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7/28/2019 Corelation Analysis
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Quantitative Methods
7/28/2019 Corelation Analysis
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Quantitative Methods
Models for Data Analysis & Interpretation:
Correlation Analysis
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Quotable Quotes
• There is a Great Correlation Between
Music and Images. – Graham Nash
• There is Little Correlation Between the
Conditions of People's Lives and How
Happy They Are. – Dennis Prager
• Even Pop Singer and Talk Show Host Talk
About Correlation.
• What Is It?
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Correlation
• Dictionary Says: Correlation is a Close
Connection Between Two Things In Which
One Thing Changes as the Other Does.
• Note the Phrase: Close Connection
• Remember: Correlation Does Not
Necessarily Mean Causation.
• Importance: Use Information About One
To Estimate Values of the Other.
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Scatter Plot
• Scatter Plot is a Visual Representation of
the Relationship Between Two Variables.
• Use the Horizontal Axis for Values of One
Variable.
• Use the Vertical Axis for Values of the
Other Variable.
• Plot the Actual Data.
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Reasoning & Creativity Scores of
Twenty Job Applicants
Apl No, RsnSc CrvSc Apl No, RsnSc CrvSc
01 15.2 11.9 11 8.1 6.8
02 9.9 13.1 12 15.2 13.0
03 7.1 8.9 13 10.9 13.904 17.9 17.4 14 17.2 19.1
05 5.1 6.9 15 8.2 10.1
06 10.0 8.8 16 10.8 15.9
07 7.2 14.0 17 12.0 12.1
08 17.1 15.8 18 13.1 16.0
09 15.2 9.7 19 17.9 19.2
10 9.2 12.1 20 7.1 11.9
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Scatter Plot
Horizontal Axis: Reasoning Scores
Vertical Axis: Creativity Scores
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Basic Patterns of Scatter Plot
Both Move Together Move In Opposite Way No Relationship
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Positive Correlation
• Both Variables Increase Simultaneously or
Decrease Simultaneously.
• Examples:
Your Income and Jeweler's Bills
Exercise and Appetite
Rainfall and Absenteeism Discount and Sales
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Negative Correlation
• As One Variables Increases the Other
Variable Decreases.
• Examples:
TV Viewing and Book Reading
Age and Sleep
Price and Demand Machine Downtime and Production
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Correlation Coefficient
• It Measures the Extent of Quantitative
Relationship Between Two Variables
• Examples:
Rainfall & Sales of Agro-Chemicals
Gold Price & Real Estate Price
Snowfall in Alps & Onion Price in Dadar • Compute Correlation Coefficient Only
Between Logically Related Factors
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Logically Related Variables
• Technical: 1.
2.
3.
• Marketing: 1.2.
3.
• Corporate: 1.2.
3.
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Features of
Correlation Coefficient
• Value Ranges Between -1 and +1.
• Perfect Positive Correlation = +1
• Perfect Negative Correlation = -1• Positive Corr. Coeff.: Two Variables Go
Up or Down Simultaneously
• Negative Corr. Coeff.: Exactly Opposite• Zero Corr. Coeff.: No Relationship At All
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Computing Correlation
• Caution: Method for ComputingCorrelation Coefficient between TwoCardinal Variables is Different from the
One for Two Ordinal Variables• Statutory Warning: Using One Formula
for the Other is Seriously Injurious toCorporate Health.
• So, First Identify the Type of the Variables At Hand: Cardinal or Ordinal.
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Correlation Coefficient
For Cardinal Variables
• Data: Actual Measurements on Both Variables
• Formula: Ratio of {Mean of Products of Values
– Product of the Two Means} to Product of the
Two Standard Deviations
Mean of Products of Values – Product of the Two Means= --------------------------------------------------------------------------
Product of the Two Standard Deviations
• Name: Pearson’s Correlation Coefficient
• But, Your Statistician Calls It Pearson’s r.
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Annual Production of 7 Plants
Plant 2004 (X) 2005 (Y) XY
A 1 4 4
B 3 7 21
C 5 10 50D 7 13 91
E 9 16 144
F 11 19 209
G 13 22 286
Total 49 91 805
Arith Mean 7 13
Std Deviation 4 6
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Pearson’s Correlation Coefficient
of Plant Production
• Formula: Ratio of (Mean of Products of
Values – Product of the Two Means) to
Product of the Two Std. Deviations
(805 / 7) – (7 x 13) 115 - 91= ------------------------ = ---------- = 1
4 x 6 24
• Interpretation: Perfect Correlation 1
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One More Example
Empl. No. Yrs in Co. Salary (‘000) Product
1 2 25 50
2 3 30 90
3 5 37 185
4 7 38 266
5 8 40 320
Total 25 170 911
Arith Mean 5 34
Std. Dev. 2.3 5.6
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Pearson’s Correlation Coefficient
Between Yrs in Co & Salary
• Formula: Ratio of (Mean of Products of Values – Product of the Two Means) toProduct of the Two Std. Deviations
(911 / 5) – (5 x 34) 182.2 - 170= ----------------------- = ------------- = 0.94
2.3 x 5.6 12.9
• Interpretation: Salary and Years of Service in the Company are StronglyCorrelated With Each Other
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One More for Practice
Month Discount% Sales Product
Nov 2 25 50
Dec 5 38 190
Jan 3 37 111
Feb 7 30 210
March 8 40 320
Total 25 170 881
Arith Mean 5 34
Std. Dev. 2.3 5.6
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Pearson’s Correlation Coefficient
Between Discount & Sales
• Formula: Ratio of (Mean of Products of
Values – Product of the Two Means) to
Product of the Two Std. Deviations
(881 / 5) – (5 x 34) 176.2 - 170= ----------------------- = ------------- = 0.48
2.3 x 5.6 12.9
• Interpretation: Sales Do Improve WithDiscounts, But Not Very Significantly.
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One More for Practice
Month M/cDowntime Production Product
Nov 8 25 200
Dec 5 30 150
Jan 7 37 259
Feb 3 38 114
March 2 40 80
Total 25 170 803
Mean 5 34
S. D. 2.3 5.6
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Pearson’s Correlation Coefficient
Between M/c Downtime & Production
• Formula: Ratio of (Mean of Products of
Values – Product of the Two Means) to
Product of the Two Std. Deviations
(803 / 5) – (5 x 34) 160.6 - 170
= ----------------------- = ------------- = -0.73
2.3 x 5.6 12.9
• Interpretation: Significant Negative
Correlation between M/c Downtime & Prod
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Correlation Coefficient
For Ordinal Variables
• Actual Measurements on Both VariablesNot Available
• Data Are In the Form of Ranks
6 x Sum Square of Rank Diff
• Formula: 1 - ---------------------------------------
n x {(Square of n) -1}
where n denotes Number of Observations
• Name: Rank Correlation Coefficient
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Rank Correlation Coefficient
Between Age & Performance
Age Rank Performance
Rank
Difference Square
1 4 3 9
2 2 0 0
3 1 2 4
4 5 1 15 3 2 4
Total 18
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Rank Correlation Coefficient
Between Age & Performance
• Formula:
6 x 18 108
1 - ------------------- = 1 - ------- = 1 - 0.9 = 0.15 x (25 -1) 120
• Interpretation: Age Has Very Little To DoWith Performance
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Frequent Blunders
• People Treat All Variables As Cardinal.
• They Use Pearson’s Formula on OrdinalVariables and Create Havoc with Wrong
Interpretations.• Even for Ranking Data on Cardinal
Variables, They Use Pearson’s Formula
and Draw Misleading Conclusions.• This is an International Disease.
• DO NOT FALL PREY TO IT.
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Tips to Busy Executives
• If One Set of Data is Cardinal and theOther Ordinal, Convert Cardinal ValuesInto Ordinal Ranks, and Then Compute
Rank Correlation Coefficient.• To Get a Quick Measure of the Extent of
Relationship Between Two CardinalVariables, Convert Both Sets of Data IntoOrdinal Ranks, and Compute RankCorrelation Coefficient.
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Rank Correlation Coefficient Between
M/c Downtime & Production
M/c Down
Rank
Prod Rank Difference Square
5 1 4 16
3 2 1 1
4 3 1 1
2 4 2 41 5 4 16
Total 38
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Rank Correlation Coefficient Between
M/c Downtime & Production
• Formula:
6 x 38 228
1 - ------------------- = 1 - ------- = 1 - 1.9 = -0.9
5 x (25 -1) 120
• Interpretation: Strong Negative
Correlation between M/c Downtime & Prod
• Recall: Pearson’s Corr. Coeff. was -0.73
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How Will You Proceed To Work Out
Correlation In Following Pairs
• Adult IQ and Annual Income
• Consumer Price Index and Sensex
• Dealer Seniority and Dealer Performance• Gold Prices and Real Estate Prices
• Birth Rate in Germany and Voter Turnout
in Kerala• WTA Ranking and Height ..