Percentiles

Learning Objectives

After completing this section, you should be able to:

1. Compute percentiles.
2. Solve application problems involving percentiles.

Finding Percentiles

There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the quart-), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root quant– means “number”).

In Mean, Median, and Mode, we defined the median as a number that is greater than no more than half of the data in a dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: The median is the value at the 50th percentile (or second quartile).

Let’s look at some examples.

Finding Percentiles

Try it.

Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22.

1. At what percentile is the value 5?
2. What value is at the 60th percentile?

Solution

Before we can answer these two questions, we must put the data in increasing order: 1, 2, 2, 5, 8, 12, 15, 16, 20, 22.

1. There are three values (1, 2, and 2) in the set that are less than 5, and there are ten values in the set. Thus, 5 is at the \(\frac{3}{10}\times 100=30\text{th}\) percentile.
2. To find the value at the 60th percentile, we note that there are ten data values, and 60% of ten is six. Thus, the number we want is greater than exactly six of the data values. Thus, the 60th percentile is 15.

So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute something involving percentiles without technology , use the technique we used in the previous example. In all other cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” definitions).

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

• The percentile rank of a data value is the percentage of all values in the dataset that are less than or equal to the given value.

Practice (3)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22.

   1. At what percentile is the value 5?
   2. What value is at the 60th percentile?
   Jawaabta muuji

   Before we can answer these two questions, we must put the data in increasing order: 1, 2, 2, 5, 8, 12, 15, 16, 20, 22.

   1. There are three values (1, 2, and 2) in the set that are less than 5, and there are ten values in the set. Thus, 5 is at the \(\frac{3}{10}\times 100=30\text{th}\) percentile.
   2. To find the value at the 60th percentile, we note that there are ten data values, and 60% of ten is six. Thus, the number we want is greater than exactly six of the data values. Thus, the 60th percentile is 15.

2. The data in “AvgSAT” contains the average SAT score for students attending every institution of higher learning in the US for which data is available.

   1. What score is at the 3rd quartile?
   2. What score is at the 40th percentile?
   3. At what percentile is Albion College in Michigan (average SAT: 1132)?
   4. At what percentile is Oregon State University (average SAT: 1205)?
   Jawaabta muuji

   1. The 3rd quartile is the 75th percentile, so we’ll use the PERCENTILE function. Click on an empty cell, and type “=PERCENTILE(“. Next, enter the data: click on the letter at the top of the column containing the average SAT scores. Then, tell the function which percentile we want; it needs to be entered as a decimal. So, type a comma (to separate the two pieces of information we’re giving this function), then type “0.75” and close the parentheses with “)”. The result should look like this (assuming the data are in column C): “=PERCENTILE(C:C, 0.75)”. When you hit the enter key, the formula will be replaced with the average SAT score at the 75th percentile: 1199.
   2. Using the PERCENTILE function, we find that an average SAT of 1100 is at the 40th percentile.
   3. Since we want to know the percentile for a particular score, we’ll use the PERCENTRANK function. Like the PERCENTILE function, we need to give PERCENTRANK two pieces of information: the data, and the value we care about. So, click on an empty cell, type “=PERCENTRANK(“, and then click on the letter at the top of the column containing the data. Next, type a comma and then the value we want to find the percentile for; in this case, we’ll type “, 1132”. Finally, close the parentheses with “)” and hit the enter key. The formula will be replaced with the information we want: an average SAT of 1132 is at the 54th percentile.
   4. Using the PERCENTRANK function, an average SAT of 1205 is at the 76.1th percentile.

3. The dataset "InState" contains the in-state tuitions of every college and university in the country that reported that data to the Department of Education. Use that dataset to answer these questions.

   1. What tuition is at the second quintile?
   2. What tuition is at the 95th percentile?
   3. At what percentile is Walla Walla University in Washington (in-state tuition: $28,035)?
   4. At what percentile is the College of Saint Mary in Nebraska (in-state tuition: $20,350)?
   Jawaabta muuji

   1. The second quintile is the 40th percentile; using PERCENTILE in Google Sheets, we get $8,400.
   2. Using PERCENTILE again, we get $44,866.
   3. Using PERCENTRANK, we get the 81.6th percentile.
   4. Using PERCENTRANK, we get the 74.8th percentile.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\approx

approximately equal
Equal to the precision shown, not exactly.

n!

factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\bar{x},\ \mu

sample mean, population mean
Average of the data; average of the whole population.

\sigma,\ s,\ \sigma^2

standard deviation, sample s.d., variance
Typical distance from the mean; its square.

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

E[X],\ \operatorname{Var}(X)

expected value, variance
Probability-weighted average of X; its spread.

N(\mu, \sigma^2),\ z

normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.

How to: Percentiles

1. Compute percentiles.
2. Solve application problems involving percentiles.
3. At what percentile is the value 5?
4. What value is at the 60th percentile?
5. There are three values (1, 2, and 2) in the set that are less than 5, and there are ten values in the set. Thus, 5 is at the
6. To find the value at the 60th percentile, we note that there are ten data values, and 60% of ten is six. Thus, the number we want is greater than exactly six of the data values. Thus, the 60th percentile is 15.
7. What score is at the 3rd quartile?
8. What score is at the 40th percentile?

Questions people ask

Mean or median — which should I use?

Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree — the gap is itself information.

What does a p-value actually say?

The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.

Why divide by n − 1 for the sample variance?

The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.

Ku day inaad ku

⌨ Type ✎ Draw

Su'aalkaaga xisaabta

Xalin

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

In ka badan Statistics & Probability

Sampling and dataDescribing data with graphsMean, median and modeProbabilityCounting: permutations and combinationsDiscrete random variablesContinuous random variablesThe normal distributionThe central limit theoremConfidence intervalsHypothesis testingComparing two samplesChi-square testsLinear regression and correlation