From a Normal-Model Value to Its Percentile
In Probability Between Two Values on a Normal Curve, you found the area between two values. A percentile question uses one value as a cutoff: it asks what proportion of a normal model lies at or below that value. The answer describes the value’s relative position in the distribution.
For example, if a test score is at the 80th percentile in a normal model, about 80% of modeled scores are at or below that score. The percentile is not the score itself, and it does not say that the score is 80 points. The remaining 20% of modeled scores lie above it.
If \(X\sim N(\mu,\sigma)\), the area to the left of \(x\) is \(P(X\leq x)\). Multiply this probability by 100 to express it as a percentile. A result of 0.80, for instance, corresponds to the 80th percentile.
This is a left-tail calculation. As in Finding Areas Above and Below a Value, you can use \(\operatorname{normalcdf}\) with a very low lower bound and \(x\) as the upper bound. Or, as in Calculating a z-Score, standardize \(x\) and use Table A to find the area to its left.
The standard normal table reports a cumulative area to the left of a z-score. Once you find that area, convert it to a percentage by multiplying by 100. Keep the probability as a decimal when using the calculator, then state the percentile as a percentage in your interpretation.
Because a normal distribution is continuous, the probability of being exactly equal to a particular value is 0. Thus, for this model, using “below” or “at or below” gives the same area. The usual percentile interpretation uses “at or below.”
A Routine for Finding a Percentile
Name the measurement \(X\), its units, and the normal model \(N(\mu,\sigma)\).
Call the value of interest \(x\). Make sure \(x\), the mean, and the standard deviation use the same units.
Use \(\operatorname{normalcdf}(-1E99,x,\mu,\sigma)\), or calculate the z-score and use Table A.
Multiply the area by 100, then state what percentage of modeled values are at or below \(x\).
A quick check is the value’s position relative to the mean. A value below the mean should have a percentile below 50; the mean itself is at the 50th percentile; and a value above the mean should have a percentile above 50. The farther a value is into a tail, the closer its percentile is to 0 or 100.
Worked Example: A Student’s Standardized Test Score
Worked Example: A Student’s Standardized Test Score
In a fictional school district, standardized test scores are modeled as normal with mean 70 points and standard deviation 8 points. Find the percentile rank of a score of 82 points and interpret it.
State. Let \(X\) be a student’s test score, in points. The model is \(X\sim N(70,8)\), and the value of interest is \(x=82\).
Plan. The score is above the mean, so its percentile should be above 50. Find the area to the left of 82 using the normal model, then convert that decimal area to a percentage. The score, mean, and standard deviation are all measured in points.
Do. Using the calculator:
Check by standardizing the score: \(z=(82-70)/8=1.5\). Table A gives a cumulative area of about 0.9332 to the left of \(z=1.50\). Converting the area to a percentage gives \(0.9332(100)=93.32\%\), rounded to two decimal places.
Conclude. A score of 82 points is at about the 93.32nd percentile in this model. About 93.32% of modeled student scores are at or below 82 points, and about 6.68% are above 82 points.
Worked Example: Daily Commute Times
Worked Example: Daily Commute Times
For a fictional group of commuters, daily one-way commute times are modeled as normal with mean 24 minutes and standard deviation 5 minutes. Find the percentile rank of a commute time of 18 minutes.
State. Let \(X\) be a commuter’s one-way travel time, in minutes. The model is \(X\sim N(24,5)\), and the value of interest is \(x=18\).
Plan. Since 18 minutes is below the mean of 24 minutes, the percentile should be below 50. Find the area to the left of 18 minutes, either directly with normalcdf or by using the corresponding z-score and Table A.
Do. The left-tail calculation is:
For a check, \(z=(18-24)/5=-1.2\). Table A gives a left area of about 0.1151 at \(z=-1.20\). Multiplying by 100 gives \(0.1151(100)=11.51\%\), rounded to two decimal places.
Conclude. A commute time of 18 minutes is at about the 11.51st percentile in this model. About 11.51% of modeled commute times are at or below 18 minutes. This is a low percentile because 18 minutes is below the model’s mean.
Worked Example: Seedling Heights
Worked Example: Seedling Heights
A fictional greenhouse models the heights of a particular kind of seedling as normal with mean 30 centimeters and standard deviation 4 centimeters. Find the percentile rank of a seedling that is 35 centimeters tall.
State. Let \(X\) be a seedling’s height, in centimeters. The model is \(X\sim N(30,4)\), and the value of interest is \(x=35\).
Plan. Because 35 centimeters is above the mean, its percentile should exceed 50. Find the area to the left of 35 centimeters under the stated normal model, then report the area as a percentage.
Do. The calculator command is:
To check, standardize the height: \(z=(35-30)/4=1.25\). Table A gives a cumulative area of about 0.8944 to the left of \(z=1.25\). Therefore, \(0.8944(100)=89.44\%\), rounded to two decimal places.
Conclude. A height of 35 centimeters is at about the 89.44th percentile in this model. About 89.44% of modeled seedling heights are at or below 35 centimeters, while about \(100\%-89.44\%=10.56\%\) are above it.
Reading and Communicating Percentiles
A percentile is a relative position within a distribution, not a distance from the mean. The z-score gives the distance from the mean in standard-deviation units; the percentile gives the percentage of the distribution at or below the value. In the test-score example, \(z=1.5\) says that 82 points is 1.5 standard deviations above the mean, while the 93.32nd percentile says that about 93.32% of modeled scores are at or below 82.
The same percentile can correspond to different actual values in different distributions. A value’s percentile depends on both the value and the model’s mean and standard deviation. Always include the model or group being described so the comparison is clear.
Percentiles describe the distribution under the model. They do not guarantee that precisely that percentage of a particular small group will fall below the value. If the model is only an approximation for real measurements, the percentile is an approximate model-based description as well.
Common Mistakes and AP Exam Tips
- Finding the area on the wrong side. A percentile rank asks for the area at or below the value, so use the left tail. The area above the value is the complement, not its percentile rank.
- Reporting a decimal without converting it. An area of 0.8944 corresponds to the 89.44th percentile, not the 0.8944th percentile.
- Calling the percentile a measurement. A score of 82 points can be at the 93rd percentile; the percentile is not 93 points. Keep the score’s units separate from the percentage rank.
- Confusing a percentile with a z-score. A z-score reports standard-deviation units from the mean. A percentile reports the percentage of the model at or below the value.
- Assuming every above-average value is at the same percentile. A value’s percentile depends on how far it is from the mean relative to the standard deviation, not just on whether it exceeds the mean.
- Leaving the context out of the conclusion. A strong interpretation names the value and measurement, identifies the modeled group, and states the percentage at or below that value.
For full-credit communication, define \(X\) in context, state the normal model and its units, show the left-tail calculation, convert the area to a percentage, and finish with a contextual interpretation. If you use a z-score and Table A, show the standardization and make clear that the table entry is the area to the left.
Key Takeaway
To find the percentile rank of a value in a normal model, calculate the area to its left and multiply by 100. Use normalcdf with the value as the upper bound, or standardize the value and look up its cumulative area in Table A. Interpret the result as the percentage of modeled values at or below the given value.
Check Your Understanding
For each question, identify the left-tail area to find and explain what the percentile means in context.
- A fictional batch of oranges has weights modeled as \(N(180,12)\), in grams. Write the normalcdf command for finding the percentile rank of an orange weighing 195 grams.
- For \(X\sim N(50,6)\), a value of 44 has z-score \(-1\). Table A gives a left area of about 0.1587. What is its percentile rank, and what does that mean?
- A measurement is at the 20th percentile in a stated normal model. What percentage of modeled measurements are at or below it, and what percentage are above it?
- Explain the difference between saying a value has a z-score of 1.2 and saying it is at the 88th percentile.
- A student finds a left-tail area of 0.9750 and reports that the value is at the 0.9750th percentile. Identify the conversion error and give the correct percentile rank.