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# Chapter 3. Describing Data: Numerical Measures - PowerPoint PPT Presentation

Chapter 3. Describing Data: Numerical Measures. http://statisticdescriptive.wordpress.com/. Numerical Measures:. 1. Measure of location. 2. Measure of dispersion. The Population Mean. Population mean = (sum of all the values in the population)/(number of values in the population)

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### Chapter 3.Describing Data: Numerical Measures

http://statisticdescriptive.wordpress.com/

Chapter 3: Describing Data: Numerical Measures

1. Measure of location.

2. Measure of dispersion.

Chapter 3: Describing Data: Numerical Measures

• 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

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

• 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

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

• Median:

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

Chapter 3: Describing Data: Numerical Measures

Prices ordered from low to high:

60000

65000

70000 ……..median

80000

275000

Chapter 3: Describing Data: Numerical Measures

• Mode

the value of the observation that appears most frequently.

Example Page 64

Chapter 3: Describing Data: Numerical Measures

• A symmetric distribution

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

Chart 3-2 Page 67

Chapter 3: Describing Data: Numerical Measures

• 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

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)

• 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

Example (continued): (continued)

s2 = 10

Chapter 3: Describing Data: Numerical Measures

• 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

• 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

Homework: (continued)

No. 81 Page 93.

Chapter 3: Describing Data: Numerical Measures