Chapter 8 Statistics
8.5 Percentiles
Learning Objectives
By the end of this section, you will be able to:
- Compute percentiles
- Solve application problems involving percentiles
A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is [latex]\frac{223}{235} \approx 95\%[/latex] of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that are below a given number can help us evaluate differences between individuals in separate populations. These proportions are called percentiles. If [latex]p\%[/latex] of the values in a dataset are less than a number [latex]n[/latex], then we say that [latex]n[/latex] is at the [latex]p\text{th}[/latex] percentile.
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 Section 8.3, 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.
Example 1
Consider the dataset 5, 8, 12, 1, 2, 16, 2, 15, 20, 22.
a) At what percentile is the value 5?
Before we can answer any questions, we must put the data in increasing order: 1, 2, 2, 5, 8, 12, 15, 16, 20, 22.
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 [latex]\frac{3}{10} \cdot 100 = 30\text{th}[/latex] percentile.
b) What value is at the 60th percentile?
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.
Exercise 1
a) What value is at the 80th percentile?
b) At what percentile is the value 12?
Solution
a) 15
b) 70th percentile
Quartiles are special percentiles. The first quartile, [latex]Q_1[/la