Section 2.5 Measures of Position Larson/Farber 4th ed. 1

Section 2.5 Objectives Determine the quartiles of a data set Determine the interquartile range of a data set Create a box-and-whisker plot Interpret other fractiles such as percentiles Determine and interpret the standard score (z-score) Larson/Farber 4th ed. 2

Quartiles Fractiles are numbers that partition (divide) an ordered data set into equal parts. Quartiles approximately divide an ordered data set into four equal parts.  First quartile, Q 1 : About one quarter of the data fall on or below Q 1.  Second quartile, Q 2 : About one half of the data fall on or below Q 2 (median).  Third quartile, Q 3 : About three quarters of the data fall on or below Q 3. Larson/Farber 4th ed. 3

Example: Finding Quartiles The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores Larson/Farber 4th ed. 4 Solution: Q 2 divides the data set into two halves Q2Q2 Lower half Upper half

Solution: Finding Quartiles The first and third quartiles are the medians of the lower and upper halves of the data set Larson/Farber 4th ed. 5 Q2Q2 Lower half Upper half Q1Q1 Q3Q3 About one fourth of the employees scored 10 or less, about one half scored 15 or less; and about three fourths scored 18 or less.

Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q 3 – Q 1 Larson/Farber 4th ed. 6

Example: Finding the Interquartile Range Find the interquartile range of the test scores. Recall Q 1 = 10, Q 2 = 15, and Q 3 = 18 Larson/Farber 4th ed. 7 Solution: IQR = Q 3 – Q 1 = 18 – 10 = 8 The test scores in the middle portion of the data set vary by at most 8 points.

Box-and-Whisker Plot Box-and-whisker plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary):  Minimum entry  First quartile Q 1  Median Q 2  Third quartile Q 3  Maximum entry Larson/Farber 4th ed. 8

Drawing a Box-and-Whisker Plot 1.Find the five-number summary of the data set. 2.Construct a horizontal scale that spans the range of the data. 3.Plot the five numbers above the horizontal scale. 4.Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q 2. 5.Draw whiskers from the box to the minimum and maximum entries. Larson/Farber 4th ed. 9 Whisker Maximum entry Minimum entry Box Median, Q 2 Q3Q3 Q1Q1

Example: Drawing a Box-and-Whisker Plot Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q 1 = 10 Q 2 = 15 Q 3 = 18 Max = 37 Larson/Farber 4th ed Solution: About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.

Percentiles and Other Fractiles FractilesSummarySymbols QuartilesDivides data into 4 equal parts Q 1, Q 2, Q 3 DecilesDivides data into 10 equal parts D 1, D 2, D 3,…, D 9 PercentilesDivides data into 100 equal parts P 1, P 2, P 3,…, P 99 Larson/Farber 4th ed. 11

Example: Interpreting Percentiles The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 72 nd percentile? How should you interpret this? (Source: College Board Online) Larson/Farber 4th ed. 12

Solution: Interpreting Percentiles The 72 nd percentile corresponds to a test score of This means that 72% of the students had an SAT score of 1700 or less. Larson/Farber 4th ed. 13

The Standard Score Standard Score (z-score) Represents the number of standard deviations a given value x falls from the mean μ. Larson/Farber 4th ed. 14

Example: Comparing z-Scores from Different Data Sets In 2007, Forest Whitaker won the Best Actor Oscar at age 45 for his role in the movie The Last King of Scotland. Helen Mirren won the Best Actress Oscar at age 61 for her role in The Queen. The mean age of all best actor winners is 43.7, with a standard deviation of 8.8. The mean age of all best actress winners is 36, with a standard deviation of Find the z-score that corresponds to the age for each actor or actress. Then compare your results. Larson/Farber 4th ed. 15

Solution: Comparing z-Scores from Different Data Sets Larson/Farber 4th ed. 16 Forest Whitaker Helen Mirren 0.15 standard deviations above the mean 2.17 standard deviations above the mean

Solution: Comparing z-Scores from Different Data Sets Larson/Farber 4th ed. 17 The z-score corresponding to the age of Helen Mirren is more than two standard deviations from the mean, so it is considered unusual. Compared to other Best Actress winners, she is relatively older, whereas the age of Forest Whitaker is only slightly higher than the average age of other Best Actor winners. z = 0.15z = 2.17

Section 2.5 Summary Determined the quartiles of a data set Determined the interquartile range of a data set Created a box-and-whisker plot Interpreted other fractiles such as percentiles Determined and interpreted the standard score (z-score) Larson/Farber 4th ed. 18

Homework p 109: 1-5, 11, 12, 21, Larson/Farber 4th ed. 19