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Chapter 3. Describing Data: Numerical Measures

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### Chapter 3.Describing Data: Numerical Measures

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Chapter 3: Describing Data: Numerical Measures

Numerical Measures:

1. Measure of location.

2. Measure of dispersion.

Chapter 3: Describing Data: Numerical Measures

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

Chapter 3: Describing Data: Numerical Measures

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.

Chapter 3: Describing Data: Numerical Measures

granted

General Motors 511

Nissan 385

Daimler 275

Toyota 257

Honda 249

Ford 234

Mazda 210

Chrysler 97

Porsche 50

Mitsubishi 36

Volvo 23

BMW 13

Is this a sample or a population?

Chapter 3: Describing Data: Numerical Measures

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

Chapter 3: Describing Data: Numerical Measures

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?

Chapter 3: Describing Data: Numerical Measures

The Median

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

Chapter 3: Describing Data: Numerical Measures

Example Page 63

Prices ordered from low to high:

60000

65000

70000 ……..median

80000

275000

Chapter 3: Describing Data: Numerical Measures

The Mode

- Mode
the value of the observation that appears most frequently.

Example Page 64

Chapter 3: Describing Data: Numerical Measures

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

Chapter 3: Describing Data: Numerical Measures

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

Chapter 3: Describing Data: Numerical Measures

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

Chapter 3: Describing Data: Numerical Measures

Dispersion (continued)

- 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.

Chapter 3: Describing Data: Numerical Measures

Why Study Dispersion? (continued)

- 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.

Chapter 3: Describing Data: Numerical Measures

Measures Of Dispersion (continued)

- Range.
- Mean deviation.
- Variance and standard deviation.

Chapter 3: Describing Data: Numerical Measures

Measures Of Dispersion (continued) (continued)

Range:

- The simplest.

- Equation 3-6 (page 73)

Range = (largest value) – (smallest value)

Chapter 3: Describing Data: Numerical Measures

Measures Of Dispersion (continued) (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

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

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.

Chapter 3: Describing Data: Numerical Measures

Example (continued) (continued)

For the Orange County:

Mean : 50 cappuccinos per day

Median : 50 cappuccinos per day

Range : 60 cappuccinos per day

Chapter 3: Describing Data: Numerical Measures

Example (continued), For Orange County (continued)

Chapter 3: Describing Data: Numerical Measures

Example (continued) For Orange County (continued)

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.

Chapter 3: Describing Data: Numerical Measures

Measures Of Dispersion (continued) (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 variance

Equation 3-8 Page 76

Example Page 77

Population standard deviation

Equation 3-9 Page 78

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

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?

Chapter 3: Describing Data: Numerical Measures

Example (continued)

Chapter 3: Describing Data: Numerical Measures

Example (continued)

m = (SX) / N = 140 / 5 = 28

s2 = {S(X-m)2} / N = (534) / 5 =106.8

Chapter 3: Describing Data: Numerical Measures

Measures Of Dispersion (continued) (continued)

Sample variance

Equation 3-10 Page 79

Example Page 79

Sample standard deviation

Equation 3-11 Page 79

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

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?

Chapter 3: Describing Data: Numerical Measures

Example (continued): (continued)

Chapter 3: Describing Data: Numerical Measures

The Mean And Standard Deviation Of Grouped Data (continued)

- Arithmetic mean of grouped data
Equation 3-12 Page 84

Example Page 84 and 85

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

Chapter 3: Describing Data: Numerical Measures

Example (continued): (continued)

Chapter 3: Describing Data: Numerical Measures

The Mean And Standard Deviation Of Grouped Data (continued)

- Standard deviation, grouped data
Equation 3-13 Page 85

Example Page 86

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

Chapter 3: Describing Data: Numerical Measures

Example: (continued)

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

Chapter 3: Describing Data: Numerical Measures

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