Chapter 3. Describing Data: Numerical Measures
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Transcript of Chapter 3. Describing Data: Numerical Measures
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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Chapter 3.Describing Data: Numerical
Measures
http://statisticdescriptive.wordpress.com/
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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Numerical Measures:
1. Measure of location.
2. Measure of dispersion.
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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
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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.
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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?
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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
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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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?
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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)
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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
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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.
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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
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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
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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.
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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
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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?
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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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
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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
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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
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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?
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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
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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Example (continued):
s2 = 10
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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
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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
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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
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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
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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
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Ir. Muhril Ardiansyah, M.Sc., Ph.D.
Chapter 3: Describing Data: Numerical Measures
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Example:
S = root of (1531.8/(80-1)) = 4.403
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Chapter 3: Describing Data: Numerical Measures
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Homework:
No. 81 Page 93.