Small Wording Differences Can Change the Meaning
A summary statistic is only useful when its interpretation matches what it actually measures. Two common sources of confusion are calling the median “the average” and treating a percentile as if it were a percent. Both errors can make a sentence sound reasonable while giving the reader the wrong information.
In Computing the Mean and Interpreting It and Finding the Median for Odd and Even Samples, you learned how to calculate and describe these measures of center. In Percentiles and Their Interpretation, you learned that a percentile describes a position within a reference group. This tutorial focuses on choosing wording that keeps those meanings separate.
Mean and Median Are Both Measures of Center—but Not the Same Measure
In everyday conversation, “average” is sometimes used loosely to mean a typical or central value. In a precise statistics response, however, average usually refers to the arithmetic mean. If a question asks for the median, name the median. If it asks for the mean, name the mean. Do not substitute one for the other just because both describe center.
The mean uses every observation in its calculation. The median depends on the middle position or positions after the values are ordered. As explained in Resistant Versus Non-Resistant Statistics, this difference matters: an extreme observation can move the mean substantially while having little or no effect on the median.
A median can still be a useful description of a “typical” value, especially for skewed data or data with outliers. But saying “the median is the average” hides which statistic you used and can suggest that you calculated the arithmetic mean. State the measure by its correct name, then interpret it in context.
Percent Is a Proportion; Percentile Is a Position
A percent tells how large a part is relative to a whole. For example, a score of 72 points out of 100 is 72% of the available points. A percentile, by contrast, describes a score’s position relative to scores in a specified reference group.
Under the at-or-below convention from Percentiles and Their Interpretation, the percentile rank of a value is the percentage of observations in the data set that are less than or equal to that value. Thus, a score at the 80th percentile is at or above about 80% of scores in its reference group. It does not mean the score itself was 80%, and it does not say the student got 80% of the available points.
The reference group matters. A student’s percentile rank describes standing among the group used for comparison; it does not, by itself, say how many questions the student answered correctly or how much of the material the student knows. Also, ties and the method used to determine a percentile can affect the exact share at or below a particular value. For AP Statistics interpretations, “about” is often important.
| Term | What it describes | Question it answers |
|---|---|---|
| Percent | A proportion relative to a whole | What share of the total is this? |
| Percentile | A value’s position in a reference group | About what percentage of the group is at or below this value? |
| Mean | Arithmetic center, using all observations | What is the sum divided by the number of observations? |
| Median | Middle position in the ordered observations | What value lies at the center of the ordered data? |
Worked Examples: Correct the Interpretation
Worked Example: Is the Median the “Average” Screen Time?
A fictional sample of eight students reported their recreational screen time on a particular day, in minutes: \(12,15,16,18,19,20,21,\) and \(79\). A student writes, “The average screen time was 18.5 minutes.” Check the statement and give accurate interpretations of the mean and median.
Find the mean. The sum is \(12+15+16+18+19+20+21+79=200\) minutes. There are eight observations, so the arithmetic mean is:
A check is to compare the observations with 25: the first seven values are below 25, and the high value of 79 pulls the arithmetic average upward. The deviations from 25 sum to zero: \(-13-10-9-7-6-5-4+54=0\), as they should for deviations from the mean.
Find the median. The data are already ordered, and there are eight values. The fourth and fifth observations are 18 and 19, so:
The student’s number is the median, not the mean. The answer is not wrong as a calculation, but calling it “the average” is imprecise if “average” means arithmetic mean. The mean of 25 minutes is higher than seven of the eight observations because the unusually large value affects the arithmetic average.
Conclusion in context. The mean recreational screen time in this sample was 25 minutes per student, while the median was 18.5 minutes. A clear report names which measure is being interpreted; it does not use “average” to blur the distinction.
Worked Example: Percent of Points Versus Percentile Rank
In a fictional class of 40 students, one student earned 70 points out of 100 on a quiz. The student’s score is at or above the scores of 32 students in the class, counting the student. Distinguish the student’s percent score from the percentile rank described by the class data.
Percent of points earned. The student earned 70 of the 100 available points:
This is the student’s percent score: 70% of the available points. Checking the fraction confirms it is \(0.70\) of the total, which is \(70\%\).
Percentile rank in the class. Under the at-or-below convention, 32 of the 40 scores are less than or equal to this score. Therefore:
The value 70 points has percentile rank 80 in this class under the stated convention: about 80% of the class scored 70 points or less. This calculation checks because \(40-32=8\) students, or \(8/40=20\%\), scored above 70; the two shares total 100%.
Conclusion in context. The student earned 70% of the quiz points and was at the 80th percentile in the class. Those statements answer different questions. The first compares points earned with points available; the second compares the student’s score with the class’s scores.
Worked Example: Interpreting a Percentile Value for Wait Times
A fictional community clinic summarizes its appointment wait times. It reports that the 90th percentile is 18 minutes. In a group of 50 waits, 45 were 18 minutes or less. Explain what the report means and identify a tempting but incorrect interpretation.
Interpret the percentile. The 90th percentile is a wait-time value with about 90% of the group at or below it. Here, the count supports that interpretation:
As a check, the other \(50-45=5\) waits are above 18 minutes, and \(5/50=10\%\). So about 10% of these waits exceeded 18 minutes, while 90% were 18 minutes or less.
Reject the confusion. The statement does not mean that a wait lasted 90 minutes, that each person waited 90% of 18 minutes, or that the clinic’s average wait was 18 minutes. It identifies a location in the distribution: 18 minutes is the reported percentile value. Nor does the statement tell us that 90% of each individual appointment’s wait has elapsed.
Conclusion in context. In this group of clinic waits, the reported 90th-percentile wait time was 18 minutes: about 90% of waits were 18 minutes or shorter. This is a percentile interpretation, not a percent score or a statement about the mean.
A Quick Method for Checking a Summary-Statistics Sentence
When reading a statement about a summary statistic, do not rely only on the number. Identify what kind of quantity the number represents, what group it refers to, and what comparison it makes. This habit helps catch errors even when two different statistics happen to have the same numerical value.
Is the number a mean, median, percent, or percentile? Do not infer the answer from the number alone.
For a percent, identify the whole. For a percentile, identify the reference group. For a mean or median, identify the variable and group.
Describe an arithmetic average, middle value, share of a whole, or relative position as appropriate.
A time percentile is a time value; a percent score is a share of available points. State units and comparison groups clearly.
This check also helps with the median. For example, “the middle half of the students had scores between 62 and 84 points” describes a range of values; “the median score was 73 points” identifies the center. Neither sentence says that the median is the arithmetic average.
Common Mistakes and AP Exam Tips
- Calling every measure of center an average. In casual speech, people may use “average” to mean typical. In a precise answer, name the statistic: mean or median. If a question uses “average” ambiguously, clarify which measure you calculated.
- Claiming the median equals the mean. They can happen to have the same value, but that does not make them the same statistic. State each name and calculation or interpretation separately.
- Reading the percentile number as a score percentage. An 80th-percentile score is a relative standing in a reference group. It does not mean 80% of the questions were correct or 80% of the possible points were earned.
- Reversing the percentile statement. Being at the 90th percentile means about 90% of the reference group is at or below the value, not that the value is greater than 90% of the group. About 10% are above it when there are no complications from ties.
- Leaving out the reference group. “The 80th percentile” is incomplete if the reader cannot tell whose values are being compared. Name the group, such as “the 80th percentile among students in this class.”
- Leaving out units or the variable. A percentile value of 18 is not fully interpreted until you say what 18 measures. Write “18 minutes” for a wait-time percentile, not merely “18.”
For full credit, a careful AP response identifies the statistic and connects it to its meaning in context. For a mean or median, name the group, variable, and units. For a percentile, state the value and explain approximately what percentage of the specified reference group is at or below it. For a percent, identify the whole being measured.
Check Your Understanding
For each item, identify the interpretation error, if any, and write a more precise statement.
- A data set has a median of 14 minutes and a mean of 19 minutes. Someone says, “The average was 14 minutes.” Which statistic is 14 minutes, and how should the sentence be revised?
- A student earns 84 points out of 100 and is at the 75th percentile in the class. What does each number describe? Why are 84 points and the 75th percentile not interchangeable?
- In a group of 60 observations, 48 are less than or equal to a particular value. What is the percentile rank of that value under the at-or-below convention? Explain what the rank means.
- A report says the 85th percentile of delivery times is 32 minutes. Write an interpretation that names the relevant group and does not confuse the percentile with a percent.
- Explain why it can be useful to call the median a measure of typical center while still avoiding the unqualified statement “the median is the average.”