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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

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Chapter 3 describing data numerical measures

Chapter 3.Describing Data: Numerical Measures

http://statisticdescriptive.wordpress.com/

Chapter 3: Describing Data: Numerical Measures


Numerical measures

Numerical Measures:

1. Measure of location.

2. Measure of dispersion.

Chapter 3: Describing Data: Numerical Measures


The population mean

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

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


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

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

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

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

Example Page 63

Prices ordered from low to high:

60000

65000

70000 ……..median

80000

275000

Chapter 3: Describing Data: Numerical Measures


The mode

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

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

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 continued1

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

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

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.

Chapter 3: Describing Data: Numerical Measures


Measures of dispersion

Measures Of Dispersion

  • Range.

  • Mean deviation.

  • Variance and standard deviation.

Chapter 3: Describing Data: Numerical Measures


Measures of dispersion continued

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 continued1

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

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

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

Example (continued), For Orange County

Chapter 3: Describing Data: Numerical Measures


Example continued for orange county1

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 continued2

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


Example1

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


Example2

Example

Chapter 3: Describing Data: Numerical Measures


Example3

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 continued3

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


Example4

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 continued1

Example (continued):

Chapter 3: Describing Data: Numerical Measures


Example continued2

Example (continued):

s2 = 10

Chapter 3: Describing Data: Numerical Measures


The mean and standard deviation of grouped data

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


Example5

Example:

Chapter 3: Describing Data: Numerical Measures


Example continued3

Example (continued):

Chapter 3: Describing Data: Numerical Measures


The mean and standard deviation of grouped data1

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


Example6

Example:

Chapter 3: Describing Data: Numerical Measures


Example7

Example:

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

Chapter 3: Describing Data: Numerical Measures


Homework

Homework:

No. 81 Page 93.

Chapter 3: Describing Data: Numerical Measures


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