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

Section 2.5

Measures of Position

Larson/Farber 4th ed.

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)


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.


Example: Finding Quartiles

The number of vacation days used by a sample of 20 employees in recent years. Find the first, second, and third quartiles of the test scores.

3 9 2 1 7 5 3 2 2 6 4 0 10 0 3 5 7 8 6 5


Solution:

• Q2 divides the data set into two halves.

0 0 1 2 2 2 3 3 3 4 5 5 5 6 6 7 7 8 9 10Q2

Lower half Upper half

= Median = (4 + 5) ÷ 2 = 4.5

Solution: Finding Quartiles

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

0 0 1 2 2 2 3 3 3 4 5 5 5 6 6 7 7 8 9 10


Q2

Lower half Upper half

Q1 Q3

About one fourth of the employees used 2 or less days of vacation, about one half used 4 or less; and about three fourths scored 6 or less.

Finding Quartiles

Solution: Finding Quartiles

• To calculate the Quartiles for employees vacations:

Interquartile Range

Interquartile Range (IQR)• The difference between the third and first quartiles.

• IQR = Q3 – Q1


Example: Finding the Interquartile Range

Find the interquartile range of the test scores.

Recall Q1 = 2, Q2 = 4.5, and Q3 = 6.5


Solution:

• IQR = Q3 – Q1 = 6.5 – 2 = 4.5

The days of vacation taken in the middle portion of the data set vary by at most 4.5 days.

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. 139

Drawing a Box-and-Whisker Plot

1. Find the five-number summary of the data set: Min, Q1, Q2,Q3, Max.

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.


Maximum entry

Minimum entry

BoxWhisker Whisker

Median, Q2Q3Q1

Example: Drawing a Box-and-Whisker Plot

Draw a box-and-whisker plot that represents the 20 employees taking vacation days.

Recall Min = 0 Q1 = 2 Q2 = 4.5 Q3 = 6.5 Max = 10


2

4.5

6.5

0 10

Solution:

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


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)


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.


The Standard Score

Standard Score (z-score)• Represents the number of standard deviations a given

value x falls from the mean μ.


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.


Solution: Comparing z-Scores from Different Data Sets


• 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


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

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)


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

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