Session - 6 Statistics and Data Analysis

M. S. Ramaiah University of Applied Sciences

1

Data Analysis

Session Speaker

K.M. Sharath Kumar

Session 6


22

Session Objectives

>_To explain the relevance of data analysis for carrying outresearch

>_To explore different types of data analysis techniques foreffective interpretation

>_To critique and recommend appropriate exploratory dataanalysis techniques for a problem


33

Session Outline

Sampling Design

Data Collection Methods

Quantitative and Qualitative Data Analysis

Stages in Data Analysis

Review of Techniques

Error Analysis


44


55

One Variant

6,200 Distinct Parts

Imported from 17 Countries

From 240 Suppliers

Assembled in 1 Plant

Within few minutes

Exported to 34 Countries

Same day

Without becoming inventory!

Suzuki Grand Vitara


6

The secret of success is to know something nobody else knows

- Aristotle Onassis


7

Turn Data into InsightInsight into Action

Action into Tangible Results

- Accenture


8

Data Analysis (1/2)

Explore relationships among the variables

Partition the total variability (by statement / variance component analysis)

Handle noisy data appropriately

Questions to be answered:

Is the process stable?

Is the process capable of meeting specifications?

What are the major sources of variation (noise, etc)?

Listen to what the data is saying


9

Data Analysis (2/2)

Data Analysis is carried out in two distinct environment

Result of a special study or Experiment

By product of some operations or Observational

Experimental Studies

Here we compare various condition and try to determinewhich condition is better. We have finite amount of data andcarry out one time analysis

Observational Studies

Here we get data from steady state process and trying to findout any unplanned change is occurred or not. Generally weperform a sequential analysis using a continuing stream ofdata


10

Quantitative vs. Qualitative

Explanation through numbers

Objective

Deductive reasoning

Predefined variables and measurement

Data collection before analysis

Cause and effect relationships

Explanation through words

Subjective

Inductive reasoning

Creativity, extraneous variables

Data collection and analysis intertwined

Description, meaning

Classification of Data Analysis


1111

Ambushed Every Where


12

Data analysis should be:

Supported by data

Shown in graphical and statistical format

Not based on intuition

Make sense from an engineering standpoint

Data and Hard Evidence!!


13

Key Components of a Data Analysis Plan

Purpose of the evaluation

Questions

What you hope to learn from the question

Analysis technique

How data will be presented


1414

Types of Data

Continuous Data

Discrete Data


1515

Continuous Data

Data generated by

Physically measuring the characteristic

Generally using an instrument

Assigning an unique value to each item

Examples:

Time to receive a shipment, Time spend per page, Time to

activate, CPU Speed, Total Minutes per Incident (TMPI),

etc.

Hardness, Strength, Weight, Diameter, etc.


1616

Discrete Data

Data generated by

Classifying the items into different groups based on

some criteria

No physical measurement is involved

Examples:

Sex, Shade variation, Surface defects etc.

% of visitors signing in for AOL messenger per day,

Number of Recharges per Month , Number of Operating

Systems, % Escalations, etc .


1717

Continuous Data: Example (Time spend per page visit (in

minutes))

SL No. Data SL No. Data

1 0.98 11 1.02

2 1.03 12 0.98

3 1.00 13 1.01

4 1.00 14 1.01

5 0.99 15 0.99

6 1.01 16 1.00

7 0.97 17 1.01

8 1.02 18 0.99

9 1.00 19 1.00

10 0.99 20 1.02


1818

Continuous Data: Example (Time spend per visit (in

minutes)) Graphical Representation

0

1

2

3

4

5

6

0.9631 0.9731 0.9831 0.9931 1.0031 1.0131 1.0231 1.0331


19

Random Variables

BBBB

BGBB

GBBB

BBBG

BBGB

GGBB

GBBG

BGBG

BGGB

GBGB

BBGG

BGGG

GBGG

GGGB

GGBG

GGGG

0

1

2

3

4

X

Sample Space

Points on the

Real Line


20

Suppose, the random variable X = 3 when any of the four outcomes BGGG, GBGG, GGBG, or GGGB occurs,

P(X = 3) = P(BGGG) + P(GBGG) + P(GGBG) + P(GGGB) = 4/16

The probability distribution of a random variable is a table that lists the possible values of the random variables and their associated probabilities.

x P(x)0 1/161 4/162 6/163 4/164 1/16

16/16=1

Random Variables (Continued)

The Graphical Display for this Probability Distributionis shown on the next Slide.


21

Random Variables (Continued)

Number of Girls, X

Pro

ba

bili

ty,

P(X

)

43210

0.4

0.3

0.2

0.1

0.0

1/16

4/16

6/16

4/16

1/16

Probability Distribution of the Number of Girls in Four Births


22

Consider the experiment of tossing two six-sided dice. There are 36 possible

outcomes. Let the random variable X represent the sum of the numbers on

the two dice:

2 3 4 5 6 7

1,1 1,2 1,3 1,4 1,5 1,6 8

2,1 2,2 2,3 2,4 2,5 2,6 9

3,1 3,2 3,3 3,4 3,5 3,6 10

4,1 4,2 4,3 4,4 4,5 4,6 11

5,1 5,2 5,3 5,4 5,5 5,6 12

6,1 6,2 6,3 6,4 6,5 6,6

x P(x)2 1/363 2/364 3/365 4/366 5/367 6/368 5/369 4/3610 3/3611 2/3612 1/36

1

12111098765432

0.17

0.12

0.07

0.02

x

p(x

)

P robab ility Dis tribution o f S um of Two Dice

Example


23

NORMAL DISTRIBUTION


2424

Generic Causes Of Variation

Machines

Materials

Methods

Measurements

Mother Nature

People

PROCESS


25

Center of the barSmooth curve interconnecting the center of each bar

Units of Measure

THE NORMAL CURVE


26

If the frequency distribution of a set of values is such that :

68.26% of the values lie within 1 from the meanAND

95.46% of the values lie within 2 from the meanAND

99.73% of the values lie within 3 from the mean

Then the distribution is normal.

NORMAL DISTRIBUTION IS CHARACTERISED BY A BELL SHAPED CURVE.

Normal Distribution


27

Standard Normal Distribution

Since each normal variables have different units of measurement

Standard Normal Distribution can tackle this

Standard Normal Variable Z = (x ) /

First convert the original problem into Z. The probability table for Zwill be available


28

Sampling Design


2929

Population (N) Sample (n)

Samples and Populations


30

Sampling Design within the Research Process

Draw

sample

Question hierarchy

Sample Type

Sampling

technique

Define Relevant

Population

Identify existing

sampling frame

Evaluate

sampling frame

Modify

sampling frame

Dont

accept

Probability

Non-Probability

Select

sampling frame


31

Types of Sampling

Probability Sampling Non-Probability

Sampling

Simple

Random

Sampling

Stratified

Random

Sampling

Systematic

Sampling

Cluster

Sampling

Convenience

Sampling

Quota

Sampling

Expert

Sampling


32

In stratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2+...+ Nm =N.

Stratified Random Sampling

2 Stratum

1 Stratum

mStratum

1N

2N

mN

The m strata are non-overlapping.

NNm

i i

1

Population


33

Systematic Random Sampling

Units are drawn from the population at regular intervals clearly defined

Steps

- Compute K =(N/n) and take integer value. K is called sampling interval

- Select a random number between 1 and k

- Starting with this number, select every kth number until all the n units are selected


34

Example

Suppose in a market survey, you have to select 5 households out of 50 households in a block.

- Number of units in the population N = 50

- Number of units in the sample n = 5

- Sampling Interval K = (N/n) = 50/5 = 10

- Select a random number between 1 and 10

Suppose the selected random number is 5. Starting with 5, select every 10th unit.


35

Example Contd.

1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 17 1819 20 21 22 23 24 25 26 2728 29 30 31 32 33 34 35 3637 38 39 40 41 42 43 44 4546 47 48 49 50


36

7654321Group

Population Distribution

In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).

Sample Distribution

In cluster sampling observations are drawn from m out of M areas or clusters of the population.

Cluster Sampling


37

Caution

None of the Non-probability sampling should be generalised about the population


38

Sampling Distribution

- A conceptual framework


39

Sampling Distribution of the Mean from Normal Population

If X1, X2,.., Xn are n independent random samples drawn from a normal population with mean and standard deviation ,

then

the sampling distribution of X follows a normal distribution with mean and standard deviation / sqrt(n)

Standard deviation of the sample mean = = standard error

nnX XXXX ni

.......21

n


40

Standard error

of statistic

Sample size = n

Sample size = 2n

Standard error

of statistic

The sample size determines the bound of a statistic, since the standard error of a statistic shrinks as the sample size increases:

Sample Size and Standard Error


41

Determining Sample Size


42

Determining Sample Size using Confidence Interval

If we know the precision (sampling error), the confidencelevel, and the standard deviation of the originalpopulation the sample size can be determined


43

Sample Size Determination Population Mean

n

xZ

Sampling Error E = X , squaring both sides we get

EZn

2

22

Where Z is the value corresponding to the area of

((1-) / 2) from the mean of the standard normal

distribution


44

Example

A marketing manager of a fast food restaurant in a city wishes toestimate the average yearly amount that families spend on fastfood restaurants. He wants the estimate to be within + or Rs. 100with a confidence interval of 99%. It is known from an earlier pilotstudy that the standard deviation of the family expenditure on fastfood restaurant is Rs. 500. How many families must be chosen forthis problem?


45

Solution

Applying the formula

n = ((2.58^2) * (500^2)) / (100^2) = 166.41

= 166 (ROUNDED OFF)

EZn

2

22


46

Sample Size Determination Population Proportion

We know

Sampling Error E = (p-P), squaring both sides and simplifying

We get:

n

pp

PpZ

)1(

EZ ppn

2

2)1(

Where Z is the value corresponding to the area of

((1-) / 2) from the proportion of the standard

normal distribution


47

Example

A company manufacturing sports goods wants to estimate theproportion of cricket players among high school students in India.The company wants the estimate to be within + or 0.03 with aconfidence interval of 99%. A pilot study done earlier reveals thatout of 80 high school students, 36 students play cricket. Whatshould be the sample size?


48

Solution

p = 36/80 = 0.45

Applying the formula

n = ((2.58^2) (0.45(1-0.45)))/(0.03^2)

n = 1831


49


Primary Data Collection

Secondary Data Collection


50


Primary Data

Observation method

Interview method

Questionnaires

Warranty cards

Mechanical devices

Secondary Data

Agency

Published material etc.


51

Scales of Measurement

Nominal Scale - groups or classes

Gender

Ordinal Scale - order matters

Ranks (top ten videos)

Interval Scale - difference or distance matters has arbitrary zero value

Temperatures (0F, 0C)

Ratio Scale - Ratio matters has a natural zero value

Salaries

Likerts Scale


52

Sample Rating Scales

Simple category scale: (data: nominal)

Ex:

I plan to purchase a laptop in next twelve months

Yes

No


53


Multiple choice Single response scale (data: nominal)

Ex:

What newspaper do you read most often?

TOI

DH

The Hindu

Mint

Others (specify:_________)


54


Likert Scale (data: interval)

Ex:

The internet is superior to traditional libraries for comprehensive searches

Strongly Agree Neutral Disagree Strongly

Agree Disagree


55


Semantic Differential Scale (data: interval)

Ex:

Lands end catalog

Fast ___: ___ : ___ : ___ : ___ : ___ : ___ : Slow


56


Numerical Scale (data: ordinal or interval)

Ex:

Extremely 5 4 3 2 1 Extremely

Favourable Unfavourable

Employees cooperation in teams___

Employees knowledge of task ___

Employees planning effectiveness ___


57


Multiple rating list scale (data: interval)

Ex:

Please indicate how important or unimportant each service characteristic is:

Important Unimportant

Fast reliable repair 7 6 5 4 3 2 1

Service at my location 7 6 5 4 3 2 1

Maintenance by manufacturer 7 6 5 4 3 2 1

Knowledgeable technicians 7 6 5 4 3 2 1

Service contract after warranty 7 6 5 4 3 2 1


58


Constant-Sum Scale (data: ratio)

Ex:

Taking all the supplier characteristics we have just discussed and now considering cost, what is their relative importance to you (dividing 100 units between)

Being one of the lowest cost suppliers

All other aspects of supplier performance

Sum 100


59


Stapel Scale (data: ordinal or interval)

Ex:

Company Name

+3 +3 +3

+2 +2 +2

+1 +1 +1

Technology Existing Reputation

Leader Products

-1 -1 -1

-2 -2 -2

-3 -3 -3


60


Graphic rating scale (data: ordinal or interval or ratio)

Ex:

How likely are you to recommend complete care to others?

Very Likely Very Unlikely


61

Data Analysis


62

The cure for boredom is curiosity ,There is no cure for curiosity

- Dorothy Parker


63

Things arent always what we think!

Six blind men go to observe an elephant. One feels the side and thinks the

elephant is like a wall. One feels the tusk and thinks the elephant is a like a

spear. One touches the squirming trunk and thinks the elephant is like a

snake. One feels the knee and thinks the elephant is like a tree. One

touches the ear, and thinks the elephant is like a fan. One grasps the tail and

thinks it is like a rope. They argue long and loud and though each was partly

in the right, all were in the wrong.

For a detailed version of this fable see:

http://www.wordinfo.info/words/index/info/view_unit/1/?letter=B&spage=3

Blind men and an elephant

- Indian fable


64

Stages in Data Analysis

Editing

coding

Data entry

Key Boarding

Data

Analysis

Descriptive

analysis

Univariate

analysis

Bivariate

analysis

Multivariate

analysis

Interpretation

Err

or

chec

kin

g

and ver

ific

atio

n


65

Descriptive Analysis Techniques

Count (frequencies)

Percentage

Mean

Mode

Median

Range

Standard deviation

Variance

Ranking


66

Overview of the Stages in Data Analysis

Editing

coding

Data entry

Key Boarding

Data

Analysis

Descriptive

analysis

Univariate

analysis

Bivariate

analysis

Multivariate

analysis

Interpretation

Err

or

chec

kin

g

and ver

ific

atio

n


67

Frequency Distributions

To what extent did you increase your skills in

putting together a household budget?

A lot Some A little Not at all

Women (N=30) 14 9 5 2

Uni-variate Analysis The analysis of a single variable, for

purposes of description (examples: frequency distribution,

averages, and measures of dispersion)


68

Percentage Distributions

To what extent did you increase your skills in

putting together a household budget?

A lot Some A little Not at all

Women (N=30) 46% 30% 17% 7%


69

Graphing Frequency DataHow did you first hear about the web site?

N Percent

Court Referral 10 24.4%

Social Worker 5 12.2%

Friend 5 12.2%

Web Search Engine 8 19.5%

Librarian 9 22.0%

Newspaper Story 3 7.3%

Other 1 2.4%

41 100.0%

How did you first hear about the web-site?

Court Referral

Social Worker

Friend or Acquaintance

Web Search EngineLibrarian

Newspaper Story

Other


70

Means and Medians

History 95

English 96

Biology 93

Latin 92

Math 98

Music 94

Gym 40

Mean = 87

Median = 94

Math 98

English 96

History 95

Music 94

Biology 93

Latin 92

Gym 40


71

Note

40 50 55 94 100 100 100

40 92 93 94 95 96 98

Mean = 81

Mean = 87


72

Histograms

0

1

2

3

4

5

6

60 70 80 90 100

Fre

qu

en

cy

0

1

2

3

4

5

6

7

60 70 80 90 100

Fre

qu

en

cy


73

Cross Tabulations

Program Type Area of Inquiry Outcome

Web site Employment law Satisfied

I & R Line Family law Not satisfied

Law clinic Immigration Pending

Web site Immigration Satisfied

I & R Line Immigration Satisfied

I & R Line Family law Not satisfied

Web site Employment law Not satisfied

Law clinic Other Satisfied

I & R Line Other Not satisfied

I & R Line Other Satisfied

Law clinic Employment law Satisfied

Web site Family law Satisfied

Law clinic Family law Satisfied

Web site Immigration Not satisfied

Law clinic Immigration Not satisfied

I & R Line Family law Satisfied

I & R Line Immigration Not satisfied

I & R Line Employment law Not satisfied

Law clinic Other Pending

Count of Outcome Outcome

Program Type Not satisfied Pending Satisfied Grand Total

I & R Line 7 5 12

Law clinic 1 3 7 11

Web site 6 5 11

Grand Total 14 3 17 34

Count of Outcome Outcome

Program Type Not satisfied Pending Satisfied Grand Total

I & R Line 58% 0% 42% 100%

Law clinic 9% 27% 64% 100%

Web site 55% 0% 45% 100%

Grand Total 41.18% 8.82% 50.00% 100.00%


74

Graphing comparisons

Satisfaction with Services

0

5

10

15

20

25

30

35

40

A B C D E

Clinic Name

Sati

sfact

ion

Sco

re


75


0

2

4

6

8

10

12

14

16

A B C D E

Clinic

Sa

tisf

act

ion

Sco

re

Staff

Advice

Facility


76


0

2

4

6

8

10

12

14

16

Staff Advice Facility

Satisfaction Component

Sa

tisf

act

ion

Sco

re

A

B

C

D

E


77


Editing

coding

Data entry

Key Boarding

Data

Analysis

Descriptive

analysis

Univariate

analysis

Bivariate

analysis

Multivariate

analysis

Interpretation

Err

or

chec

kin

g

and ver

ific

atio

n


78

Bi-variate Analysis

The analysis of two variables simultaneously fordetermining the empirical relationship between them

Y = f (X)


79

Few Techniques Available

Correlation

Regression

Chi-square Test and Cramers rule

Hypothesis Test for two population means/proportions

Paired T-tests comparing two groups


80

Measure of Correlation: Coefficient of Correlation Symbol : r

Range : -1 to 1

Sign : Type of correlation

Value : Degree of correlation

Examples:

r = 0.6 , 60 % positive correlation

r = -0.82, 82% negative correlation

r = 0, No correlation


81

Regression

Regression helps

To identify the exact form of the relationship

To model output in terms of input or process variables

y = a + b x

Examples:

Yield = 5 + 3 x Time

Y = 2 - 5x


82

Coefficient of Regression

Measure of degree of Relationship

Symbol : R2

Range of R2 : 0 to 1

If R2 > 0.6, the Model is reasonably good


8383

Error or Residual Analysis

Root Mean Square Error for Prediction

(MSEP)

Regression Statistics

Multiple R 0.594159006

R Square 0.353024925

Adjusted R Square 0.191281156

Standard Error 27.80337004

Observations 6

Coefficients

Intercept 83.00449781

x -0.605970474

x y

65 69

8 78

89 8

88 21

50 24

73 72


8484

Root Mean Square Error:

Predicted y = 83.0045 0.6059 x

Error = y predicted y

Mean Square Error = 3092.11 / 6 = 515.35

Root Mean Square Error = 22.70

x y Predicted y Error Error Square

65 69 43.62 25.38 644.33

8 78 78.16 -0.16 0.02

89 8 29.07 -21.07 444.08

88 21 29.68 -8.68 75.33

50 24 52.71 -28.71 824.03

73 72 38.77 33.23 1104.32

3092.11Sum


85

Difference between Observed Values Yi and model predicted

values f(Xi) for n datasets

Decomposition of MSEP has been carried out using mean bias (UM), slope bias (UR) and random error (UD)

MSEPYXfU iM /)(

2_

MSEPbSU jXR /)1(* 2

2

MSEPSrU YD /*122


86

Objective

To develop a mathematical model for an attribute or response metric

(Y) in terms of other available attributes (Xs).

When to Use

Xs : Continuous

Y : Discrete binary

Logistic Regression


87

Hypothesis Test for Difference between Two Means

Objective

To test hypothesis that compare the population mean of interestfor two separate populations (independent samples)

Test Statistic (Large Sample) Test Statistic (Small Sample)

nn

XXZ

2

2

2

1

2

1

21

nnS

XXt

21

2

21

11


88

Chi-Square Test

Objective:

To test whether two variables which have frequency data are related or not

Usage:

When both the variables ( X & Y) are categorical (grouped)

Cramers Rule: To quantify the relationship between X & Y


89


Editing

coding

Data entry

Key Boarding

Data

Analysis

Descriptive

analysis

Univariate

analysis

Bivariate

analysis

Multivariate

analysis

Interpretation

Err

or

chec

kin

g

and ver

ific

atio

n


90

Multivariate Analysis

The analysis of the simultaneous relationships among several variables

Analyse the data covariance structure to understand it orto reduce the data dimension

Assign observations to groups

Explore relationships among categorical variables


91

Few Techniques Available

Multiple Linear Regression

Cluster Analysis

Factor Analysis

ANOVA

MANOVA

Conjoint Analysis

Optimisation Techniques .


92

Multiple Regression

To model output variable y in terms of two or more variables

General Form:

Y = a + b1X1 + b2X2 + - - - + bkXk

Two variable case:

Y = a + b1X1 + b2X2

Adjusted R2

If Adj R2 > 0.6, then the model is reasonably good

P value from coefficient table

If p value < 0.05, the corresponding term has strong relationship with output


93

Residual Plots: Error Analysis

Y = 44+0.19X1-2.55X2


94

Evidence of a strong Shift to Shift Effect

TimeShiftDay

43212121

0.038

0.033

0.028

0.023

0.018

Impurity

Main Effects Plot - Data Means for Impurity

Main Effects Plot


95

Validation Tests for Model Adequacy

Mean Square Error (MSE) for checking Model Precision

Mean Bias (MB) for checking Model Accuracy

where, f(Xi)= ith model Prediction

2

1

2^

n

YY

MSE

n

i

ii

n

XfY

MB

n

i

ii

1)(


9696

First Factor

Seco

nd F

acto

r

1.00.80.60.40.20.0-0.2-0.4

0.75

0.50

0.25

0.00

-0.25

-0.50

Home

Health

Employ

School

Pop

Loading Plot of Pop, ..., Home

Explain the presence of each variable with the sign (+ or -). This

way we can reduce the number of variables

Factor Analysis


97

Predictors Selection


98

P = 0.001


99

Classification Methods

Example:

Attribute 1 x1

Attribute 2 x2

Label : y y1 (Red) , y2 (Blue)

20

22

24

26

28

30

32

34

36

38

40

10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

x1

x2

x2

x1y1 y2

> 35 < 28

y2 y1

< 15.5 > 15.5


100100

CLASSIFICATION METHODS

Example: Rules

Attribute 1 x1

Attribute 2 x2

Label : y y1 (Red) , y2 (Blue)

x2

x1y1 y2

> 35 < 28

y2 y1

< 15.5 > 15.5

If x2 > 35 then y = y1

If x2 < 28, then y = y2

If 28 > x2 > 35 & x1 > 15.5, then y = y1

If 28 > x2 > 35 & x1 < 15.5, then y = y2


101

Cluster Analysis

Objective

To classify the records or items into a smaller number of groups based

on the values of available attributes.

When to Use

When there is no Y attribute

All attributes are considered as Xs only


102

K-Nearest Neighbors Cluster Analysis

Weight in kg

Weight in kg

Acc

eler

ati

on

in

m/s

2

Acc

elera

tion

in

m/s

2


103

ANOVA or Experimental Design

Sometimes, an investigator would like to compare more than twopopulation means in a problem situation

ANOVA decomposes the total variation into components ofvariation

1 2 3

Population 1 Population 2 Population 3


104

MANOVA and Conjoint Analysis

MANOVA is similar to the ANOVA with added ability to handleseveral dependent variables

The most common applications of conjoint analysis are marketresearch and product development for making trade-offs


105

Optimisation Methods

Objective

To identify the best values of a set of variables (Xs) which will optimize an objective function satisfying a given set of constraints

For n variables in m constraints

Max / Min Z = C1x1 + C2x2 + .CnxnSubject to

a11 x1 + a12x2 + . + a1nxn < /> = b1a21 x1 + a22x2 + . + a2nxn < /> = b2

am1 x1 + am2x2 + . + amnxn < /> = bm

And xi > 0, I = 1,2,.n


106

You never know what is enough unless you know what is more than enough

- William Blake


107

Session Summary (1/2)

Statistical Techniques and Tools:

Completely dependent on type of data used (continuous ordiscrete)

Normal Distribution:

Describes many natural phenomena, industrial and scientificsituations. A normal curve is a graphical representation to describethe normal distribution

Data Analysis is carried out in two distinct environment:

Result of a special study or Experiment

By product of some operations or Observational


108

Session Summary (2/2)

Uni-variate Analysis:

The analysis of a single variable, for purposes of description(examples: frequency distribution, averages, and measures ofdispersion)

Bi-variate Analysis:

The analysis of two variables simultaneously for determining theempirical relationship between independent and dependentvariables

Multi-variate Analysis:

The analysis of the simultaneous relationships among severalvariables

Documents

Session - 6 Statistics and Data Analysis