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212 Part IV: Quality Assurance3. MEASURES OF PROPORTION OR PERCENTAGECalculate these measures for various datasets. (Application)Body of Knowledge IV.A.3In this section we will study measures of proportion, which divide the data intovarious parts of certain percentages or help to locate the place of any data value inthe whole data set. For example, more commonly used measures of proportion arepercentiles and quartiles. Percentiles divide the data into one hundred parts suchthat each part contains at the most one percent of the data, and quartiles divide thedata into four parts such that each part contains at the most 25 percent of the data.Then from quartiles we can derive another measure, called interquartile range,which gives the range of the middle 50 percent of the data values obtained byfirst writing the data in ascending order and then trimming 25 percent of the datavalues from each of the lower and the upper ends.PercentilesPart IV.A.3Percentiles divide the data into 100 equal parts, each part containing at the mostone percent of the data, and numbered from 1 to 99. For example, the median ofa data set is the 50th percentile, which divides the data into two equal parts suchthat at the most 50 percent of the data fall below the median and at the most 50percent of the data fall above it. The procedure for determining the percentiles issimilar to the procedure used for determining the median. We compute the percentilesas follows:Step 1. Write the data values in ascending order and rank them from1 to n.Step 2. Find the rank of the pth percentile (p = 1, 2, . . . , 99), which isgiven by:Rank of the pth percentile = p × [(n + 1)/100]Step 3. Find the data value that corresponds to the rank of the pthpercentile. We illustrate this procedure with the followingexample.In our above discussion we determined the value x of a given percentile p. Nowwe would like to find the percentile p corresponding to a given value x. This canbe done by using the following formula:p =(# of data values x)n + 1( )100 (18.11)× ( )

Chapter 18: A. Basic Statistics and Applications 213EXAMPLE 18.13The following data give the salaries (in thousands of dollars) of 15 engineers of acorporation:62, 48, 52, 63, 85, 51, 95, 76, 72, 51, 69, 73, 58, 55, 54Find the 70th percentile for these data.Solution:Write the data values in ascending order and rank them from 1 to 15, since n is equalto 15.Step 1. Salaries: 48, 51, 51, 52, 54, 55, 58, 62, 63, 69, 72, 73, 76, 85, 95Rank: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Step 2. Find the rank of the 70th percentile, which is given by 70 ×((15 + 1)/100) = 11.2Step 3. Find the data value that corresponds to the rank 11.2, which willbe the 70th percentile. From Figure 18.6, we can easily see that thevalue of the 70th percentile is given by:70th percentile = 72(.8) + 73(.2) = 72.2Thus, the 70th percentile of the salary data is $72,200.That is, at the most 70 percent of the engineers are making less than $72,200 and at themost 30 percent of the engineers are making more than $72,200.11 11.272.2 .8Figure 18.6 Salary data.1273Part IV.A.3For instance, in Example 18.13 the percentile corresponding to the salary of$60,000 isP = 7/(15 + 1)100 = 44.Thus, the engineer who makes a salary of $60,000 is at the 44th percentile. In otherwords, at most 44 percent of the engineers are making less than $60,000 or at most56 percent are making more than $60,000.QuartilesIn the previous discussion we studied the percentiles, which divide the data into100 equal parts. Some of the percentiles have special importance. These importantpercentiles are the 25th, 50th, and 75th percentiles and are known as thefirst, second, and third quartiles (denoted by Q 1 , Q 2 , and Q 3 ) respectively. These

212 Part IV: Quality Assurance

3. MEASURES OF PROPORTION OR PERCENTAGE

Calculate these measures for various data

sets. (Application)

Body of Knowledge IV.A.3

In this section we will study measures of proportion, which divide the data into

various parts of certain percentages or help to locate the place of any data value in

the whole data set. For example, more commonly used measures of proportion are

percentiles and quartiles. Percentiles divide the data into one hundred parts such

that each part contains at the most one percent of the data, and quartiles divide the

data into four parts such that each part contains at the most 25 percent of the data.

Then from quartiles we can derive another measure, called interquartile range,

which gives the range of the middle 50 percent of the data values obtained by

first writing the data in ascending order and then trimming 25 percent of the data

values from each of the lower and the upper ends.

Percentiles

Part IV.A.3

Percentiles divide the data into 100 equal parts, each part containing at the most

one percent of the data, and numbered from 1 to 99. For example, the median of

a data set is the 50th percentile, which divides the data into two equal parts such

that at the most 50 percent of the data fall below the median and at the most 50

percent of the data fall above it. The procedure for determining the percentiles is

similar to the procedure used for determining the median. We compute the percentiles

as follows:

Step 1. Write the data values in ascending order and rank them from

1 to n.

Step 2. Find the rank of the pth percentile (p = 1, 2, . . . , 99), which is

given by:

Rank of the pth percentile = p × [(n + 1)/100]

Step 3. Find the data value that corresponds to the rank of the pth

percentile. We illustrate this procedure with the following

example.

In our above discussion we determined the value x of a given percentile p. Now

we would like to find the percentile p corresponding to a given value x. This can

be done by using the following formula:

p =

(# of data values x)

n + 1

( )

100 (18.11)

× ( )

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