Correlation Quantitative A Aptitude & & Business S - PowerPoint PPT Presentation

Correlation Quantitative A Aptitude & & Business S Statistics

Correlation • Correlation is the relationship that exists betw een tw o or more variables. • If tw o variables are related to each other in such a w ay that change increases a corresponding change in other, then variables are said to be correlated. Quantitative Aptitude & Business 2 Statistics: Correlation

Examples • Relationship between the heights and weights. • Relationship between the quantum of rainfall and the yield of wheat. • Relationship between the Price and demand of commodity. • Relationship between the dose of insulin and blood sugar. Quantitative Aptitude & Business 3 Statistics: Correlation

Uses of Correlation • Economic theory and business studies relationship between variables like price and quantity demand. • Correlation analysis helps in deriving precisely the degree and the direction of such relationships. Quantitative Aptitude & Business 4 Statistics: Correlation

• The effect of correlation is to reduce the range of uncertainty of our prediction . • The prediction based on correlation analysis will more reliable and near to reality. Quantitative Aptitude & Business 5 Statistics: Correlation

Positive correlation • If both the variables are vary in the same direction ,correlation is said to be positive . • If one variable increases ,the other also increases or ,if one variable decreases ,the other also decreases ,then the tw o variables are said to be positive . Quantitative Aptitude & Business 6 Statistics: Correlation

Negative correlation • If both the variables are vary in the opposite direction ,correlation is said to be Negative . • If one variable increases ,the other decrease or ,if one variable decreases ,the other also increases ,then the tw o variables are said to be Negative . Quantitative Aptitude & Business 7 Statistics: Correlation

Types of Correlation • Simple correlation • Multiple correlation • Partial Multiple correlation Quantitative Aptitude & Business 8 Statistics: Correlation

Methods of studying correlation Method of studying Correlation Graphic Algebraic 1.Karl Pearson Scatter Diagram 2.Rank method Method 3.Concurrent Deviation Quantitative Aptitude & Business 9 Statistics: Correlation

Scatter Diagram Method • Scatter diagrams are used to demonstrate correlation betw een tw o quantitative variables. Quantitative Aptitude & Business 10 Statistics: Correlation

Scatter Plots of Data w ith Various Correlation Coefficients Y Y Y X X X r = -1 r = -Ve r = 0 Y Y X X Quantitative Aptitude & Business 11 r = +Ve r = 1 Statistics: Correlation

Features of Correlation Coefficient • Ranges betw een –1 and 1 • The closer to –1, the stronger the negative linear relationship • The closer to 1, the stronger the positive linear relationship • The closer to 0, the w eaker any positive linear relationship Quantitative Aptitude & Business 12 Statistics: Correlation

The value of r lies betw een - 1 and +1 • If r=0 There exists no relationship between the variables • If +0.75 ≤r ≤ +1 There exists high positive relationship between the variables . • If -0.75 ≥ r ≥ -1 There exists high negative relationship between the variables Quantitative Aptitude & Business 13 Statistics: Correlation

• If +0.5 ≤r ≤ 0.75 There exists Moderate positive relationship between the variables . • If -0.50 ≥ r >-0.75 There exists moderate negative relationship between the variables. • If r > -0.50 There exists low negative relationship between the variables • If r <0.5 There exists low positive relationship between the variables . Quantitative Aptitude & Business 14 Statistics: Correlation

Covariance • Definition : Given a n pairs of observations (X 1 ,Y 1 ),(X 2 ,Y 2 ) .,,,,,, (X n ,Y n ) relating to tw o variables X and Y ,the Covariance of X and Y is usually represented by Cov(X,Y) ( )( ) ∑ − − . X X Y Y = ( , ) Cov X Y N ∑ xy = Quantitative Aptitude & Business 15 N Statistics: Correlation

Properties of Co-Variance • Independent of Choice of origin • not Independent of Choice of Scale. • Co-variance lies betw een negative infinity to positive infinity. • In other w ords co-variance may be positive or negative or Zero. Quantitative Aptitude & Business 16 Statistics: Correlation

From the follow ing Data Calculate Co-Variance X 1 2 3 4 5 Y 10 20 30 50 40 Quantitative Aptitude & Business 17 Statistics: Correlation

Calculation of Covariance X X-X=x Y Y-Y=y x.y 1 -2 10 -20 40 2 -1 20 -10 10 3 0 30 0 0 4 1 50 20 20 5 2 40 10 20 =15 =0 =150 =0 =90 Quantitative Aptitude & Business 18 Statistics: Correlation

• N= number of pairs =5 = ∑ = ∑ 15 = 150 = X Y = = X 3 Y 30 N 5 N 5 ( )( ) ∑ − − . X X Y Y = ( , ) Cov X Y N ∑ xy 90 = = = 18 Quantitative Aptitude & Business 19 N 5 Statistics: Correlation

Karl Pearson's Correlation • The most w idely used mathematical method for measuring the intensity or the magnitude of linear relationship betw een tw o variables w as suggested by Karl Pearson's Quantitative Aptitude & Business 20 Statistics: Correlation

Coefficient of Correlation • Measures the strength of the linear relationship betw een tw o quantitative variables n ( )( ) ∑ − − X X Y Y i i = = i 1 r n ( ) n ( ) ∑ ∑ 2 2 − − X X Y Y i i = = 1 1 i i Quantitative Aptitude & Business 21 Statistics: Correlation

Properties of KralPear son’s Coefficient of Correlation • Independent of choice of origin • Independent of Choice Scale • Independent of units of Measurement Quantitative Aptitude & Business 22 Statistics: Correlation

Assumptions of Karl Pearson’s Coefficient of Correlation • Linear relationship between variables. • Cause and effect relationship. • Normality. Quantitative Aptitude & Business 23 Statistics: Correlation

• The correlation coefficient lies betw een -1 and +1 • The coefficient of correlation is the geometric mean of tw o regression coefficients. Quantitative Aptitude & Business 24 Statistics: Correlation

Merits of Karl Pear son’s Coefficient of Correlation • Coefficient of Correlation gives direction as well as degree of relationship between variables • Coefficient of Correlation along with other information helps in estimating the value of the dependent variable from the known value of independent variable . Quantitative Aptitude & Business 25 Statistics: Correlation

Limitations of KralPear son’s Coefficient of Correlation • Assumptions of Linear Relationship • Time consuming • Affected by extreme values • Requires careful Interpretation Quantitative Aptitude & Business 26 Statistics: Correlation

From the follow ing Data Calculate Coefficient of correlation X 1 2 3 4 5 Y 10 20 30 50 40 Quantitative Aptitude & Business 27 Statistics: Correlation

x 2 X X-X=x 1 -2 4 2 -1 1 3 0 0 4 1 1 5 2 4 =15 =0 =10 Quantitative Aptitude & Business 28 Statistics: Correlation

Y Y-Y=y y 2 x.y 10 -20 400 40 20 -10 100 10 30 0 0 0 50 20 400 20 40 10 100 20 =150 =0 =1000 =90 Quantitative Aptitude & Business 29 Statistics: Correlation

• N= number of pairs =5 = ∑ 15 = X ∑ = 3 X xy = N 5 r ∑ ∑ = ∑ × 2 2 150 = x y Y = 30 Y N 5 90 90 = = = + 0 . 9 100 10000 • r=0.9 there exists high degree of positive correlation Quantitative Aptitude & Business 30 Statistics: Correlation

Correlation for Bivariate analysis ( ) ( ) ∑ ∑ f . d f . d ∑ − x y fd . d x y N = r ( ) ( ) ∑ ∑ 2 2 f . dx f . dx ∑ ∑ − − 2 2 f . d f . d x y N N Quantitative Aptitude & Business 31 Statistics: Correlation

Standard error • Standard error of co efficient of correlation is used foe ascertaining the probable error of coefficient of correlation • Where r=Coefficient of correlation • N= No. of Pairs of observations 1 − 2 r = SE N Quantitative Aptitude & Business 32 Statistics: Correlation

Probable Error • The Probable error of coefficient of correlation is an amount w hich if added to and subtracted from value of r gives the upper and low er limits w ith in w hich coefficients of correlation in the population can be expected to lie. It is 0.6745 times of standard error . Quantitative Aptitude & Business 33 Statistics: Correlation

− 2 1 r = Pr obableErro r 0 . 6745 . N Quantitative Aptitude & Business 34 Statistics: Correlation

Uses of Probable Error • PE is used to for determining reliability of the value of r in so far as it depends on the condition of random sampling. Quantitative Aptitude & Business 35 Statistics: Correlation