Basic concept about correlations and its type. Formulae for calculation of correlation coefficient (r) Derivation of regression equation and calculation of unknown variable using that equation

1. Correlation:
Basics of correlations, types of correlations,
Correlation coefficient ,
calculation based and graphical method for correlation
Regression:
Basics of regression
Calculation of Y on X and X on Y regression coefficientCalculation of Y on X and X on Y regression coefficient

2. Correlation:
Analysis of relationship between two or more
variables
Regression analysis:Regression analysis:
It is the study about that “how change in one
variable (independent variable or predictor) affect
the another variable (dependent variable or
response)”.

3. Correlation:
Analysis of relationship between two or more variables.
There are two types of variables
(Independent variable or Predictor)
(Dependent variable or Response)(Dependent variable or Response)
Example:
Adding fertilizer to plants (independent) and plant growth (dependent)
Taking medicine (independent) and being cured/healthy (dependent)
Watching TV till late night (independent) and awaking late (dependent)

4. Types of Correlation:
Positive correlation
Negative correlation
Zero correlation
Linear correlationLinear correlation
Non linear correlation

5. Positive:
One increase
Other also increases
One decreases
Other also decreases
Negative:
One increase
Other decreases
One decreases
Other increase
But not the perfect
r=+0.8
to
+0.99
r=-0.8
to
-0.99

6. Perfect alsor=+1 r=-1

7. Positive:
One increase
Other may increases or decrease
One decreases
Other may increases or decrease
Zero correlation
r=0

8. Linear correlation: The correlation between two
variables is said to be linear if one unit change in one
variable result in the corresponding change in the other
variable over the entire range of values. If a graph is plotted
between two variables then it will be a straight line.
In linear correlation there one variable is dependent upon
other by a simple linear equation like
Y= aX+bY= aX+b
Example: Log phase growth of bacterial growth curve

9. Non Linear correlation: If one unit change in one
variable does not cause a constant change but cause
fluctuating change than its called non linear correlation.
line curve. If a graph is plotted between two variables then
it will not be a straight line.
Y= a+ bX2+cX3
Example: Radioactive decayExample: Radioactive decay

10. Formula for Correlation coefficient

11. Another formula for Correlation coefficient

12. Fertilizers
(in Kg)
Yield
(in Quintal)
X Y XY X2 Y2
2 5 10 4 25
5 9 45 25 81
4 7 28 16 49
6 9 54 36 81
3 5 15 9 25
6 10 60 36 1006 10 60 36 100
1 2 2 1 4
5 9 45 25 81
SUM 32 56 259 152 446
ƩXY ƩX2 ƩY2
=+0.9722

13. Fertilizers yield
X Y X-Ẍ Y-Ῡ
(X-Ẍ)
.(Y-Ῡ) (X-Ẍ)2 (Y-Ῡ)2
2 5 -2 -2 4 4 4
5 9 1 2 2 1 4
4 7 0 0 0 0 0
6 9 2 2 4 4 4
3 5 -1 -2 2 1 4
6 10 2 3 6 4 9
1 2 -3 -5 15 9 251 2 -3 -5 15 9 25
5 9 1 2 2 1 4
SUM ƩX= 32 ƩY=56 0 0 35 24 54
Mean Ẍ= 4 Ῡ= 7 *** Ʃ(X-Ẍ)2 Ʃ(Y-Ῡ)2
r=+0.9722

14. Graph method:
Plot one variable on X-axis and another variable on Y-axis.
Based on slope of line and distance of dots from it, type and
Approximate degree of correlation can be determined.
For calculation of value of r calculations are required
y = 1.458x + 1.166
R² = 0.945
10
12
yield
0
2
4
6
8
0 1 2 3 4 5 6 7
yield
Linear (yield)

15. Regression analysis
It is the study about that
“how change in one variable affect the another variable
(independent variable/ or predictor/or covariance/or feature)(independent variable/ or predictor/or covariance/or feature)
(dependent variable /or response/ or outcome/ or explanatory)”.

16. Regression analysis
Like correlation analysis, simple linear regression is a technique
that is used to explore the nature of the relationship between two
continuous random variables. The primary difference between
these two analytical methods is that regression enables us to
investigate the change in one variable, called the response, whichinvestigate the change in one variable, called the response, which
corresponds to a given change in the other, known as the
explanatory variable. Correlation analysis makes no such
distinction; the two variables involved are treated symmetrically.
While in regression one is dependent on other.

17. Fertilizers yield
X Y X-Ẍ Y-Ῡ
(X-Ẍ)
.(Y-Ῡ) (X-Ẍ)2 (Y-Ῡ)2
2 5 -2 -2 4 4 4
5 9 1 2 2 1 4
4 7 0 0 0 0 0
6 9 2 2 4 4 4
3 5 -1 -2 2 1 4
6 10 2 3 6 4 96 10 2 3 6 4 9
1 2 -3 -5 15 9 25
5 9 1 2 2 1 4
20 ???
Yield (Y) depending upon X (fertilizers)
Y on X
??? 25
X depends upon Y
X on Y

18. Y on X
=byx
X on Y
=bxy
byx= coefficient of regression
equation for Y on X
Y on X X on Y
bxy=coefficient of regression
equation for X on Y
Y on X
(Y-Ῡ) =byx (X-Ẍ)
For calculation of
value of Y
X on Y
(X-Ẍ) =bxy (Y-Ῡ)
For calculation of
value of X

19. Y on X
(Y-Ῡ) =byx (X-Ẍ)
For calculation of
value of Y
byx=
X on Y
(X-Ẍ) =bxy (Y-Ῡ)
For calculation of
value of X
bxy=
n(Ʃxy) - (ƩX)(ƩY)
nƩX2 - (ƩX)2
n(Ʃxy) - (ƩX)(ƩY)
nƩY2 - (ƩY)2yx
byx= r
byx=
bxy=
bxy= r
bxy=
nƩX2 - (ƩX)2 nƩY2 - (ƩY)2
CoV (X,Y)
(σX)2
σX
σY
CoV (X,Y)
(σY)2
σY
σX

20. Y=a + byxX
byx= r
σY
σX
X=a + bxyY
bxy= r
σX
σY
a= Ῡ - bẌ a= Ῡ - bẌ