2.2 Measures of the Location of the Data

The common measures of location are quartiles and percentiles.

Quartiles divide an ordered data set into four equal parts. The three quartiles of a data set are labeled as Q1, Q2, and Q3.

• About one-fourth of the data falls on or below the first quartile Q1.
• About one-half of the data falls on or below the second quartile Q2.
• About three-fourths of the data falls on or below the first quartile Q3.

In the same way, percentiles divide a data set into 100 equal parts.

To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the 90^th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75^th percentile. That translates into a score of at least 1220.

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.

The median is a number that measures the "center" of the data. You can think of the median as the "middle value," but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data.
1; 11.5; 6; 7.2; 4; 8; 9; 10; 6.8; 8.3; 2; 2; 10; 1

Ordered from smallest to largest:
1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5

Since there are 14 observations, the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two.

The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.

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, Q₁, is the middle value of the lower half of the data, and the third quartile, Q₃, 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 median or second quartile is seven. The lower half of the data are 1, 1, 2, 2, 4, 6, 6.8. The middle value of the lower half is two.
1; 1; 2; 2; 4; 6; 6.8

The number two, which is part of the data, is the first quartile. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.

The upper half of the data is 7.2, 8, 8.3, 9, 10, 10, 11.5. The middle value of the upper half is nine.

The third quartile, Q3, is nine. Three-fourths (75%) of the ordered data set are less than nine. One-fourth (25%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.

The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile (Q₃) and the first quartile (Q₁).

IQR = Q₃ – Q₁

The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile. Potential outliers always require further investigation.

A Formula for Finding the kth Percentile

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

k = the k^th percentile. It may or may not be part of the data.

i = the index (ranking or position of a data value)

n = the total number of data points, or observations

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

Example 2.17

Problem

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

1. Find the 70^th percentile.
2. Find the 83^rd percentile.

Solution

1. • k = 70
   • i = the index
   • n = 29
   i = $\frac{k}{100}$ (n + 1) = ($\frac{70}{100}$)(29 + 1) = 21. Twenty-one is an integer, and the data value in the 21^st position in the ordered data set is 64. The 70^th percentile is 64 years.
2. • k = 83^rd percentile
   • i = the index
   • n = 29
   i  = $\frac{k}{100}$ (n + 1) = ($\frac{83}{100}$)(29 + 1) = 24.9, which is NOT an integer. Round it down to 24 and up to 25. The age in the 24^th position is 71 and the age in the 25^th position is 72. Average 71 and 72. The 83^rd percentile is 71.5 years.

Try It 2.17

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.

18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
Calculate the 20^th percentile and the 55^th percentile.

NOTE

You can calculate percentiles using calculators and computers. There are a variety of online calculators.

Example 2.18

Problem

Using Table 2.24:

1. Find the 80^th percentile.
2. Find the 90^th percentile.
3. Find the first quartile. What is another name for the first quartile?

Solution

Using the data from the frequency table, we have:

1. Notice there are 50 data values in the table, so n = 50.
   Calculate the index i as follows:
   $i=\frac{80}{100}(50+1)=40.8$
   Since i = 40.8, calculate the mean of the 40^th and 41^st data values. The 40^th data value is 8, the 41^st data value is 9, and the mean of these two data values is 8.5. Thus, the 80^th percentile is 8.5.
2. Calculate the index i as follows:
   $i=\frac{90}{100}(50+1)=45.9$
   Since i = 45.9, calculate the mean of the 45^th and 46^th data values. The 45^th data value is 9, the 46^th data value is 9, and the mean of these two data values is 9. Thus, the 90^th percentile is 9.
3. Another name for the first quartile is the 25^th percentile. Proceed to calculate the 25^th percentile:
   Calculate the index i as follows:
   $i=\frac{25}{100}(50+1)=12.75$
   Since i = 12.75, calculate the mean of the 12^th and 13^th data values. The 12^th data value is 6, the 13^th data value is 6, and the mean of these two data values is 6. Thus, the first quartile is 6.

Try It 2.18

Refer to the Table 2.25. Find the third quartile. What is another name for the third quartile?

Collaborative Exercise

Your instructor or a member of the class will ask everyone in class how many sweaters they own. Answer the following questions:

1. How many students were surveyed?
2. What kind of sampling did you do?
3. Construct two different histograms. For each, starting value = _____ ending value = ____.
4. Find the median, first quartile, and third quartile.
5. Construct a table of the data to find the following:
   1. the 10^th percentile
   2. the 70^th percentile
   3. the percent of students who own less than four sweaters

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

• Order the data from smallest to largest.
• x = 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.
• y = the number of data values equal to the data value for which you want to find the percentile.
• n = the total number of data.
• Calculate $\frac{x+0.5y}{n}$(100). Then round to the nearest integer.

Example 2.19

Problem

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

1. Find the percentile for 58.
2. Find the percentile for 25.

Solution

1. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58.

   x = 18 and y = 1.$\frac{x+0.5y}{n}$(100) = $\frac{18+0.5(1)}{29}$(100) = 63.80. 58 is the 64^th percentile.

2. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25.

   x = 3 and y = 1.$\frac{x+0.5y}{n}$(100) = $\frac{3+0.5(1)}{29}$(100) = 12.07. Twenty-five is the 12^th percentile.

Try It 2.19

Listed are 30 ages for Academy Award winning best actors in order from smallest to largest.

18; 21; 22; 25; 26; 27; 29; 30; 31, 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
Find the percentiles for 47 and 31.

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 pth percentile. For example, 15% 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 applies. 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.

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