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Measurement of central tendency

Measurement of dispersionCorrelation Regression

Statistical methods

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Data ts types

Definition ofData: Facts, figures, enumerations & other materials, pastand present, serving as basis for study and analysis; they are raw

material for analysis; provide basis for testing hypothesis, developingscales and tables Data help researchers draw inferences on specific issues/

problems Quality of findings depend on relevance, adequacy & reliability of

data Types of data (Not in statistical sense)

A.1. Personal data (Individual as a source) Demographic & socio-economic Characteristics Behaviour variables Attitude, behaviour, opinions Awareness, preferences, knowledge

Practices, intensions

2. Organisational data (Organisational sources) Archives ,Manuscript library, museums

3. Territorial data Economic structure, occupation pattern

B. I Secondary (Paper method)

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Methods & Techniques ofData Collection

I-Secondary data

How to scrutinize

Published & unpublished

Methods where used

A-Meta analysis

B- Historical method

C-Content analysis D-Informetrics

E-Use studies

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II-Primary data

A-Records & relics B-Observation C-Experimentation D-Simulation E-Ask people orally F-Ask people in writing G-Panel study H-Projective techniques I -Sociometry

J -Case study-Interview / Depth interview / Schedule-Mail survey / questionnaire-Mechanical devices

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Primary Data

Secondary Data-1. Internet sites /webpage of different companies and

organizations2. Central and local govt. studies and reports,3. Rules on international trading, import and exports,

state budgets4. FICCI(federation ofIndian chambers of conference

and industry),CII(Confederation ofINDIANINDUSTRY),ASSOCAM(Associated chamber ofcommerce and Industry).

5. Policies on foreign direct investment

Data Sources

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Skewness and Kurtosis: someexamples

Edu ational Attainment

7.06.05.0

.0

.0

.01.0

Edu ational Attainment

Frequen

1

0

100

80

60

0

0

0

Std. De

= 1.81

ean =

.8

N =

.00

Reason or ermination

17.515.01

.510.07.55.0

.50.0

Reason or ermination

Frequen

80

60

0

0

0

Std. De

= 5.

6

ean =

.6

N = 1

.00

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Pictogram

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Annotated box plot

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Describing Data Numerically

Arithmetic Mean

Median

Mode

Describing Data Numerically

Variance

Standard Deviation

Coefficient of Variation

Range

Interquartile Range

Central Tendency Variation

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Measures of Central Tendency

Central Tendency

Mean Median Mode

n

n

1i

i!!

Overview

Midpoint ofranked values

Most fre uentlyobserved value

Arithmeticaverage

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Arithmetic Mean

The arithmetic mean (mean) is the mostcommon measure of central tendency

For a population ofN values:

For a sample of size n:

Sample size

nnn1

n

1ii

!!

!

. Observedvalues

N

xxx

N

x

N21

N

1ii

!!

! .

Population size

Populationvalues

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Arithmetic Mean

The most common measure of central tendency

Mean sum of values divided by the number of values

Affected by extreme values (outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

35

15

5

54321!!

4

5

2

5

104321!!

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Median

In an ordered list, the median is the middlenumber(50% above, 50% below)

Not affected by extreme values Median L+[(1/2N-C)/f ]h Q2 Compare knowledge level in Two subjects for a

group of students by median

0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

Median = 3

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Quartiles, Deciles.Percentiles

Similar to median which divides data in to parts , Quartiles (dividesdata in four parts), Deciles(divides data in ten parts) and percentiles(divides data in 1000 parts)

Mode 3median-2mode

3,2,1,..4

Qj !

! jhf

fcpjN

L

9,....2,1

..10

Dj

!

!

j

hf

fcpjN

L

99...2,1

..100

Pj

!

!

j

hf

fcpjN

L

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Finding the Median

The location of the median:

If the number of values is odd, the median is the middle number

If the number of values is even, the median is the average ofthe two middle numbers

Note that is not the value of the median, only the

position of the median in the ranked data

dataorderedtheinosition

1n

ositionedian

!

2

1n

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Mode

A measure of central tendency

Value that occurs most often

Not affected by extreme values

Used for either numerical or categorical data

There may be several modes

Mode L+[(f-f-1)/(2f-f-1-f1 )]h

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

Frequency after modalclass

Frequency beforemodal class

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Five houses on a hill by the beach

Review xample

$

$

$

$

$

House Prices:

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

100,000

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Review xample:Summary Statistics

Mean: ($3,000,000/5)

$600,000

Median: middle value of ranked data$300,000

Mode: most fre uent value$100,000

House Prices:

$2,000,000

500,000300,000100,000100,000

Sum 3,000,000

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Example

5 1 Class Freque C.F less C.F More Than

9 2 19 5-10 5 5 49

7 3 20 10-15 6 11 44

9 4 22 15-20 15 26 38

10 5 22 20-25 10 36 23

9 7 17 25-30 5 41 135 7 30-35 4 45 8

Mean 7.714286 4.142857 20 35-40 2 47 4

mode 9 7 22 40-45 2 49 2median 9 4 20

SD 4.238095 5.47619 4.5

Median=L+[(1/2N-C)/f ]h e ( - - ( - - -Median Class=Total Freq/2 Class MODAL CLASS= Max Frequency class

Median Class='15-20 i.e 15 is max fre in freq G21

i.e 26 in Cumulative frequency

Median=15+[((1/2)49-11)/15 ]5 Mode 15+[(15-6)/(2x15-6-10 )]5

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Mean is generally used, unlessextreme values (outliers) exist

Then median is often used, sincethe median is not sensitive toextreme values.

Example: Median home prices may be

reported for a region less sensitive tooutliers

Which measure of locationis the best?

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Geometric mean & Harmonicmean

Geometric mean is nth root of product of n observations ( ex: averagepercent increase in sales, production, ), Best considered in case ofconstructing index number.

Harmonic mean: restricted use such as average rate of increase of

profits average price at which an article has been sold

NX

anti !log

logG.M

,H.M,1

H.M

!

!

X

f

X

N

21

2211 loglog.log NN

GNGN

!

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Same center,

different variation

Measures of Variability

Variation

Variance Standard

Deviation

Coefficient

of Variation

Range Interquartile

Range

Measures of variation give

information on the spreadorvariability of the datavalues.

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Range

Simplest measure of variation

Difference between the largest and the smallest

observations:Range Xlargest Xsmallest

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

Range = 14 - 1 = 13

Example:

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Ignores the way in which data are distributed

Sensitive to outliers

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

Disadvantages of the Range

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119

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