The document presents an overview of correlation methods, including definitions and applications of Karl Pearson's coefficient of correlation and multiple correlation. It emphasizes the distinction between correlation and causation, detailing how correlation measures the strength and direction of relationships between variables. Additionally, the document includes examples and methods for calculating correlation coefficients, as well as an explanation of scatter plots and visual representations of data.

1. Unit: 1
Subtopic:4
Correlation
Ravinandan A P
Assistant Professor,
Sree Siddaganga College of Pharmacy
In association with
Siddaganga Hospital
Tumkur-02

2. Presentation Outlines……….
1. Definition
2. Karl Pearson’s coefficient of
correlation
3. Multiple correlation

3. Relationships between Variables
• Common aim of research is to try to relate a
variable of interest to one or more other
variables demonstrate an association
between variables (not necessarily causal)
• Ex: dose of drug and resulting systolic blood
pressure, advertising expenditure of a
company and end of year sales figures, etc.
• Establish a theoretical model to predict the
value of one variable from a number of
known factors

4. Dependent and Independent Variables
• The independent variable is the variable
which is under the investigator’s control
(denoted x)
• the dependent variable is the one which
the investigator is trying to estimate or
predict (denoted y)
• Can a relationship be used to predict what
happens to y as x changes (ie what
happens to the dependent variable as the
independent variable changes)?

6. Exploring Relationships

7. The relationship between x and y
• Correlation: is there a relationship between 2
variables?
• Regression: how well a certain independent variable
predict dependent variable?
• Regression: a measure of the relation between
the mean value of one variable (e.g. output) and
corresponding values of other variables (e.g.
time and cost).
• CORRELATION  CAUSATION
– In order to infer causality: manipulate independent variable
and observe effect on dependent variable

8. Correlation
• Correlation is the study of relationship between two or
variables
• Correlation methods are used to measure the association of
two or more variables
• Two values are related means, that one variable may be
predicted from the knowledge of the other.
• Ex: One predict the dissolution of tablets based on the
hardness
• Correlation is applied to the relationship of continuous
variables.

9. Correlation
A correlation is a relationship between two variables.
The data can be represented by the ordered pairs (x, y)
where x is the independent (or explanatory) variable,
and y is the dependent (or response) variable.
x
2 4
–2
– 4
y
2
6
x 1 2 3 4 5
y – 4 – 2 – 1 0 2
Example:

11. Correlation key concepts:
Types of correlation Methods of studying
correlation
a) Scatter diagram
b) Karl pearson’s coefficient of correlation
c) Spearman’s Rank correlation coefficient
d) Method of least square

12. Scatter Plot
• Visual representation of scores.
• Each individual score is represented by a single
point on the graph.
• Allows you to see any patterns of trends that exist
in the data.
• A scatter plot can be used to determine whether
a linear (straight line) correlation exists between
two variables.

13. Scatter Plots
Y
X
Y
X
Y
X
Y
Y Y
Positive correlation Negative correlation No correlation

18. Linear Correlation
x
y
Negative Linear Correlation
x
y
No Correlation
x
y
Positive Linear Correlation
x
y
Nonlinear Correlation
As x increases, y
tends to decrease.
As x increases, y
tends to increase.

21. Correlation Coefficient
Statistic showing the degree of relation between two
variables
The correlation coefficient is a measure of the strength and the
direction of a linear relationship between two variables.
The symbol r represents the sample correlation coefficient.
The formula for r is
( )( )
( ) ( )
2 2
2 2
.
n xy x y
r
n x x n y y
 −  
=
 −   − 
The range of the correlation coefficient is −1 to 1. If x and y have a
strong positive linear correlation, r is close to 1. If x and y have a
strong negative linear correlation, r is close to −1. If there is no linear
correlation or a weak linear correlation, r is close to 0.

22. Linear Correlation
x
y
Strong negative correlation
x
y
Weak positive correlation
x
y
Strong positive correlation
x
y
Nonlinear Correlation
r = −0.91 r = 0.88
r = 0.42
r = 0.07

23. Calculating a Correlation Coefficient
1. Find the sum of the x-values.
2. Find the sum of the y-values.
3. Multiply each x-value by its corresponding
y-value and find the sum.
4. Square each x-value and find the sum.
5. Square each y-value and find the sum.
6. Use these five sums to calculate the
correlation coefficient.
Continued.
Calculating a Correlation Coefficient
In Words In Symbols
x

y

xy

2
x

2
y

( )( )
( ) ( )
2 2
2 2
.
n xy x y
r
n x x n y y
 −  
=
 −   − 

24. Correlation Coefficient
Example:
Calculate the correlation coefficient r for the following data.
x y xy x2 y2
1 – 3 – 3 1 9
2 – 1 – 2 4 1
3 0 0 9 0
4 1 4 16 1
5 2 10 25 4
15
x
 = 1
y
 = − 9
xy
 = 2
55
x
 = 2
15
y
 =
( )( )
( ) ( )
2 2
2 2
n xy x y
r
n x x n y y
 −  
=
 −   − 
( )( )
( )2
2
5(9) 15 1
5(55) 15 5(15) 1
− −
=
− − −
60
50 74
= 0.986

There is a strong positive
linear correlation between x
and y.

25. Correlation Coefficient
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50
Example:
The following data represents the number of hours 12
different students watched television during the
weekend and the scores of each student who took a test
the following Monday.
a.) Display the scatter plot.
b.) Calculate the correlation coefficient r.
Continued.

26. Correlation Coefficient
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50
Example continued:
100
x
y
Hours watching TV
Test
score
80
60
40
20
2 4 6 8 10
Continued.

27. Correlation Coefficient
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50
xy 0 85 164 222 285 340 380 420 348 455 525 500
x2 0 1 4 9 9 25 25 25 36 49 49 100
y2 9216 7225 6724 5476 9025 4624 5776 7056 3364 4225 5625 2500
Example continued:
( )( )
( ) ( )
2 2
2 2
n xy x y
r
n x x n y y
 −  
=
 −   − 
( )( )
( )2
2
12(3724) 54 908
12(332) 54 12(70836) 908
−
=
− −
0.831
 −
There is a strong negative linear correlation.
As the number of hours spent watching TV increases, the test
scores tend to decrease.
54
x
 = 908
y
 = 3724
xy
 = 2
332
x
 = 2
70836
y
 =

28. Simple Correlation coefficient (r)
➢It is also called Pearson's correlation or
product moment correlation coefficient.
➢It measures the nature and strength between
two variables of the quantitative type.
The sign of r denotes the nature of
association
while the value of r denotes the strength of
association.

29. ➢If the sign is +ve this means the relation
is direct (an increase in one variable is
associated with an increase in the
other variable and a decrease in one
variable is associated with a
decrease in the other variable).
➢While if the sign is -ve this means an
inverse or indirect relationship (which
means an increase in one variable is
associated with a decrease in the other).

30. ➢ The value of r ranges between ( -1) and ( +1)
➢ The value of r denotes the strength of the
association as illustrated by the following
diagram.
-1 1
0
-0.25
-0.75 0.75
0.25
strong strong
intermediate intermediate
weak weak
no relation
perfect
correlation
perfect
correlation
Direct
indirect

31. If r = Zero this means no association or
correlation between the two variables.
If 0 < r < 0.25 = weak correlation.
If 0.25 ≤ r < 0.75 = intermediate correlation.
If 0.75 ≤ r < 1 = strong correlation.
If r = l = perfect correlation.

33. Karl Pearson's correlation coefficient
• Karl Pearson's correlation coefficient measures
degree of linear relationship between two variables
• If the relationship between two variables X and Y is
to be ascertained through Karl Pearson method ,
then the following formula is used:
• The value of the coefficient of correlation always
lies between ±1

35. Multiple correlation
• Coefficient of multiple correlation is a
measure of how well a given variable can be
predicted using a linear function of a set of
other variables.
• It is the correlation between the variable's
values and the best predictions that can be
computed linearly from the predictive
variables.
• The coefficient of multiple correlation is known
as the square root of the coefficient of
determination,

36. Multiple correlation
• The coefficient of multiple correlation takes
values between 0 and 1.
• Higher values indicate higher predictability of
the dependent variable from the independent
variables, with a value of 1 indicating that the
predictions are exactly correct and a value of
0 indicating that no linear combination of the
independent variables is a better predictor
than is the fixed mean of the dependent
variable

39. Thank you