: 88% of the students scored 80 or lower. Because five students tied at 80, the percentage strictly below the score is only 68%. Naming which count is used makes the interpretation unambiguous.
Common Mistakes and AP Exam Tips
- Confusing percentile with percent correct. A score at the 85th percentile describes relative standing among a reference group. It does not mean that 85% of the questions were answered correctly.
- Turning an approximate percentile into an exact count. A report of the 85th percentile suggests about 85% at or below the score; it does not by itself prove that exactly 85% of the individuals in a particular group are at or below it. Ties and the reporting convention matter. Use exact counts only when the data establish them.
- Forgetting tied observations. Under the convention used here, include every observation equal to the value. State “at or below” in the calculation and interpretation.
- Using the wrong denominator. Divide by the total number of observations, not by the number below the value. In a frequency table, first add the frequencies to verify the total.
- Leaving out the reference group. A percentile rank depends on the group being compared. Name the group and variable, and do not generalize a sample’s percentile rank to a different population without supporting information.
- Assuming one tie rule applies everywhere. Percentile conventions can vary, particularly for tied values or for locating a percentile value from a data set. Follow any convention stated in the question; otherwise, identify the rule you used.
For a complete response, show the count included in the numerator, the total count in the denominator, and the resulting percentage. Then interpret it in context with the phrase “at or below” when using this tutorial’s convention. For a reported percentile, keep the interpretation approximate unless exact data and a specified convention support exact counts.
Check Your Understanding
For each question, state what is being counted and use the at-or-below convention unless a different convention is specified.
- A runner’s time is reported at the 70th percentile among runners in a particular race group. What does this say about the runner’s position, and what does it not say about the percent of the race completed?
- In an ordered sample of 10 values, 6 are at or below 42. Calculate the percentile rank of 42 under the convention used in this tutorial.
- In a data set of 20 observations, 4 are below a value and 3 are equal to it. Find its percentile rank under the at-or-below convention. What percentage is strictly below it?
- A report says a score is at the 85th percentile in a group of 60 people. Why should you avoid claiming an exact count at or below the score without further information?
- A frequency table gives 12 observations at or below a value out of 16 total. Find the percentile rank and explain what it means for the group summarized by the table.