lec-1 final

8/2/2019 lec-1 final

1/48

M.B.A. SEMM.B.A. SEM--11

QuantitativeQuantitative AnalysisAnalysis

Devina UpadhyayDevina Upadhyay

M.Sc. , M.phil., PhD (pursuing)M.Sc. , M.phil., PhD (pursuing)


2/48

Statistics

Descriptive theory Inferential theory Decision theory


3/48

Introduction to DescriptiveIntroduction to Descriptivestatisticsstatistics


4/48

Descriptive theory

Measures of

central tendency

Measures of

Dispersion

Measures of

Variation


5/48

Ungrouped VersusUngrouped Versus

Grouped DataGrouped Data

Ungrouped dataUngrouped data

have not been summarized in anyhave not been summarized in anywayway

are also calledare also called raw dataraw data

Grouped dataGrouped data

have been organized into ahave been organized into a

frequency distributionfrequency distribution


6/48

Example ofUngroupedExample ofUngrouped

DataData42

30

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

54

Ages of a Sample of

Managers from

Urban Child CareCenters in the

United States


7/48

Frequency Distribution ofFrequency Distribution of

Child Care Managers AgesChild Care Managers Ages

Class IntervalClass Interval FrequencyFrequency

2020--under30under30 66







8/48

Data RangeData Range

4

230

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

54

Smallest

Largest

51=

23-74=

Smallest-Largest=Range


9/48

Relative FrequencyRelative Frequency

RelativeRelativeClass IntervalClass Interval FrequencyFrequency FrequencyFrequency

2020--under30under30 66 .12.12

3030--under40under40 1818 .36.36

4040--under50under50 1111 .22.22

5050--under60under60 1111 .22.22

6060--under70under70 33 .06.06

7070--under80under80 11 .02.02

TotalTotal 5050 1.001.00

6

50!

18

50!


10/48

Cumulative FrequencyCumulative Frequency

CumulativeCumulativeClass IntervalClass Interval FrequencyFrequency FrequencyFrequency

2020--under30under30 66 66

3030--under40under40 1818 2424

4040--under50under50 1111 3535

5050--under60under60 1111 4646

6060--under70under70 33 4949

7070--under80under80 11 5050

TotalTotal 5050

18 + 6

11 + 24


11/48

Class Midpoints, RelativeClass Midpoints, Relative

Frequencies, and CumulativeFrequencies, and Cumulative

FrequenciesFrequencies

Relative CumulativeRelative Cumulative

Class IntervalClass IntervalFrequencyFrequency MidpointMidpoint FrequencyFrequency FrequencyFrequency

2020--under30under30 66 2525 .12.12 663030--under40under40 1818 3535 .36.36 2424

4040--under50under50 1111 4545 .22.22 3535

5050--under60under60 1111 5555 .22.22 4646

6060--under70under70 33 6565 .06.06 4949

7070--under80under80 11 7575 .02.02 5050

TotalTotal 5050 1.001.00


12/48

Common Statistical GraphsCommon Statistical Graphs

HistogramHistogram ---- vertical bar chart ofvertical bar chart of

frequenciesfrequencies

Frequency PolygonFrequency Polygon ---- line graph ofline graph of


Pie ChartPie Chart ---- proportionalproportional

representation for categories of arepresentation for categories of a

whole.whole.


13/48

HistogramHistogram ---- vertical bar chart ofvertical bar chart of


Class Interval Frequency

20-under 30 6

30-under 40 18

40-under 50 11

50-under 60 11

60-under 70 3

70-under 80 1 0

10

20

0 10 20 30 40 50 60 70 80

Years

Freq

uen

cy


14/48

Histogram ConstructionHistogram Construction

Class IntervalClass Interval FrequencyFrequency

2020--under 30under 30 66

3030--under 40under 40 1818

4040--under 50under 50 1111

5050--under 60under 60 1111

6060--under 70under 70 33

7070--under 80under 80 110

10

20

0 10 20 30 40 50 60 70 80

Years

Freq

u

en

cy


15/48

Frequency PolygonFrequency Polygon ---- line graphline graph

of frequenciesof frequencies

Class Interval Frequency

20-under 30 6

30-under 40 18

40-under 50 11

50-under 60 11

60-under 70 3

70-under 80 10

10

20

0 10 20 30 40 50 60 70 80

Years

Frequency


16/48

TruckTruckProduction inProduction in

the U.S. inthe U.S. in

last yearlast year

(Hypothetical(Hypothetical

values)values)

TruckProduction

Company

A

B

C

D

E

Totals

357,411

354,936

160,997

34,099

12,747

920,190


17/48

39%

39%

17%

4%1%

A B C D E

. . ruc ro uc on. . ruc ro uc on---- propor onapropor ona

representation for categories of arepresentation for categories of a

whole.whole.


18/48

Descriptive statistics, measure ofDescriptive statistics, measure of

central tendency, Measure ofcentral tendency, Measure of

Variability, For group andVariability, For group andUngrouped data, Measures ofUngrouped data, Measures of

shapeshape ..


19/48

Measures of Central Tendency:Measures of Central Tendency:

Ungrouped Data

Ungrouped DataMeasures of central tendency meansMeasures of central tendency means

measures of location.measures of location.

Common Measures of LocationCommon Measures of Location

ModeMode

MedianMedian MeanMean

PercentilesPercentiles

QuartilesQuartiles


20/48

MeanMean

Arithmetic mean : Simple averageArithmetic mean : Simple average

Geometric mean: Relative percentageGeometric mean: Relative percentage

Weighted mean: Weights associated withWeighted mean: Weights associated with

every units.every units.


21/48

ModeMode

The most frequently occurring value in aThe most frequently occurring value in a

data setdata set

BimodalBimodal ---- Data sets that have two modesData sets that have two modes

MultimodalMultimodal ---- Data sets that contain moreData sets that contain morethan two modesthan two modes


22/48

The mode is 44.The mode is 44.

There are more 44sThere are more 44s

than any other value.than any other value.

35

37

37

39

40

40

41

41

43

43

43

43

44

44

44

44

44

45

45

46

46

46

46

48

ModeMode ---- ExampleExample


23/48

MedianMedian

Middle value in an ordered array ofMiddle value in an ordered array of

numbers.numbers.

Unaffected by extremely large andUnaffected by extremely large and

extremely small values.extremely small values.


24/48

Median: ComputationalMedian: Computational

ProcedureProcedureFirst ProcedureFirst Procedure

Arrange the observations in an ordered array.Arrange the observations in an ordered array.

If there is an odd number of terms, theIf there is an odd number of terms, themedian is the middle term of the orderedmedian is the middle term of the ordered

array.array.

If there is an even number of terms, theIf there is an even number of terms, the

median is the average of the middle twomedian is the average of the middle twoterms.terms.

Second ProcedureSecond Procedure

The medians position in an ordered array isThe medians position in an ordered array is

iven b n+1 /2.iven b n+1 /2.


25/48

Median: Example

with an Even Number of Terms

Ordered Array

3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21

There are 16 terms in the ordered array. Position of median = (n+1)/2 = (16+1)/2 = 8.5

The median is between the 8th and 9th terms,14.5.

If the 21 is replaced by 100, the median is14.5.

If the 3 is replaced by -88, the median is 14.5.


26/48

Median: ExampleMedian: Example

with an Odd Number ofT

ermswith an Odd Number ofT

ermsOrdered ArrayOrdered Array3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 213 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21

2222

There are 17 terms in the ordered array.There are 17 terms in the ordered array.Position of median = (n+1)/2 = (17+1)/2 =Position of median = (n+1)/2 = (17+1)/2 =99T

he median is the 9th term, 15.T

he median is the 9th term, 15.If the 22 is replaced by 100, the median isIf the 22 is replaced by 100, the median is15.15.If the 3 is replaced byIf the 3 is replaced by --103, the median is103, the median is15.15.


27/48

VariabilityVariability

No Variability

Variability


28/48

Measures of Variability:Measures of Variability:

Ungrouped Data

Ungrouped DataMeasures of variability describe theMeasures of variability describe the

spread or the dispersion of a set of data.spread or the dispersion of a set of data.

Common Measures of VariabilityCommon Measures of Variability RangeRange

Interquartile RangeInterquartile Range

VarianceVariance

Standard DeviationStandard Deviation

Coefficient of VariationCoefficient of Variation


29/48

RangeRange

The difference between the largest andThe difference between the largest and

the smallest values in a set of datathe smallest values in a set of data

Simple to computeSimple to compute

Ignores all data points exceptIgnores all data points except thethe

two extremestwo extremes

Example:Example:

RangeRange ==LargestLargest -- SmallestSmallest ==

4848 -- 35 = 1335 = 13

35

37

37

39

40

40

41

41

43

43

43

43

44

44

44

44

44

45

45

46

46

46

46

48

35

48


30/48

Interquartile RangeInterquartile Range

Range of values between the first and thirdRange of values between the first and third

quartilesquartiles

Less influenced by extremesLess influenced by extremes

Interquartile Range Q Q! 3 1


31/48

Population VariancePopulation Variance

Average of theAverage of the squaredsquared deviations fromdeviations from

the arithmetic mean.the arithmetic mean.

mean is 13.mean is 13.Observations:5,9,16,17,18.Observations:5,9,16,17,18.

59

16

17

18

-8-4

+3

+4

+5

0

6416

9

16

25

130

XX Q 2X Q 2

2

130

5

260

W

Q!

!

!

XN

.


32/48

Population Standard DeviationPopulation Standard Deviation

Square root of theSquare root of thevariancevariance

2

2

2

130

5

26 0

26 0

51

WQ

WW

!

!

!

!

!

!

X

N

.

.

.

5

9

16

17

18

-8

-4

+3

+4

+5

0

64

16

9

16

25

130

XX Q 2X Q


33/48

Sample VarianceSample Variance

Average of theAverage of the squaredsquared deviations fromdeviations fromthe arithmetic mean.the arithmetic mean.Mean:1773Mean:1773

Observations:2398,1844,1539,1311.Observations:2398,1844,1539,1311.

2,398

1,8441,539

1,311

7,092

625

71-234

-462

0

390,625

5,04154,756

213,444

663,866

X X X 2X X 2

2

1

663 866

3

221 288 67

SX X

n

!

!

!

,

, .


34/48

Sample Standard DeviationSample Standard Deviation

Square root of theSquare root of the

sample variancesample variance 22

2

1

663866

3

22128867

22128867

47041

SX X

S

n

S

!

!

!

!

!

!

,

, .

, .

.

2,398

1,844

1,539

1,311

7,092

625

71

-234

-462

0

390,625

5,041

54,756

213,444

663,866

XX X 2X X


35/48


Ratio of the standard deviation to theRatio of the standard deviation to the

mean, expressed as a percentagemean, expressed as a percentage

Measurement ofMeasurement of relativerelative dispersiondispersion

100..Q

W!VC


36/48

Less c.v more consistencyLess c.v more consistency

Less c.v more uniformityLess c.v more uniformity

Less c.v less riskLess c.v less risk


37/48


A BA B

129

4 6

100

4 629

100

1586

1

1

1

1

Q

W

WQ

!

!

!

!

!

.

.

.

. .CV

284

10

100

1084

100

1190

2

2

2

2

Q

W

WQ

!

!

!

!

!

CV. .

.


38/48

Measures of Central TendencyMeasures of Central Tendency

and Variability: Grouped Dataand Variability: Grouped Data

Measures of Central TendencyMeasures of Central Tendency

MeanMean MedianMedian

ModeMode

Measures of VariabilityMeasures of Variability

VarianceVariance

Standard DeviationStandard Deviation


39/48

Mean of Grouped DataMean of Grouped Data

Weighted average of class midpointsWeighted average of class midpoints

Class frequencies are the weightsClass frequencies are the weights

Q !

!

!

fMf

fM

N f M f M f M f M

f f f f

i i

i

1 1 2 2 3 3

1 2 3


40/48

Calculation of Grouped MeanCalculation of Grouped Mean

Class Interval Frequency Class Midpoint fM

20-under 30 6 25 150

30-under 40 18 35 630

40-under 50 11 45 49550-under60 11 55 605

60-under 70 3 65 195

70-under 80 1 75 75

50 2150

Q ! ! !fM

f

2150

5043 0.


41/48

Median of Grouped DataMedian of Grouped Data

Median L

Ncf

fW

Where

p

med

!

!

2

:

L the lower limit of the median class

cf = cumulative frequency of class preceding the median class

f = frequency of the median classW = width of the median class

N = total of frequencies

p

med


42/48

Median of Grouped DataMedian of Grouped Data ----

ExampleExampleCumulative

Class Interval Frequency Frequency

20-under 30 6 630-under 40 18 24

40-under 50 11 35

50-under 60 11 46

60-under 70 3 49

70-under 80 1 50

N = 50

Md L

Ncf

fW

p

med

!

!

!

2

40

50

224

1110

40 909.


43/48

Mode of Grouped DataMode of Grouped Data

Midpoint of the modal classMidpoint of the modal class

Modal class has the greatestModal class has the greatest

frequencyfrequencyClass Interval Frequency20-under 30 6

30-under 40 18

40-under 50 11

50-under 60 1160-under 70 3

70-under 80 1

1579.33

10*11636

61830

*2012

01mod

352

4030

!

!

!

!

!

Wfff

ffLe

Mode

ar ance an tan arar ance an tan ar


44/48

ar ance an tan ar ar ance an tan ar

DeviationDeviation

of Grouped Dataof Grouped Data

22

2

WQ

WW

!

!

fN

M

Population

22

2

1S

M X

S

f

n

S

!

!

Sample

opu at on ar ance anopu at on ar ance an


45/48

opu at on ar ance anopu at on ar ance an

Standard Deviation of GroupedStandard Deviation of Grouped

DataData

1944

115244

1584

1452

1024

7200

20-under 30

30-under 4040-under 50

50-under 60

60-under 70

70-under 80

Class Interval

6

1811

11

3

1

50

f

25

3545

55

65

75

M

150

630495

605

195

75

2150

fM

-18

-82

12

22

32

M Q f M2

Q

324

644

144

484

1024

2

MQ

2

2

7200

50144W

Q! ! !

fN

M W W! ! !2

144 12


46/48

Measures of ShapeMeasures of Shape

SkewnessSkewness Absence of symmetryAbsence of symmetry Extreme values in one side of a distributionExtreme values in one side of a distribution

KurtosisKurtosis Peakedness of a distributionPeakedness of a distribution

Dispersion:Dispersion: Spread of the dataSpread of the data


47/48

SkewnessSkewness

Symmetric skewedSymmetric skewed


48/48

Dispersion KurtosisDispersion Kurtosis

lec-1 final

Documents

Transcript of lec-1 final