qarm (lecture-04 and 05)

8/10/2019 QARM (Lecture-04 and 05)

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Mean Absolute Deviation

Average of the absolute deviations from themean

5

9

16

1718

-8

-4

+3

+4+5

0

+8

+4

+3

+4+5

24

X X X

M A D X

N. . .

.

24

5

4 8

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The empirical rule approximates the variationof data in a bell-shaped distribution

Approximately 68% of the data in a bell

shaped distribution is within 1 standarddeviation of the mean or

The Empirical Rule

1

68%

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Approximately 95% of the data in a bell-shaped distributionlies within two standard deviations of the mean, or 2

Approximately 99.7% of the data in a bell-shaped distributionlies within three standard deviations of the mean, or 3

The Empirical Rule

3

99.7%95%

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Using the Empirical Rule

Suppose that the variable Math SAT scores isbell-shaped with a mean of 500 and astandard deviation of 90. Then,

68% of all test takers scored between 410 and 590(500 90).

95% of all test takers scored between 320 and 680(500 180).

99.7% of all test takers scored between 230 and770 (500 270).

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Empirical Rule

Data are normally distributed (or approximatelynormal)

1

2 3

95

99.7

68

Distance fromthe Mean Percentage of ValuesFalling Within Distance

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Measures of Variation:The Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Can be used to compare the variability of two

or more sets of data measured in different

units100%

X

SCV

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Measures of Variation:Comparing Coefficients of Variation

Stock A:

Average price last year = $=50

Standard deviation = $5

Stock B:

Average price last year = $100


Both stocks have

the same

standard

deviation, but

stock B is lessvariable relative

to its price

10%100%$50$5100%

XSCVA

5%100%$100

$5100%

X

SCVB

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Measures of Variation:Comparing Coefficients of Variation

Stock A:



Stock C:



Stock C has a

much smaller

standard

deviation but a

much highercoefficient of

variation

10%100%$50$5100%

XSCVA

25%100%$8

$2100%

X

SCVC

(continued)

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ZScore Example

The mean time to assemble aproduct is 22.5 minutes with astandard deviation of 2.5minutes.

Find the zscore for an itemthat took 20 minutes toassemble.

Find the zscore for an itemthat took 27.5 minutes toassemble.

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ZScore Example

x = 20, = 22.5 = 2.5

x 20 22.5

z = =

2.5= 1.0

x = 27.5, = 22.5 = 2.5

x 27.5 22.5

z = =

2.5

= 2.0

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Locating Extreme Outliers:

Z-Score

To compute the Z-scoreof a data value, subtract the meanand divide by the standard deviation.

The Z-score is the number of standard deviations a datavalue is from the mean.

A data value is considered an extreme outlier if its Z-scoreis less than -3.0 or greater than +3.0.

The larger the absolute value of the Z-score, the farther thedata value is from the mean.

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Locating Extreme Outliers:

Z-Score

Suppose the mean math SAT score is 490,with a standard deviation of 100.

Compute the Z-score for a test score of 620.

3.1100

130

100

490620

S

XXZ

A score of 620 is 1.3 standard deviations above the

mean and would not be considered an outlier.

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Ungrouped Versus Grouped

Data

Ungrouped data have not been summarized in any way

are also called raw data Grouped data

have been organized into a frequencydistribution

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Example of Ungrouped Data

42

30

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

54

Ages of a Sample of

Managers from

Urban Child CareCenters in the

United States

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Frequency Distribution of

Child Care Managers Ages

Class Interval Frequency

20-under 30 6

30-under 40 18

40-under 50 11

50-under 60 11

60-under 70 3

70-under 80 1

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Data Range

42

30

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

54

Smallest

Largest

Range = Largest - Smallest

= 74 - 23

= 51

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Number of Classes and Class Width

The number of classes should be between 5 and 15. Fewer than 5 classes cause excessive summarization.

More than 15 classes leave too much detail.

Class Width

Divide the range by the number of classes for an

approximate class width Round up to a convenient number

10=WidthClass

8.5=6

51

=WidthClasseApproximat

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Class Midpoint

Class Midpoint =beginning class endpoint + ending class endpoint

2

= 30 + 402

= 35

Class Midpoint = class beginning point +1

2class width

= 30 +1

210

= 35

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Relative Frequency

RelativeClass Interval Frequency Frequency

20-under 30 6 .12

30-under 40 18 .36

40-under 50 11 .22

50-under 60 11 .22

60-under 70 3 .06

70-under 80 1 .02

Total 50 1.00

6

50

18

50

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Cumulative Frequency

CumulativeClass Interval Frequency Frequency

20-under 30 6 6

30-under 40 18 24

40-under 50 11 35

50-under 60 11 46

60-under 70 3 49

70-under 80 1 50

Total 50

18 + 6

11 + 24

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Class Midpoints, Relative Frequencies,

and Cumulative Frequencies

Relative CumulativeClass Interval Frequency Midpoint Frequency Frequency

20-under 30 6 25 .12 630-under 40 18 35 .36 24

40-under 50 11 45 .22 35

50-under 60 11 55 .22 46

60-under 70 3 65 .06 49

70-under 80 1 75 .02 50

Total 50 1.00

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Cumulative Relative Frequencies

Cumulative

Relative Cumulative Relative

Class Interval Frequency Frequency Frequency Frequency

20-under 30 6 .12 6 .12

30-under 40 18 .36 24 .48

40-under 50 11 .22 35 .70

50-under 60 11 .22 46 .92

60-under 70 3 .06 49 .98

70-under 80 1 .02 50 1.00Total 50 1.00

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Measures of Central Tendency

and Variability: Grouped Data Measures of Central Tendency

Mean

MedianMode

Measures of VariabilityVariance

Standard Deviation

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Mean of Grouped Data

Weighted average of class midpoints Class frequencies are the weights

fMf

fM

Nf M f M f M f M

f f f f

i i

i

1 1 2 2 3 3

1 2 3

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Calculation of Grouped Mean

Class Interval Frequency Class Midpoint fM20-under 30 6 25 150

30-under 40 18 35 630

40-under 50 11 45 495

50-under 60 11 55 605

60-under 70 3 65 195

70-under 80 1 75 75

50 2150

fM

f

2150

5043 0.

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Median of Grouped Data

Median L

Ncf

f W

Where

p

med

2

:

L the lower limit of the median class

cf = cumulative frequency of class preceding the median class

f = frequency of the median classW = width of the median class

N = total of frequencies

p

med

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Median of Grouped Data -- Example

Cumulative

Class Interval Frequency Frequency

20-under 30 6 630-under 40 18 24

40-under 50 11 35

50-under 60 11 46

60-under 70 3 49

70-under 80 1 50

N = 50

Md L

Ncf

f

Wp

med

2

40

50

224

1110

40 909.

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Mode of Grouped Data

Midpoint of the modal class

Modal class has the greatest frequency

Class Interval Frequency20-under 30 6

30-under 40 18

40-under 50 11

50-under 60 1160-under 70 3

70-under 80 1

Mode 30 402

35

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Variance and Standard Deviation

of Grouped Data

22

2

fN

M

Population

22

2

1S M X

S

f

n

S

Sample

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Population Variance and Standard

Deviation of Grouped Data

1944

1152

44

1584

1452

1024

7200

20-under 30

30-under 4040-under 50

50-under 60

60-under 70

70-under 80

Class Interval

6

18

11

11

3

1

50

f

25

35

45

55

65

75

150

630

495

605

195

75

2150

fM

-18

-8

2

12

22

32

M f M2

324

64

4

144

484

1024

2

M

2

2

7 2 0 0

5 01 4 4

f

N

M 2

144 12

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