CORRELATION.pptx

Correlation- If two variables are so inter-related in such a manner that change in one variable brings about change in the other variable, then this type of relation of variable is known as correlation.
Types of Correlation.
1.Based on the direction of change of variables
a. Positive
correlation
b. Negative
correlation
2. Based upon the number of variables studied
a. Simple
correlation
b. Partial correlation
c. Multiple correlation
3. Based upon the constancy of the ratio of change between the variables
a. Linear correlation
b. Non-linear correlation
METHODS OF STUDYING CORRELATION
1) GRAPHIC
METHODS
A) SCATTER DIAGRAM
B) CORRELATION
GRAPH
2). ALGEBRIC METHOD
A) KARL PEARSON COEFFICIENT OF CORRELATION
B) RANK CORRELATION METHOD
C) CONCURRENT DEVIATION METHOD
Uses of Correlation.
Merits of Correlation.
Demerits of Correlation.

1. BIOSTATISTICS AND RESEARCH METHODOLOGY Unit-1: CORRELATION PRESENTED BY Gokara Madhuri B. Pharmacy IV Year UNDER THE GUIDANCE OF Gangu Sreelatha M.Pharm., (Ph.D) Assistant Professor CMR College of Pharmacy, Hyderabad. email: [email protected]

2. CORRELATION If two variables are so inter-related in such a manner that change in one variable brings about change in the other variable, then this type of relation of variables is known as correlation. If we change the value of one variable that will make corresponding change in the value of other variable on an average, then we can say that the two variables are in correlation. The value of correlation coefficient will vary from -1 to +1. According to Croxton and Cowden, “When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in brief formula is known as correlation.” According to E. Davenport “It refer to that interrelation between separate characters by which they tend to move together at least in some degree”.

3. TYPES OF CORRELATION

4. A. POSITIVE, NEGATIVE AND CORRELATION If the values of two variables move in the same direction i.e. if the value of one variable increases (or decrease), then value of other variable also increases (or decreases) on an average, then that is said to be positive correlation. Example: Height and Weight (as height increases weight also increases). If the value of one variable increases (or decreases), then the value of other variable decreases (or increases) on an average or in a simple manner, if the value of both variable moves in opposite direction, then it is said to be negative correlation. If the change in the value of one variable will not affect the value of other variable then the correlation is zero. Example of negative correlation X 1 2 3 4 5 6 Y 70 60 50 40 30 20 Example of positive correlation X 1 2 3 4 5 6 Y 10 20 30 40 50 60

5. B. SIMPLE, PARTIAL AND MULTIPLE CORRELATION If we study the relationship between two variables x and y then it is called simple correlation . Example: height and weight. If we study the relationship between two variables, keeping all the other variables as constant, then it is called as partial correlation . If we study the relationship between more than two variables then it is called as multiple correlation . In multiple correlation we measure the degree of relationship between one variable on one side and combined effect of all other variable on the other side. Note Since the value of correlation lies between -1 and +1(i.e. -1≤ r ≤ + 1) If r > 0, we say that a positive correlation occur between variables. If r < 0, we say that a negative correlation occur between variables. If r = 0, we say that no correlation occur between variables.

6. C. LINEAR AND NON – LINEAR CORRELATION If the change in values of one variable makes a constant ratio with the change in value of other variable, then such type of relation is known as linear correlation. Example: In scatter diagram, all the points lies in a straight line. x 1 2 3 4 5 6 y 0.8 1.5 3.8 4.4 4.5 6 NON- LINEAR CORRELATION The correlation is said to be non-linear if the value in one variable does not make a constant ratio with change in the value of other variable. Example: In scatter diagram, all the points does not lies in a straight line. x 10 20 30 40 50 60 y 1 1 5 2 3 1

7. METHODS OF STUDYING CORRELATION

8. a) SCATTERED DIAGRAM In the study of correlation between two variables by using graphical method, first we draw scatter diagram, for which we take the value of one variable on x-axis and the value of other variable on y-axis. The resulting graph is a scattered point or dot in a graph sheet known as scatter diagram. There are various types of scatter diagrams: Perfect positive correlation All the points are in correlation. The straight line in upward direction (left bottom to right up), the correlation scatter diagram showing positive correlation is a perfect positive. Highly positive If all the points are very near to straight line in upward direction, then we say it as a highly positive correlation. Positive correlation: I f all points are near to the straight line(but not very near) the correlation is positive (r = +1). Perfect negative: I f all the points in a scattered diagram lies in a straight line in downward direction(left top to right bottom), the correlation is perfect negative(r = -1 ). Highly negative : If the points are very close to straight line in downward direction, the correlation is high negative.

9. Negative: If the points are close to straight line(not very close) in downward direction, the correlation is negative. Zero correlation : If all the points are widely scattered in a graph, the correlation is said to be zero. b) KARL PEARSON’S COEFFICIENT OF CORRELATION Karl pearson’s coefficient of correlation is used to measure the degree of linear relationship between two variables. It is also called as moment correlation coefficient. It is denoted by ‘r’ . There are 2 methods. Where X = x – and Y = y – N = number of pairs of values of variables = Standard deviation Another form of correlation coefficient is r = Where Cov(XY) = S.D.(X) = Standard deviation for x series  

10. a) DIRECT METHOD If x and y are two variates having their means and respectively, then Where )2 It can also be written as r xy = Where n is number of observations in X or Y series x, y are standard deviation of X and Y respectively. The following formula can also be deduced from above Where n is the number of observations  

11. Working Rule The coefficient of correlation is calculated by the following steps: Step – 1: Denote one series by x and the other series by y. Step – 2: Calculate and of the x and y series respectively. Step – 3: Take the deviations of the observations in x series from and write it under the column headed by dx = x– . Take the deviations of the observations in y series from y and it in a column headed by dy = y – Step – 4: Square these deviations and write them under the columns headed by dx2 and dy2. Step – 5: Multiply the respective dx and dy and write it under the column headed by dxdy. Step – 6: Apply the following formula to calculate r or r xy ,the coefficient or correlation. or  

12. Where n is the number of observations in x and y series, σ x and σ y are standard deviations of x and y. The above method is illustrated by the following example. Example: Find the coefficient of correlation between the weights of tablets and capsules from the following data: Solution: Let the weights of tablets be denoted by x and that of capsules be y, then = = = 69   Weight of tablets in grams (x) 65 66 67 68 69 70 71 Weight of capsules (y) 67 68 66 69 72 72 69

13. Let us prepare the following table   dx= (x – 68) dx2=(x – 68)2 y dy=(y – 68) dy2=(y-68)2 dxdy 65 -3 9 67 -2 4 6 66 -2 4 68 -1 1 2 67 -1 1 66 -3 9 3 68 69 69 1 1 72 3 9 3 70 2 4 72 3 9 6 71 3 9 69 dx= (x – 68) dx2=(x – 68)2 y dy=(y – 68) dy2=(y-68)2 dxdy 65 -3 9 67 -2 4 6 66 -2 4 68 -1 1 2 67 -1 1 66 -3 9 3 68 69 69 1 1 72 3 9 3 70 2 4 72 3 9 6 71 69

14. b) SHORT – CUT METHOD The above direct method for calculating r is not convenient when (i) the terms of the series x and y are big and the calculation of and becomes difficult or (ii) the means and are not integers. In these cases we apply the formula of assumed mean Working Formula: Step 1: Take any term a (preferably the middle one)of x series as assumed mean and any term b (preferably middle one ) as assumed mean for y series Step 2 : Take deviations of the observations in x series from a, i.e., dx = x-a. Take deviations of the observations in y series from b, i.e., dy = y-b Step 3 : Find dx2 and dy2 and write it under columns dx2 and dy2. Step 4: Find dxdy and write it under the column dxdy. Step 5: Apply the formula.  

15. Example : Calculate the coefficient of correlation between x and y for follow Solution: Let 5 be the assumed mean for the values of x and 14 be assumed mean for the values of y x y dx dy dx 2   dy 2   dxdy 1 2 -4 -12 16 144 48 3 6 -2 -8 4 64 16 4 8 -1 -6 1 36 6 5 10 -4 16 7 14 2 4 8 16 3 2 9 4 6 10 20 5 6 25 36 30 n=7 n=7 x y dx dy dx 2   dy 2   dxdy 1 2 -4 -12 16 144 48 3 6 -2 -8 4 64 16 4 8 -1 -6 1 36 6 5 10 -4 16 7 14 2 4 8 16 3 2 9 4 6 10 20 5 6 25 36 30 n=7 n=7 X: 1 3 4 5 7 8 10 Y: 2 6 8 10 14 16 20

16. Let dx = x-5, dx 2 = (x-5)2, dy = (y – 14), dy 2 = (y – 14) 2 we have the table. Since we have taken the deviations from assumed mean, so we shall apply the following formula for correlation for assumed mean: = = = = 1  

17. USES OF CORRELATION The correlation studies are used for a variety of purposes and considered to basic tools for detailed analysis and interpretation of biostatical data relating to two or more variables. The study of correlation is of immense use in practical life because of the following reasons. Sampling error can be calculated. In correlation analysis, we can measure in one figure, the degree of relationship existing between the variables. Correlation study help in identifying such factors which can stabilize a disturbed economic situation. Interrelationship studies between different variables are very helpful tools in promoting research and opening new frontiers of knowledge. Correlation analysis helps in measuring the degree of relationship between the variables like supply and demand, price and supply, income and expenditure. Correlation analysis helps us in locating such variables on which other variables depend. Example: We can find out the factors responsible for price rise or low productivity.

18. MERITS AND DEMERITS OF KARL’S PEARSONS COEFFICIENT METHOD MERITS: It is important method to give a precise and quantitative result with a meaningful interpretation. It also gives a direction ( i.e., positive or negative) as well as the degree of the correlation between the variables. It is most widely used method of studying relationship between interrelated phenomenon. DEMERITS: This method is time consuming. It is tedious to calculate. It is unduly affected by the values of extremities. It is liable to be misinterpreted, as high degree of correlation doesn’t necessarily mean very close relationship between the variable.