This presentation is the property of its rightful owner.
1 / 37

# 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)

Chapter 3. Describing Data: Numerical Measures

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## Chapter 3.Describing Data: Numerical Measures

http://statisticdescriptive.wordpress.com/

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

Company Number of patents

granted

General Motors 511

Nissan385

Daimler275

Toyota257

Honda249

Ford234

Mazda210

Chrysler97

Porsche50

Mitsubishi36

Volvo23

BMW13

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

• 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?

• 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

• Range.

• Mean deviation.

• Variance and standard deviation.

Chapter 3: Describing Data: Numerical Measures

### Measures Of Dispersion (continued)

Range:

- The simplest.

- Equation 3-6 (page 73)

Range = (largest value) – (smallest value)

Chapter 3: Describing Data: Numerical Measures

### 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

Chapter 3: Describing Data: Numerical Measures

### 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.

Chapter 3: Describing Data: Numerical Measures

### Example (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

Chapter 3: Describing Data: Numerical Measures

### 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.

Chapter 3: Describing Data: Numerical Measures

### 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 variance

Equation 3-8 Page 76

Example Page 77

Population standard deviation

Equation 3-9 Page 78

Chapter 3: Describing Data: Numerical Measures

### 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?

Chapter 3: Describing Data: Numerical Measures

### Example

Chapter 3: Describing Data: Numerical Measures

### Example

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)

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:

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):

Chapter 3: Describing Data: Numerical Measures

### Example (continued):

s2 = 10

Chapter 3: Describing Data: Numerical Measures

### The Mean And Standard Deviation Of Grouped Data

• Arithmetic mean of grouped data

Equation 3-12 Page 84

Example Page 84 and 85

Chapter 3: Describing Data: Numerical Measures

### Example:

Chapter 3: Describing Data: Numerical Measures

### Example (continued):

Chapter 3: Describing Data: Numerical Measures

### The Mean And Standard Deviation Of Grouped Data

• Standard deviation, grouped data

Equation 3-13 Page 85

Example Page 86

Chapter 3: Describing Data: Numerical Measures

### Example:

Chapter 3: Describing Data: Numerical Measures

### Example:

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

Chapter 3: Describing Data: Numerical Measures

### Homework:

No. 81 Page 93.

Chapter 3: Describing Data: Numerical Measures