Descriptive measures.ppt

8/21/2019 Descriptive measures.ppt

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2003 Prentice-Hall, Inc. Chap 2-1

Business StatisticsDescriptive Statistics


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Chapter Topics

Organizing Numerical Data

The Ordered Array and Stem-Leaf Display

Tabulating and Graphing Univariate Numerical Data Frequency Distributions: Tables, Histograms, Polygons

Relative Frequency Distribution

Measures of Central Tendency

Mean, Median, Mode


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2 144677

3 028

4 1

Numerical Data

Ordered Array

Stemand Leaf

Display

Frequency Distributions

Histograms

Polygons

Ogive

Tables

41, 24, 32, 26, 27, 27, 30, 24, 38, 21

21, 24, 24, 26, 27, 27, 30, 32, 38, 41


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Data in RawForm (as collected):24, 26, 24, 21, 27, 27, 30, 41, 32, 38

Data inOrdered Arrayfrom Smallest to Largest:21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Stem-and-Leaf Display:


(continued)

2 1 4 4 6 7 7

3 0 2 8

4 1


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Summarizing Numerical Data

2 144677

3 028

4 1

Numerical Data

Ordered Array

StemandLeaf

Display

Histograms Ogive

Tables

41, 24, 32, 26, 27, 27, 30, 24, 38, 21

21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Frequency Distributions

Polygons


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Tabulating Numerical Data: Frequency

Distributions

Sort Raw Data in Ascending Order12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Find Range: 58 - 12 = 46

Select Number of Classes: 5(usually between 5 and 15)

Compute Class Interval (width): 10 (46/5 then round up)

Determine Class Boundaries (limits): 10, 20, 30, 40, 50, 60 Compute Class Midpoints: 15, 25, 35, 45, 55

Count Observations & Assign to Classes


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Frequency Distributions, Relative Frequency

Distributions and Percentage Distributions

Class Frequency

10 but under 20 3 .15 15

20 but under 30 6 .30 30

30 but under 40 5 .25 25

40 but under 50 4 .20 20

50 but under 60 2 .10 10

Total 20 1 100

RelativeFrequency

Percentage

Data in ordered array:

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58



Graphing Numerical Data:

The Histogram

Histogram

0

3

6

5

4

2

0

0

1

2

3

4

5

6

7

5 15 25 35 45 55 More

Fre

quency


12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

No Gaps

Between

Bars

Class MidpointsClass Boundaries



Graphing Numerical Data:

The Frequency Polygon

Frequency

0

1

23

4

5

6

7

5 15 25 35 45 55 More

Class Midpoints


12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58



Measures of Central Tendency or Averages



Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range



Measures of Central Tendency

Central Tendency

Mean Median

Mode1

1

n

i

i

N

i

i

X

Xn

X

N



Mean (Arithmetic Mean) for

Ungrouped Data

Mean (Arithmetic Mean) of Data Values

Sample mean

Population mean

1 1 2

n

i

i n

XX X X

Xn n

1 1 2

N

i

i N

XX X X

N N

Sample Size

Population Size



Mean (Arithmetic Mean)

The Most Common Measure of CentralTendency

Affected by Extreme Values (Outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6



Arithmetic Mean for Grouped Data

A.M = fiXi

fi

Both for discrete and continuousfrequency distributions



Median for Ungrouped Data

Robust Measure of Central Tendency

Not Affected by Extreme Values

In an Ordered Array, the Median is the

Middle Number If n or N is odd, the median is the middle number

If n or N is even, the median is the average of the

2 middle numbers

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5


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Median for Grouped Data(Continuous Frequency Distribution)

Median = L1 + C.I (mc )

f

Where L1 is the lower class boundary of MCC.I is the class interval of MC

f is the frequency of MC

m is the location of median

c is the cumulative frequency of class precedingMC

MC is the median class


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Mode for Ungrouped Data

A Measure of Central Tendency

Value that Occurs Most Often

Not Affected by Extreme Values There May Not be a Mode

There May be Several Modes

Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode


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Mode for Grouped Data

In a discrete frequency table Mode is theobservation with the highest frequency

In a continuous frequency table it is workedout using the following interpolation formula

Mode =L1+(fmf1) *C.I

(fm-f1-f2)


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Mode for Grouped Data

Where L1 is the lower boundary of model class

fm is the frequency of MC

f1 is the frequency of the class preceding theMC

f2 is the frequency of the class following the MC

C.I is the class interval of the MC

MC is the modal class


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Chapter Summary

Organized Numerical Data

The Ordered Array and Stem-Leaf Display

Tabulated and Graphed Univariate NumericalData

Frequency Distributions: Tables, Histograms,

Polygon

Relative Frequency Distribution


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2003 Prentice Hall Inc Chap 2 22

Chapter Summary

Described Measures of Central Tendency

Mean, Median, Mode

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