Читать книгу Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP - Bhisham C. Gupta, Irwin Guttman - Страница 92

Percentiles divide the data into one hundred equal parts; each part contains at the most 1% of the data and is 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 so that at most 50% of the data fall below the median and at most 50% 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:

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

2 Step 2. Find the rank of the pth percentile (), which is given by(2.7.1)

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

We illustrate this procedure with the following example.

Example 2.7.1 (Engineers' salaries) The following data give the salaries (in thousands of dollars) of 15 engineers in a corporation:

62 48 52 63 85 51 95 76 72 51 69 73 58 55 54

1 Find the 70th percentile for these data.

2 Find the percentile corresponding to the salary of $60,000.

Solution: (a) We proceed as follows:

1 Step 1. Write the data values in the ascending order and rank them from 1 to 15.Salaries485151525455586263697273768595Ranks123456789101112131415

2 Step 2. Find the rank of the 70th percentile, which from (2.7.1) is given by

3 Step 3. Find the data value that corresponds to the ranks 11 and 12, which in this example are 72 and 73, respectively. Then, the 70th percentile is given byThus, the 70th percentile of the salary data is $72,200.That is, at most 70% of the engineers are making less than $72,200 and at most 30% of the engineers are making more than $72,200.

(b) Now we want to find the percentile corresponding to a given value . This can be done by using the following formula:

(2.7.2)

Thus, the percentile corresponding to the salary of $60,000 is

Hence, the engineer who makes a salary of $60,000 is at the 44th percentile. In other words, at most 44% of the engineers are making less than $60,000, or at most 56% are making more than $60,000.