Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

1

Chapter 3.Describing Data: Numerical

Measures

http://statisticdescriptive.wordpress.com/

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

2

Numerical Measures:

1. Measure of location.

2. Measure of dispersion.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Population Mean

Population mean = (sum of all the values in the population)/(number of values in the population)

Population mean Equation 3-1 Page 57

Parameter: a characteristic of a population

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example Page 57

There are 12 automobile manufacturing

companies in the United States. Listed

below is the number of patents granted

by the United States government to

each company in a recent year.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Company Number of patents grantedGeneral Motors 511Nissan 385Daimler 275Toyota 257Honda 249Ford 234Mazda 210Chrysler 97Porsche 50Mitsubishi 36Volvo 23BMW 13

Is this a sample or a population?

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Sample Mean

Sample mean = (sum of all the values in the sample)/(number of values in the sample)

Sample mean Equation 3-2 Page 58

Statistic: a characteristic of a sample

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

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Example Page 58

SunCom is studying the number of minutes used by clients in a particular cell phone rate plan. A random sample of 12 clients showed the following number of minutes used last month.

90, 77, 94, 89, 119, 112, 91, 110, 92, 100, 113, 83,

Mean?

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Median

Median:

the midpoint of the values after they have been ordered from the smallest to the largest.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example Page 63

Prices ordered from low to high:

60000

65000

70000 ……..median

80000

275000

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Mode

Mode

the value of the observation that appears most frequently.

Example Page 64

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Relative Positions Of The Mean, Median, And Mode

A symmetric distribution

Mound-shaped distribution. Mean, median, and mode are equal.

Chart 3-2 Page 67

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Relative Positions Of The Mean, Median, And Mode (continued)

A skewed distribution is not symmetrical

A positively skewed distribution, - the arithmetic mean is the largest of the three measures (mean, median, mode). - the median is generally the next largest measure. - the mode is the smallest. - mode > median > mean. Chart 3-3 Page 68

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Relative Positions Of The Mean, Median, And Mode (continued)

A negatively skewed distribution: - the mean is the lowest of the three measures. - the median is greater than the mean. - the mode is the largest of the three measures. - mode > median > mean.

Chart 3-4 Page 68

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Dispersion Why study dispersion: - the spread of the data. - to know variation. - A small value for a measure dispersion indicates that the data are clustered closely around the arithmetic mean.

The mean considered as representative of the data.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Why Study Dispersion?

To know about the spread data A small value a measure of dispersion

indicates that the data are clustered closely.

A large measure of dispersion indicates that the mean is not reliable.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Measures Of Dispersion

1. Range.

2. Mean deviation.

3. Variance and standard deviation.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Measures Of Dispersion (continued)

Range:

- The simplest.

- Equation 3-6 (page 73)Range = (largest value) – (smallest value)

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Measures Of Dispersion (continued)

Mean deviation (MD):- The arithmetic mean of the absolute

values of the deviations from the arithmetic mean.

- Equation 3-7 Page 73

Example Page 74

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example:

The number of cappuccinos sold at the Starbuck location in the Orange County Airport between 4 and 7 pm for sample of 5 days last year were 20, 40, 50, 60 and 80. In the LAX airport in Los Angeles, the number of cappuccinos sold at a Starbuck location between 4 and 7 pm for a sample of 5 days last year were 20, 49, 50, 51, and 80. Determine the mean, median, range, and mean deviation for each location. Compare the difference.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued)

For the Orange County:

Mean : 50 cappuccinos per day

Median : 50 cappuccinos per day

Range : 60 cappuccinos per day

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued), For Orange County

Number of cappuccinos sold daily

(X – X bar) Absolute deviation

20 (20-50) = -30 30

40 (40-50) = -10 10

50 (50-50)=0 0

60 (60-50)=10 10

80 (80-50)=30 30

TOTAL 80

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued) For Orange County

MD = (80)/(5) = 16

The mean deviation is 16 cappuccinos per

day, and shows that the number of

cappuccinos sold deviates, on average, by

16 from the mean of 50 cappuccinos per

day.

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Measures Of Dispersion (continued)

Variance and standard deviation:- Based on the deviation from the mean- Variance: the arithmetic mean of the squared

deviations from the mean- Standard deviation: the square root of the variance

Population varianceEquation 3-8 Page 76Example Page 77

Population standard deviationEquation 3-9 Page 78

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example:

The number of traffic citations issued

during the last five months in Beaufort

County, South Carolina, is 38, 26, 13, 41,

and 22. What is the population variance?

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

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Example

Number (X) X- (X-)2

38 +10 100

26 -2 4

13 -15 225

41 +13 169

22 -6 36

Total: 140 Total: 0 Total: 534

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example

= (X) / N = 140 / 5 = 28

X-2} / N = (534) / 5 =106.8

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Measures Of Dispersion (continued)

Sample variance

Equation 3-10 Page 79

Example Page 79

Sample standard deviation

Equation 3-11 Page 79

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example:

The hourly wages for a sample of part time

employees at Home Depot are : $12, 20,

16, 18 and 19. What is the sample

variance?

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued):

Hourly Wage (X) X – X bar (X-Xbar)2

12 -5 25

20 3 9

16 -1 1

18 1 1

19 2 4

Total: 85 Total: 0 Total: 40

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued):

s2 = 10

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Mean And Standard Deviation Of Grouped Data

Arithmetic mean of grouped data

Equation 3-12 Page 84

Example Page 84 and 85

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example:

Selling Price Frequency

15 up to 18 8

18 up to 21 23

21 up to 24 17

24 up to 27 18

27 up to 30 8

30 up to 33 4

33 up to 36 2

TOTAL 80

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example (continued):

Selling Price f M fM

15 up to 18 8 16.5 132

18 up to 21 23 19.5 448.5

21 up to 24 17 22.5 382.5

24 up to 27 18 25.5 459.0

27 up to 30 8 28.5 228

30 up to 33 4 31.5 126

33 up to 36 2 34.5 69

Total: 80 Total: 1845

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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The Mean And Standard Deviation Of Grouped Data

Standard deviation, grouped data

Equation 3-13 Page 85

Example Page 86

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Example:

Selling Price f M (M-Xbar) (M-Xbar)2 f(M-Xbar)2

15 up to 18 8 16.5 -6.6 43.56 348.48

18 up to 21 23 19.5 -3.6 12.96 298.08

21 up to 24 17 22.5 -0.6 0.36 6.12

24 up to 27 18 25.5 2.4 5.76 103.68

27 up to 30 8 28.5 5.4 29.16 233.28

30 up to 33 4 31.5 8.4 70.56 282.24

33 up to 36 2 34.5 11.4 129.96 259.92

TOTAL 80 1531.8

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

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Example:

S = root of (1531.8/(80-1)) = 4.403

Ir. Muhril Ardiansyah, M.Sc., Ph.D.

Chapter 3: Describing Data: Numerical Measures

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Homework:

No. 81 Page 93.