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

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

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

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

Section 2.5

Measures of Position

Larson/Farber 4th ed. 1

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Section 2.5 Objectives

• Determine the quartiles of a data set

• Interpret other fractiles such as percentiles

• Determine and interpret the standard score (z-score)

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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.

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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

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Solution:

• Q2 divides the data set into two halves.

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

Q2

Lower half Upper half

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Solution: Finding Quartiles

• The first and third quartiles are the medians of the lower and upper halves of the data set.

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

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Q2

Lower half Upper half

Q1 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.

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Solution: Finding Quartiles

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 5

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

9.5 14.5 18

Q1 Q2 Q3

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Percentiles and Other Fractiles

Fractiles Summary Symbols

Quartiles Divides data into 4 equal parts

Q1, Q2, Q3

Deciles Divides data into 10 equal parts

D1, D2, D3,…, D9

Percentiles Divides data into 100 equal parts

P1, P2, P3,…, 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)

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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.

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The Standard Score

Which x has the larger percentage of its population below it?

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The Standard Score

Which x has the larger percentage of its population below it?

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The Standard ScoreTo answer this question, we need to “translate” the two xs to

the same distribution, the standard distribution.

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The Standard ScoreHow do we do the translation?

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The Standard ScoreHow do we do the translation?

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The Standard ScoreDo the same thing with the other x.

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The Standard ScoreDo the same thing with the other x.

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The Standard ScoreNote that an x value less than the mean of its distribution

will translate to a negative z score.

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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.

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value - mean

standard deviation

xz

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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.

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Solution: Comparing z-Scores from Different Data Sets

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• Forest Whitaker45 43.7

0.158.8

xz

• Helen Mirren61 36

2.1711.5

xz

0.15 standard deviations above the mean

2.17 standard deviations above the mean

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Solution: Comparing z-Scores from Different Data Sets

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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 = 0.15 z = 2.17

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Section 2.5 Summary

• Determined the quartiles of a data set

• Interpreted other fractiles such as percentiles

• Determined and interpreted the standard score(z-score)

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