A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10^th 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 25^th 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 60^th 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 50^th percentile (or second quartile).

Let’s look at some examples.

Quartiles are special percentiles. The first quartile, [latex]Q_1[/latex], is the same as the 25^th percentile, and the third quartile, [latex]Q_3[/latex], is the same as the 75^th percentile. The median, [latex]M[/latex], is called both the second quartile and the 50^th percentile.

Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, [latex]Q_1[/latex], is the middle value of the lower half of the data, and the third quartile, [latex]Q_3[/latex], is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5

The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as [latex]\frac{6.8+7.2}{2}[/latex] and equals 7.

So, the median, or second quartile ([latex]Q_2[/latex]), is 7.

The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ([latex]Q_1[/latex]) will be the middle value, or 2. Using the same procedure, we can see that the median of the upper half, or the third quartile ([latex]Q_3[/latex]) will be the middle value of the upper half, or 9.

The quartiles are illustrated below:

The interquartile range is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile ([latex]Q_3[/latex]) and the first quartile ([latex]Q_1[/latex])

[latex]\text{IQR}=Q_3-Q_1[/latex]. The [latex]\text{IQR}[/latex] for this data set is calculated as [latex]9 - 2[/latex], or 7.

The [latex]\text{IQR}[/latex] can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than [latex]1.5 \times \text{IQR}[/latex] below the first quartile or more than [latex]1.5 \times \text{IQR}[/latex] above the third quartile. Potential outliers always require further investigation.

Note: A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality, or they may be a key to understanding the data.

In the example above, you just saw the calculation of the median, first quartile, and third quartile. These three values are part of the five number summary. The other two values are the minimum value (or min) and the maximum value (or max). The five number summary is used to create a box plot.

A Formula for Finding the [latex]k^{th}[/latex] Percentile

If you were to do a little research, you would find several formulas for calculating the [latex]k^{th}[/latex] percentile. Here is one of them.

[latex]k = \text{the } k^{th} \text{ percentile. It may or may not be part of the data.}[/latex]

[latex]i = \text{the index (ranking or position of a data value)}[/latex]

[latex]n = \text{the total number of data}[/latex]

• Order the data from smallest to largest.
• Calculate [latex]i=\frac{k}{100}(n+1)[/latex].
• If [latex]i[/latex] is an integer, then the [latex]k^{th}[/latex] percentile is the data value in the [latex]i^{th}[/latex] position in the ordered set of data.
• If [latex]i[/latex] is not an integer, then round [latex]i[/latex] up and round [latex]i[/latex] down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example.

A Formula for Finding the Percentile of a Value in a Data Set

• Order the data from smallest to largest.
• [latex]x =[/latex] the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.
• [latex]y =[/latex]the number of data values equal to the data value for which you want to find the percentile.
• [latex]n =[/latex] the total number of data.
• Calculate [latex]\frac{x+.5y}{n}(100)[/latex]. Then round to the nearest integer.

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the [latex]p^{th}[/latex] percentile. For example, 15 percent of data values are less than or equal to the 15^th percentile.

• Low percentiles always correspond to lower data values.
• High percentiles always correspond to higher data values.

A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person’s height doesn’t carry any value judgment.

Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.

Example 7

A middle school is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown:

0 minutes, 40 minutes, 60 minutes, 30 minutes, 60 minutes,

10 minutes, 45 minutes, 30 minutes, 300 minutes, 90 minutes,

30 minutes, 120 minutes, 60 minutes, 0 minutes, 20 minutes

Find the five values that make up the five number summary.

[latex]Min = 0[/latex]

[latex]Q_1 = 20[/latex]

[latex]Med = 40[/latex]

[latex]Q_3 = 60[/latex]

[latex]Max = 300[/latex]

Listing the data in ascending order gives the following:

The minimum value is 0.

The maximum value is 300.

Since there are an odd number of data values, the median is the middle value of this data set as it is arranged in ascending order, or 40.

The first quartile is the median of the lower half of the scores and does not include the median. The lower half has seven data values; the median of the lower half will equal the middle value of the lower half, or 20.

The third quartile is the median of the upper half of the scores and does not include the median. The upper half also has seven data values; so the median of the upper half will equal the middle value of the upper half, or 60.

If you were the principal, would you be justified in purchasing new fitness equipment? Since 75 percent of the students exercise for 60 minutes or less daily, and since the [latex]\text{IQR}[/latex] is 40 minutes ([latex]60 – 20 = 40[/latex]), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.

However, the principal needs to be careful. The value 300 appears to be a potential outlier.

[latex]Q_3 + 1.5(\text{IQR}) = 60 + (1.5)(40) = 120[/latex].

The value 300 is greater than 120, so it is a potential outlier. If we delete it and calculate the five values, we get the following values:

• [latex]Min = 0[/latex]
• [latex]Q_1 = 20[/latex]
• [latex]Q_3 = 60[/latex]
• [latex]Max = 120[/latex]

We still have 75 percent of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample, and the principal should survey more students to be sure of his survey results.