Section 12.5 Measures of Position Math in Our World

Learning Objectives Compute the percentile rank for a data value. Find a data value corresponding to a given percentile. Use percentile rank to compare values from different data sets. Compute quartiles for a data set.

Percentile A percentile, or percentile rank, of a data value indicates the percent of data values in a set that are below that particular value.

EXAMPLE 1 Finding the Percentile Rank of a Data Value Suppose you score 77 on a test in a class of 10 people, with the 10 scores listed below. What was your percentile rank? 93 82 64 75 98 52 77 88 90 71

EXAMPLE 1 Finding the Range of a Data Set SOLUTION Step 1 Arrange the scores in order. 52 64 71 75 77 82 88 90 93 98 Step 2 Find the number of data values below 77. There are 4 values below 77. Step 3 Divide the number below the score by the total number of data values and change the answer to a percent. A test score of 77 is equivalent to the 40th percentile.

The number of words in each of the last 10 presidential inaugural addresses is listed below. Find the length that corresponds to the 30th percentile. 2,406 2,073 1,571 2,170 1,507 2,283 2,546 2,463 1,087 1,668 EXAMPLE 2 Finding a Data Value Corresponding to a Given Percentile

EXAMPLE 2 Finding a Data Value Corresponding to a Given Percentile SOLUTION Step 1 We are asked to find the number on the list that has 30% of the numbers below it. There are 10 numbers, and 30% of 10 is 3. Step 2 Arrange the data in order from smallest to largest, and find the value that has 3 values below it. 1,087 1,507 1,571 1,668 2,073 2,170 2,283 2,406 2,463 2,546 The 30th percentile is the speech that consisted of 1,668 words.

Two students are competing for one remaining spot in a law school class. Miguel ranked 51st in a graduating class of 1,700, while Dustin ranked 27th in a class of 540. Which student’s position was higher in his class? EXAMPLE 3 Using Percentiles to Compare Data from Different Sets

EXAMPLE 3 Interpreting Standard Deviation This is an ideal application of percentile rank. Miguel ranked 51st out of 1,700, so there were 1,700 – 51 = 1,649 students ranked below him. His percentile rank is SOLUTION Dustin had 540 – 27 = 513 students ranked below him, so his percentile rank is Both are excellent students, but Miguel’s ranking is higher.

Percentile A quartile divides a data set into quarters. The second quartile is the same as the median, and divides a data set into an upper half and a lower half. The first quartile is the median of the lower half, and the third quartile is the median of the upper half. We use the symbols Q1, Q2, and Q3 for the first, second, and third quartiles respectively.

Find Q1, Q2, and Q3 for the number of aircraft stolen during a recent 8-year period. 14 11 20 21 42 24 36 35 Source: USA Today EXAMPLE 4 Finding Quartiles for a Data Set Step 1 Arrange the data in order. 11 14 20 21 24 35 36 42 Step 2 Find the median. This is Q2. 11 14 20 21 24 35 36 42 ↑ Q2 = 22.5 SOLUTION

EXAMPLE 4 Finding Quartiles for a Data Set Step 3 Find the median of the data values less than Q2. This is Q1. 11 14 20 21 ↑ Q1 = 17 SOLUTION In summary, Q1 = 17, Q2 = 22.5, and Q3 = 35.5. Step 4 Find the median of the data values above Q2. This is Q3. 24 35 36 42 ↑ Q3 = 35.5