Measures of Position

Larson/Farber 4th ed.

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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.

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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, Q1: About one quarter of the data fall on or below Q1.  Second quartile, Q2: About one half of the data fall on or below Q2 (median).  Third quartile, Q3: About three quarters of the data fall on or below Q3. Larson/Farber 4th ed.

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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. 13 9 18 15 14 21 7 10 11 20 5 18 37 16 17 Solution: â&#x20AC;˘ Q2 divides the data set into two halves. Lower half

Upper half

5 7 9 10 11 13 14 15 16 17 18 18 20 21 37

Q2 Larson/Farber 4th ed.

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Solution: Finding Quartiles â&#x20AC;˘ The first and third quartiles are the medians of the lower and upper halves of the data set. Lower half

Upper half

5 7 9 10 11 13 14 15 16 17 18 18 20 21 37

Q1

Q2

Q3

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. Larson/Farber 4th ed.

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Interquartile Range Interquartile Range (IQR) • The difference between the third and first quartiles. • IQR = Q3 – Q1

Larson/Farber 4th ed.

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Example: Finding the Interquartile Range Find the interquartile range of the test scores. Recall Q1 = 10, Q2 = 15, and Q3 = 18 Solution: • IQR = Q3 – Q1 = 18 – 10 = 8 The test scores in the middle portion of the data set vary by at most 8 points.

Larson/Farber 4th ed.

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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 Q1  Median Q2  Third quartile Q3  Maximum entry Larson/Farber 4th ed.

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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 Q1 to Q3 and draw a vertical line in the box at Q2. 5. Draw whiskers from the box to the minimum and maximum entries. Box Whisker Minimum entry Larson/Farber 4th ed.

Whisker

Q1

Median, Q2

Q3

Maximum entry

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Example: Drawing a Box-and-Whisker Plot Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37 Solution: 5

10

15

18

37

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. Larson/Farber 4th ed.

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Percentiles and Other Fractiles Fractiles Quartiles Deciles Percentiles

Larson/Farber 4th ed.

Summary Divides data into 4 equal parts Divides data into 10 equal parts Divides data into 100 equal parts

Symbols Q1, Q2, Q3

D1, D2, D3,â&#x20AC;Ś, D9 P1, P2, P3,â&#x20AC;Ś, P99

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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 72nd percentile? How should you interpret this? (Source: College Board Online)

Larson/Farber 4th ed.

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Solution: Interpreting Percentiles

The 72nd percentile corresponds to a test score of 1700. This means that 72% of the students had an SAT score of 1700 or less.

Larson/Farber 4th ed.

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The Standard Score Standard Score (z-score) • Represents the number of standard deviations a given value x falls from the mean μ.

value - mean x  • z standard deviation 

Larson/Farber 4th ed.

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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 11.5. Find the z-score that corresponds to the age for each actor or actress. Then compare your results.

Larson/Farber 4th ed.

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Solution: Comparing z-Scores from Different Data Sets • Forest Whitaker

z

x

• Helen Mirren

z

Larson/Farber 4th ed.

x

45  43.7   0.15 8.8

0.15 standard deviations above the mean

61  36   2.17 11.5

2.17 standard deviations above the mean

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Solution: Comparing z-Scores from Different Data Sets

z = 0.15

z = 2.17

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. Larson/Farber 4th ed.

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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.

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