4561 2658-2-1909 69 descriptive statisticsppt

7/28/2019 4561 2658-2-1909 69 Descriptive Statisticsppt

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DESCRIPTIVE STATISTICS-II


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Measures of dispersion

are important for describing the spreadof the data, or its variation around acentral value.

It is numerical expression of spread orcompactness of the data

Small dispersion indicates high

uniformity of the items, while largedispersion indicates less uniformity.

Reliability and representative of centraltendency

Compare series


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Range

Difference between maximum andminimum values

Interquartile Range

Difference between third and first quartile(Q3 - Q1)

Variance

Average of the squared deviations from themean

Standard Deviation

Square root of the variance

Measures of Variability orDispersion


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Can eliminate some outlier problems byusing the inter-quartile range

Eliminate high- and low-valuedobservations and calculate the range of themiddle 50% of the data

Inter-quartile range = 3rd quartile 1stquartile

IQR = Q3 Q1


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Numbers of days1,2,6,4,2,4,8

Arranging in ascending order1,2,2,4,4,6,8

Q1=2

Q3=6 IQR=upper quartile-lower quartile

=6-2=4 days

Inter-quartile range calculation


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Variance

Based on all data

Average of squared deviations aroundmean

Unit is square of observation unit likeprice of share is in Rs the variance is inRs2

22

2

2

1

variance

xxn

meann

nobservatio


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Example 2.1

No of days(x) xsquare

1 1

1 1

6 36

4 16

2 4

4 16

33.33*3)74(6

11variance 22 xx

n


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Standard Deviation

Positive square root of variance()

Unit is same as observation unit

Standard deviation=

day82.133.3var iance


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Comparing Standard Deviations

Mean = 15.5

s = 3.33811 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.570

Data C

Dispersion measure suggest that data values of group Bfluctuate less than that of other groups


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Example 2.2

The operations manager of a plant thatmanufactures tyres wants to compare theactual inner diameters of two grades of tyres,each of which expected to be 575 mm. A

sample of five tyres of each grade was selectedand the results representing the inner diameterof the tyres, ranked from smallest to largestare as under.

Grade 1 568 570 575 578 584Grade 2 573 574 575 577 578

Which grade of tyre is providing better quality?
http://localhost/var/www/apps/conversion/tmp/scratch_3/Book1.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_3/Book1.xls


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Coefficient of Variation (C.V)

To study relative dispersion of two ormore series of data

It is defined as ratio of standard

deviation to the mean

C.V= S.D/Mean x 100

Also called measure of dispersion or

relative dispersion Less C.V implies more consistency in

data and vice-versa


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Two variables having different unite.g. One is in Rs and other is in days

Two variables having quite differentmeans. Ex Price of gold and silver


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Example 2.3

The manager of a package delivery service isdeciding whether to purchase a new fleet of trucks.When packages are stored in the trucks inpreparation for delivery, you need to consider two

major constraints- the weights( in kgs) and thevolume (in cubic feet) for each item. Theoperations manager samples 200 packages andfind that the mean weight is 100 kgs, with astandard deviation of 3.9 kgs and the meanvolume is 8.8 cubic feet with a standard deviation

of 2.2 cubic feet. How can the operations managerfind (compare) which one has more variation, theweight or the volume?


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Comparing Coefficient of Variation

Stock A:

Average price last year = $50

Standard deviation = $5

Stock B:

Average price last year = $100 Standard deviation = $5

Both stocks

have the same

standard

deviation, but

stock B is lessvariable relative

to its price

10%100%$50

$5100%

x

sCV

A

5%100%

$100

$5100%

x

sCV

B


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Example 2.4

Investment A B

Expected 1000 4000

returnStandard 300 500

Deviation

Which has more risk per rupee ofinvestment?


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Example 2.5

Which organization has more uniformwages?

Organization A Organization B

Number of employees 100 200

Average wage

per employee (Rs) 5000 8000Variance of wages per

employee 6000 10000


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Jyoti and Anuj are two eligible candidates forthe position of chief Manager (HR). Theiroverall rating (expressed in %) for the last five

years are summarized below. Which one ismore consistent performer?

Year Jyoti Anuj

2005 70% 81%

2006 81% 82%

2007 90% 79%

2008 74% 78%

2009 85% 75%

EXAMPLE 2.6

C.V. 10 3


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Studying the per capita income of differentstates of India or different countries in theworld measure of disparity

Studying the return on equity capital

invested in the some share- measure ofvolatility or risk

Studying the workload in different countersin banks/booking offices

Studying wages in different organization Compensation offered to MBA students indifferent institutions-Measure of uniformity


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Skewness

Measure of asymmetry in distribution of data

Skewed to left

Symmetric or unskewed

Skewed to right

Kurtosis

Measure of flatness or peakedness in distribution ofdata

Platykurtic (relatively flat)

Mesokurtic (normal)

Leptokurtic (relatively peaked)

Skewness and Kurtosis


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Relationship Between Mean,

Median, and Mode

A comparison of Mean, Median, and Mode for three Distributional Shapes

Median = Mean = Mode

(a) Symmetrical (b) Skewed to the Right

Median MeanMode

(c) Skewed to the Left

Median ModeMean


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Variation indicates the amount of spread ordispersion of individual values in a data setaround a central value, while skewnessindicates the direction of dispersion, that is,away from symmetry.

Variation is helpful in finding out the extentof variation among individual values in adata set, while skewness gives an

understanding about the concentration ofhigher or lower values around the meanvalue.


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Shape of Three Different Curves

(a) Leptokurtic (b) Platykurtic (c) Mesokurtic

Comparison of Three Data Sets Differing in Shape


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Using Microsoft Excel

Descriptive Statistics can beobtained from Microsoft Excel

Use menu choice:

tools / data analysis / descriptive

statistics

Enter details in dialog box


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Excel outputMicrosoft Excel

descriptive statistics output,

using the house price data:

House Prices:

$2,000,000500,000300,000100,000100,000

4561 2658-2-1909 69 descriptive statisticsppt

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