Correlation 1
Correlation 1
Correlation 1
Correlation
Correlation is a statistical tool that helps to measure and analyze the degree of relationship between the two variables of a bivariate (multivariate) random variable. Correlation analysis deals with the association between two or more variables.
Positive relationship Variables change in the same direction. As X increases, Y increases Indicated by As X decreases, Y decreases sign; (+) or (-). E.g., As height increases, so does weight. As study time increases , so do grades. Negative relationship Variables change in opposite directions. As X increases, Y decreases As X decreases, Y increases E.g., As TV time increases, grades decrease As alcohol consumption increases, driving ability decreases
Multiple
Total
Simple correlation: Only two variables are studied. Multiple Correlation: Three or more variables are studied. Partial correlation: Analysis recognizes more than two variables but considers only two variables keeping the other constant. Total correlation: is based on all the relevant variables, which is normally not feasible.
LINEAR
NON LINEAR
Linear correlation: Correlation is said to be linear when the amount of change in one variable tends to bear a constant ratio to the amount of change in the other. The graph of the variables having a linear relationship will form a straight line. Ex X = 1, 2, 3, 4, 5, 6, 7, 8, Y = 5, 7, 9, 11, 13, 15, 17, 19, Y = 3 + 2X Non Linear correlation: The correlation would be non linear if the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable (e.g. Y = X2).
Scatter Diagram Method Graphic Method Karl Pearsons Coefficient of Correlation Spearmans Coefficient of Correlation Concurrent Deviation Method
Scatter Diagram is a graph of observed plotted points where each points represents the values of X & Y as a coordinate. It portrays the relationship between these two variables graphically.
A linear relationship
Height of A
Height of B
Height
Positive relationship
r = +.80
Weight
Height
Degree of correlation
Shoe Size
Weight
Degree of correlation
Exam score
Degree of correlation
Exam score
Degree of correlation
Weight
Degree of correlation
Height
Simple and non mathematical method Not influenced by the size of extreme item First step in investigating the relationship between two variables
Graphic Method
Separate curves are drawn for the X and Y variables on the same graph paper. The values of the variable are taken as the ordinates of the points plotted. From the direction and closeness of the curves we can infer whether the variables are related. If both the curves move in the same direction (upward or downward) correlation is positive. If the curves move in the opposite direction correlation is negative.
r=
N xy - x y
{N x-(x)} {N y-(y)}
(x x)(y y) [ (x x) ][ (y y)
2
Example
Trunk Diameter
x 8
Tree Height
y 35 xy 280 x2 64 y2 1225
9
7 6 13
49
27 33 60
441
189 198 780
81
49 36 169
2401
729 1089 3600
7
11 12 =73
21
45 51 =321
147
495 612 =3142
49
121 144 =713
441
2025 2601 =14111
Solution
r [n( x 2 ) ( x) 2 ][n( y 2 ) ( y)2 ] 8(3142) (73)(321) [8(713) (73)2 ][8(14111) (321)2 ] n xy x y
0.886
r(x, y) = dxdy / dx dy
N dx-(dx) N dy-(dy)
Calculate the deviations x andy in two series from the assumed/their respective mean. Square each deviation of x &y then obtain the sum of the squared deviation i.e.dx2 and dy2 Multiply each deviation under x with each deviation under y and obtain their product. Then obtain the sum of the product of x , y i.e. dxdy Substitute the value in the formula.
The value of correlation coefficient r ranges from -1 to +1 If r = +1, then the correlation between the two variables is said to be perfect and positive If r = -1, then the correlation between the two variables is said to be perfect and negative If r = 0, then there exists no correlation between the variables
The correlation coefficient lies between -1 & +1 symbolically ( - 1 r 1 ) The correlation coefficient is independent of the change of origin & scale. The coefficient of correlation is the geometric mean of two regression coefficient. r = bxy * byx If one of the regression coefficients is positive the other regression coefficient is also positive and correlation coefficient is positive.
There is linear relationship between two variables, i.e. when the two variables are plotted on a scatter diagram a straight line will be formed by the points
It summarizes in one value, the degree of correlation and the direction of correlation also.
Always assumes linear relationship* Interpreting the value of r is difficult. Value of correlation coefficient is affected by the extreme values. Time consuming methods
Coefficient of Determination
The convenient way of interpreting the value of correlation coefficient is to use the square of coefficient of correlation which is called Coefficient of Determination. The Coefficient of Determination = r2. Suppose: r = 0.9, r2 = 0.81 this would mean that 81% of the variation in the dependent variable has been explained by the independent variable.
Coefficient of Determination
The maximum value of r2 is 1 because it is possible to explain all of the variation in y but it is not possible to explain more than all of it.
Suppose: r = 0.60 r = 0.30 It does not mean that the first correlation is twice as strong as the second; the r can be understood by computing the value of r2 . When r = 0.60 r2 = 0.36 -----(1) r = 0.30 r2 = 0.09 -----(2) This implies that in the first case 36% of the total variation is explained whereas in second case 9% of the total variation is explained .
When statistical series in which the variables under study are not capable of quantitative measurement but can be arranged in serial order, the Pearsons correlation coefficient can not be used. In such cases the Spearman Rank correlation can be used. R = 1- {(6 D2 ) / N (N2 1)} R = Rank correlation coefficient D = Difference of rank between paired item in two series. N = Total number of observation.
The value of rank correlation coefficient, R ranges from -1 to +1 If R = +1, then there is complete agreement in the order of the ranks and the ranks are in the same direction If R = -1, then there is complete agreement in the order of the ranks and the ranks are in the opposite direction If R = 0, then there is no correlation
Equal Ranks or tie in Ranks: In such cases average ranks should be assigned to each individual. R = 1- [{(6 D2 ) + CF }/ N (N2 1)]
This method is simpler to understand and easier to apply compared to Karl Pearsons correlation method. This method is useful where we can give the ranks and not the actual data. (qualitative terms) This method is to use where the initial data in the form of ranks.
Cannot be used for finding out correlation in a grouped frequency distribution. This method should be applied where N exceeds 30.
Show the amount (strength) of relationship present Can be used to make predictions about the variables under study. Can be used in many places, including natural settings, libraries, etc. Easier to collect corelational data
Regression Analysis
Regression Analysis is a very powerful tool in the field of statistical analysis in predicting the value of one variable, given the value of another variable, when the 2 variables are related to each other.
Regression Analysis
Regression Analysis is a mathematical measure of average relationship between two or more variables. Regression analysis is a statistical tool used in prediction of value of unknown variable from known variable.
Regression analysis provides estimates of values of the dependent variables from the values of independent variables. Regression analysis also helps to obtain a measure of the error involved in using the regression line as a basis for estimations . Regression analysis helps in obtaining a measure of the degree of association or correlation that exists between the two variable.
Existence of actual linear relationship. The regression analysis is used to estimate the values within the range for which it is valid. The relationship between the dependent and independent variables remains the same till the regression equation is calculated. The dependent variable takes any random value but the values of the independent variables are fixed. In regression, we have only one dependant variable in our estimating equation. However, we can use more than one independent variable.
Regression line
Regression line is the line which gives the best estimate of one variable from the value of any other given variable. The regression line gives the average relationship between the two variables in mathematical form.
Regression line
For two variables X and Y, there are always two lines of regression Regression line of X on Y : gives the best estimate for the value of X for any specific given values of Y X=a+bY a = X - intercept
b = Slope of the line X = Dependent variable Y = Independent variable
Regression line
For two variables X and Y, there are always two lines of regression Regression line of Y on X : gives the best estimate for the value of Y for any specific given values of X Y = a + bX a = Y - intercept
b = Slope of the line Y = Dependent variable X= Independent variable
In case of perfect correlation (positive or negative) the two line of regression coincide. If the two regression lines are far from each other then degree of correlation is less, and vice versa. The mean values of X and Y can be obtained as the point of intersection of the two regression line. The higher degree of correlation between the variables, the angle between the lines is smaller and vice versa.
Regression Equation of Y on X Y = a + bX In order to obtain the values of a & b y = na + bx xy = ax + bx2 Regression Equation of X on Y X = c + dY In order to obtain the values of c & d x = nc + dy xy = cy + dy2
Regression Equation of Y on X: Y Y = byx (X X) byx = (nxy - x y ) / (nx2 - (x) 2 ) byx = r (y / x ) Regression Equation of X on Y: X X = bxy (Y Y) bxy = (nxy - x y ) / (ny2 - (y) 2 ) bxy = r (x / y )
The coefficient of correlation is geometric mean of the two regression coefficients. r = byx * bxy If byx is positive than bxy should also be positive & vice versa. If one regression coefficient is greater than one the other must be less than one. The coefficient of correlation will have the same sign as that our regression coefficient. Arithmetic mean of byx and bxy is equal to or greater than coefficient of correlation. byx + bxy / 2 r Regression coefficient are independent of origin but not of scale.
Standard Error of Estimate is the measure of variation around the computed regression line. It measures the reliability of the regression equation. If standard error is small the equation gives us a good estimate of the dependent variable.
Y = Observed value of y Ye = Estimated values from the estimated equation that correspond to each y value e = The error term (Y Ye) n = Number of observation in sample.
X = Value of independent variable. Y = Value of dependent variable. a = Y intercept. b = Slope of estimating equation. n = Number of data points.
Regression is the average relationship between two variables Correlation need not imply cause & effect relationship between the variables understudy.- R A clearly indicate the cause and effect relation ship between the variables. There may be non-sense correlation between two variables.- There is no such thing like non-sense regression.