Chapter 4

Comprehensive index

Comprehensive index

• From its roles and the angle of the method characteristics,comprehensive index can be summarized into three categories:

The conception and function of total amount

index Total amount index is social

economic phenomenon must reflect the time, the place, the total scale, under the condition of the level of statistics.

Total amount index form is JueDuiShu, may also display to absolute difference.

Effect:

Total amount index can reflect a country's basic national conditions and National strength, reflect a department, unit and so on human, financial,The basic data of the content.

Total amount index is making decisions and the basis of scientific management.

Total amount index is the the foundation of calculated relative index and average index.

Total amount index calculation

Calculation principle: 1. The phenomenon of similar nature. 2. Clear statistical meaning. 3. Measurement unit shall be consistent.

The concept of relative index

Opposite index is two contact index, the result of the numerical contrast reflects the number of things characteristics and quantity

Opposite index role

• Can the specific social and economic phenomenon that the proportion between the relationship.

• Can make some can't direct comparison to find out the things together the basis of comparison

• Opposite index is easy to remember, easy to confidential

Structure relative index

• 1. Can reflect the overall the internal structure of the features• 2. Through the different period of relative change, we can see that the changes of things process and its development trend• 3. Can reflect on the human, material and financial resources utilization degree and the production and business operation effect quality• 4. Structure in the application of the relative average

Measures of Central TendencyMeasures of Central Tendency

xx

n

Most frequently used measure of central tendency

Strongly influenced by outliers- very large or very small values

Mean Arithmetic average

Sum of all data values divided by the number of data values within the array

x

Measures of Central TendencyMeasures of Central Tendency

xx

n

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Determine the mean value of

( 48 63 62 49 58 2 63 5 60 59 55)x

11

524x

11

x 47.64

Measures of Central TendencyMeasures of Central TendencyMedian

Data value that divides a data array into two equal groups

Data values must be ordered from lowest to highest

Useful in situations with skewed data and outliers (e.g., wealth management)

Measures of Central TendencyMeasures of Central TendencyDetermine the median value of

Organize the data array from lowest to highest value.

59, 60, 62, 63, 63

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

Select the data value that splits the data set evenly.

2, 5, 48, 49, 55, 58,

Median = 58

What if the data array had an even number of values?

60, 62, 63, 635, 48, 49, 55, 58, 59,

Measures of central tendencyMeasures of central tendency

• Usually the highest point of curve

ModeMost frequently occurring response within a data array

May not be typical

May not exist at all

Mode, bimodal, and multimodal

Measures of Central TendencyMeasures of Central TendencyDetermine the mode of

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Mode = 63

Determine the mode of

48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55Mode = 63 & 59 Bimodal

Determine the mode of

48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55Mode = 63, 59, & 48 Multimodal

Data VariationData Variation

Range

Standard Deviation

Variance

Measure of data scatter

Difference between the lowest and highest data value

Square root of the variance

Average of squared differences between each data value and the mean

RangeRange

R 63 2

Calculate by subtracting the lowest value from the highest value.

R h l

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the range for the data array.

R h l

R 61

Standard DeviationStandard Deviation 2x xs

( N 1)

1. Calculate the mean .

2. Subtract the mean from each value.

3. Square each difference.

4. Sum all squared differences.

5. Divide the summation by the number of values in the array minus 1.

6. Calculate the square root of the product.

x

Standard DeviationStandard Deviation 2x xs

( N 1)

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the standard deviation for the data array.

x

x

n

52411

1. 47.64

2. 2 - 47.64 = -45.64

5 - 47.64 = -42.64

48 - 47.64 = 0.36

49 - 47.64 = 1.36

55 - 47.64 = 7.36

58 - 47.64 = 10.36

59 - 47.64 = 11.36

60 - 47.64 = 12.36

62 - 47.64 = 14.36

63 - 47.64 = 15.36

63 - 47.64 = 15.36

x x

Standard DeviationStandard Deviation 2x xs

( N 1)

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the standard deviation for the data array.

3.

-45.642 = 2083.01

-42.642 = 1818.17

0.362 = 0.13

1.362 = 1.85

7.362 = 54.17

10.362 = 107.33

11.362 = 129.05

12.362 = 152.77

14.362 = 206.21

15.362 = 235.93

15.362 = 235.93

2x x

Standard DeviationStandard Deviation 2x xs

( N 1)

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the standard deviation for the data array.

4.

2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33 + 129.05 + 152.77 + 206.21 + 235.93 + 235.93

2x x

= 5,024.555.( N 1)

11-1 = 10

6. 2( 1

x x

N )

5,024.5510

502.46

7. 2x xs

( N 1)

502.46S = 22.42

VarianceVariance 22x x

s( N 1)

1.Calculate the mean.

2.Subtract the mean from each value.

3.Square each difference.

4.Sum all squared differences.

5.Divide the summation by the number of values in the array minus 1.

Average of the square of the deviations

VarianceVariance

2 5024.55s

(50

1 )46

02.

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the variance for the data array.

22x x

s( N 1)

Graphing Frequency DistributionGraphing Frequency DistributionNumerical assignment of each outcome of a chance experiment

A coin is tossed 3 times. Assign the variable X to represent the frequency of heads occurring in each toss.

Toss Outcome X Value

HHH

HHT

HTH

THH

HTT

THT

TTH

TTT

3

2

2

2

1

1

1

0

X =1 when?

HTT,THT,TTH

Graphing Frequency DistributionGraphing Frequency DistributionThe calculated likelihood that an outcome variable will occur within an experiment

Toss Outcome X value

HHH

HHT

HTH

THH

HTT

THT

TTH

TTT

3

2

2

2

1

1

1

0

x P(x)

0

1

2

3

xx

a

FP

F

0

1P

8

1

3P

8

2

3P

8

3

1P

8

Graphing Frequency DistributionGraphing Frequency Distribution

x P(x)

0

1

2

3

0

1P

8

1

3P

8

2

3P

8

3

1P

8 x

HistogramHistogram

HistogramHistogramOpen airplane passenger seats one week before departure

What information does the histogram provide the airline carriers?

What information does the histogram provide prospective customers?

Measures of Central TendencyMeasures of Central Tendency

xx

n

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Determine the mean value of

( 48 63 62 49 58 2 63 5 60 59 55)x

11

524x

11

x 47.64

Measures of Central TendencyMeasures of Central TendencyMedian

Data value that divides a data array into two equal groups

Data values must be ordered from lowest to highest

Useful in situations with skewed data and outliers (e.g., wealth management)

Measures of Central TendencyMeasures of Central TendencyDetermine the median value of

Organize the data array from lowest to highest value.

59, 60, 62, 63, 63

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

Select the data value that splits the data set evenly.

2, 5, 48, 49, 55, 58,

Median = 58

What if the data array had an even number of values?

60, 62, 63, 635, 48, 49, 55, 58, 59,

Measures of central tendencyMeasures of central tendency

• Usually the highest point of curve

ModeMost frequently occurring response within a data array

May not be typical

May not exist at all

Mode, bimodal, and multimodal

Measures of Central TendencyMeasures of Central TendencyDetermine the mode of

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Mode = 63

Determine the mode of

48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55Mode = 63 & 59 Bimodal

Determine the mode of

48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55Mode = 63, 59, & 48 Multimodal

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