Sec 3.3 Navidi

Transcript Sec 3.3 Navidi

SECTION 3.3

MEASURES OF POSITION

Section 3.3 - Objectives

Compute and interpret z-scores Compute percentiles of a data set Compute the quartiles of a data set Compute the five-number summary for a data set Understand the effects of outliers Construct boxplots to visualize the five-number summary and outliers Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Objective 1

Compute and interpret z-scores

z-Score

Who is taller, a man 73 inches tall or a woman 68 inches tall? The obvious answer is that the man is taller. However, men are taller than women on the average. Suppose the question is asked this way: Who is taller relative to their gender, a man 73 inches tall or a woman 68 inches tall? One way to answer this question is with a

z-score

z-Score

The z-score of an individual data value tells how many standard deviations that value is from its population mean. For example, a value one standard deviation above the mean has a z-score of 1. A value two standard deviations below the mean has a z-score of –2.

Definition:

Let x be a value from a population with mean

The z-score for x is and standard deviation σ .



Example 3.22

A National Center for Health Statistics study states that the mean height for adult men in the U.S. is

= 69.4 inches, with a standard deviation of σ = 3.1 inches. The mean height for adult women is

= 63.8 inches, with a standard deviation of σ = 2.8 inches. Who is taller relative to their gender, a man 73 inches tall, or a woman 68 inches tall?

Solution

We compute the z-scores for the two heights.

Man's Height 

    3.1

 1.16

Woman's Height 

    2.8

 1.50

Taller, relative to the population of women ’ s heights.

z-Scores & The Empirical Rule

Since the z-score is the number of standard deviations from the mean, we can easily interpret the z-score for bell-shaped populations using The Empirical Rule.

When a population has a histogram that is approximately bell-shaped, then  Approximately 68% of the data will have z-scores between –1 and 1.

 Approximately 95% of the data will have z-scores between –2 and 2.

 All, or almost all of the data will have z-scores between –3 and 3.

z = –3 z = –2 z = –1 z = 1 z = 2 z = 3

Objective 2

Compute percentiles of a data set

Objective 3

Compute the quartiles of a data set

Quartiles

There are three special percentiles which divide a data set into four pieces, each of which contains approximately one quarter of the data. These values are called the

quartiles

 The first quartile, denoted Q 1 , is the 25 th Q 1 percentile. separates the lowest 25% of the data from the highest 75%.

 The second quartile, denoted Q 2 , is the 50 th Q 2 is the same as the median.

percentile. separates the lower 50% of the data from the upper 50%. Q 2  The third quartile, denoted Q 3 , is the 75 th Q 3 percentile. separates the lowest 75% of the data from the highest 25%.

Example 3.25

The following table presents the annual rainfall, in inches, in Los Angeles during the month of February from 1965 to 2006. Compute the first and third quartiles.

0.23

3.06

6.61

1.51

12.75

3.21

0.11

1.48

1.30

0.49

0.70

4.94

8.03

4.37

0.08

2.58

0.00

13.68

0.67

2.84

0.56

0.13

6.10

5.54

7.89

1.22

8.87

0.14

1.72

0.29

3.54

1.90

4.64

3.71

3.12

4.89

0.17

4.13

11.02

8.91

7.96

2.37

Solution

Step 1 : Arrange the data in increasing order.

0.00

1.48

4.64

0.08

1.51

4.89

0.11

1.72

4.94

0.13

1.90

5.54

0.14

2.37

6.10

0.17

2.58

6.61

0.23

2.84

7.89

0.29

3.06

7.96

0.49

3.12

8.03

0.56

3.21

8.87

0.67

3.54

8.91

0.70

3.71

1.22

4.13

1.30

4.37

11.02 12.75 13.68

Step 2 : There are 42 values in the data set. We compute L = (p/100)·n for both p = 25 and p = 75:

25 = (25/100)·42 = 10.5

75 = (75/100)·42 = 31.5

Step 3 : We round these values up to 11 and 32. The first quartile, Q 1 the 11 th position. The third quartile, Q see that

1 = 0.67

and

3 = 5.54

Quartiles on the TI-84 PLUS

On the TI-84 PLUS Calculator, the 1-Var Stats command, also used for means, medians, and standard deviations, will compute quartiles.

We enter the data into list L1 and run the 1-Var Stats command.

First Quartile Third Quartile

Objective 4

Compute the five-number summary for a data set

Five-Number Summary

The

five-number summary

of a data set consists of the median, the first quartile, the third quartile, the smallest value, and the largest value. These values are generally arranged in order.

Definition:

The five-number summary of a data set consists of the following quantities:

Minimum First Quartile Median Third Quartile Maximum

Example – Five-Number Summary

Consider again the Los Angeles rainfall data.

0.00

1.48

4.64

0.08

1.51

4.89

0.11

1.72

4.94

0.13

1.90

5.54

0.14

2.37

6.10

0.17

2.58

6.61

0.23

2.84

7.89

0.29

3.06

7.96

0.49

3.12

8.03

0.56

3.21

8.87

0.67

3.54

8.91

0.70

1.22

1.30

3.71

4.13

4.37

11.02 12.75 13.68

The Minimum is 0.00 and the Maximum is 13.68. The Median is easily computed as 2.95. We have already computed Q 1 = 0.67 and Q 3 = 5.54.

The five-number summary for this data set is:

0.00

0.67

2.95

5.54

13.68

Objective 5

Understand the effects of outliers

Outliers

outlier

is a value that is considerably larger or considerably smaller than most of the values in a data set. Some outliers result from errors; for example a misplaced decimal point may cause a number to be much larger or smaller than the other values in a data set.

Some outliers are correct values, and simply reflect the fact that the population contains some extreme values.

Example 3.28

The temperature in a downtown location in a certain city is measured for eight consecutive days during the summer. The readings, in degrees Fahrenheit, are 81.2, 85.6, 89.3, 91.0, 83.2, 8.45, 79.5, and 87.8. Which reading is an outlier? Is it certain that the outlier is an error, or is it possible that it is correct? Should the outlier be deleted?

Solution:

The outlier is 8.45, which is much smaller than the rest of the data. This outlier is certainly an error; it is likely that a decimal point was misplaced. The outlier should be corrected if possible, or deleted.

Interquartile Range

One method for detecting outliers involves a measure called the

Interquartile Range

Definition:

The interquartile range is found by subtracting the first quartile from the third quartile.

IQR = Q 3 – Q 1

IQR Method for Detecting Outliers

The most frequent method used to detect outliers in a data set is the

IQR Method

. The procedure for the IQR Method is: Step 1 : Find the first quartile Q 1 , and the third quartile Q 3 .

Step 2 : Compute the interquartile range: IQR = Q 3 – Q 1 .

Step 3 : Compute the outlier boundaries. These boundaries are the cutoff points for determining outliers: Lower Outlier Boundary = Q 1 – 1.5(IQR) Upper Outlier Boundary = Q 3 + 1.5(IQR) Step 4 : Any data value that is less than the lower outlier boundary or greater than the upper outlier boundary is considered to be an outlier.

Example 3.30

The following table presents the number of students absent in a middle school in northwestern Montana for each school day January 2008. Identify any outliers.

Jan. 2 Jan. 3 Jan. 4 Jan. 7 Jan. 8 Jan. 9 65 67 71 57 51 49 Jan. 10 Jan. 11 Jan. 14 Jan. 15 Jan. 16 Jan. 17 44 41 59 49 42 56 Jan. 18 Jan. 21 Jan. 22 Jan. 23 Jan. 24 Jan. 25 45 77 44 42 45 46 Jan. 28 Jan. 29 Jan. 30 Jan. 31 100 59 53 51 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution

We may use the TI-84 PLUS, or other technology, to compute the quartiles.

We see that Q 1 = 45 and Q 3 = 59.

Solution

The Interquartile Range is IQR = Q 3 – Q 1 = 59 – 45 = 14.

The outlier boundaries are: Lower Outlier Boundary = Q 1 – 1.5(IQR) = 45 – 1.5(14) = 24 Upper Outlier Boundary = Q 3 + 1.5(IQR) = 59 + 1.5(14) = 80 There are no values in the data set less than the lower boundary of 24. There is one value, 100, which is greater than the upper boundary of 80. Thus there is one outlier, 100.

Objective 6

Construct boxplots to visualize the five-number summary and outliers

Any Data Set

boxplot

is a graph that presents the five-number summary along with some additional information about a data set. There are several different kinds of boxplots. The one we describe here is sometimes called a

modified boxplot

* *

Procedure for Drawing a Boxplot

Following is the procedure for drawing a boxplot: Step 1 : Compute the first quartile, the median, and the third quartile.

Step 2 : Draw vertical lines at the first quartile, the median, and the third quartile. Draw horizontal lines between the first and third quartiles to complete the box.

Step 3 : Compute the lower and upper outlier boundaries.

Step 4 : Find the largest data value that is less than the upper outlier boundary. Draw a horizontal line from the third quartile to this value. This horizontal line is called a

whisker

Step 5 : Find the smallest data value that is greater than the lower outlier boundary. Draw a horizontal line (whisker) from the first quartile to this value.

Step 6 : Determine which values, if any, are outliers. Plot each outlier separately.

Example 3.31

The following table presents the number of students absent in a middle school in northwestern Montana for each school day January 2008. Construct a boxplot.

Solution

Step 1

1 : We may use the TI-84 PLUS, or other technology, to compute the median and quartiles. We see that Median = 51, = 45, and Q 3 = 59.

Solution

Step 2 box.

: We draw vertical lines at 45, 51, and 59, then horizontal lines to complete the Step 3 : We compute the outlier boundaries: Lower Outlier Boundary = Q 1 – 1.5(IQR) = 24 Upper Outlier Boundary = Q 3 + 1.5(IQR) = 80 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution

Step 4 : The largest data value that is less than the upper boundary is 77. We draw a horizontal line from 59 up to 77.

Solution

Step 5 : The smallest data value that is greater than the lower boundary is 41. We draw a horizontal line from 45 down to 41.

Solution

Step 6 : The data value 100 lies outside of the outlier boundaries. Therefore, 100 is an outlier. We plot this point separately.

Determining Skewness

Boxplots can help determine the skewness of a data set.  If the median is closer to the first quartile than to the third quartile, or the upper whisker is longer than the lower whisker, the data are skewed to the right.

Determining Skewness

 If the median is closer to the third quartile than to the first quartile, or the lower whisker is longer than the upper whisker, the data are skewed to the left.

Determining Skewness

 If the median is approximately halfway between the first and third quartiles, and the two whiskers are approximately equal in length, the data are approximately symmetric Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Do You Know…

• • • • • • How to compute and interpret z-scores?

How to compute percentiles of a data set?

How to compute the quartiles of a data set?

How to compute the five-number summary for a data set?

The effects of outliers?

How to construct boxplots?

Sec 3.3 Navidi

Transcript Sec 3.3 Navidi

SECTION 3.3

MEASURES OF POSITION

Section 3.3 - Objectives

Objective 1

Compute and interpret z-scores

z-Score

z-Score

Example 3.22

Solution

z-Scores & The Empirical Rule

Objective 2

Compute percentiles of a data set

Objective 3

Compute the quartiles of a data set

Quartiles

Example 3.25

Solution

Quartiles on the TI-84 PLUS

Objective 4

Compute the five-number summary for a data set

Five-Number Summary

Example – Five-Number Summary

Objective 5

Understand the effects of outliers

Outliers

Example 3.28

Interquartile Range

IQR Method for Detecting Outliers

Example 3.30

Solution

Solution

Objective 6

Construct boxplots to visualize the five-number summary and outliers

Any Data Set

* *

Procedure for Drawing a Boxplot

Example 3.31

Solution

Solution

Solution

Solution

Solution

Determining Skewness

Determining Skewness

Determining Skewness

Do You Know…

Directory