Correlation IN STATISTICS

Quantitative techniques of management
Project assignment
Submitted by:
Sakshi Tiwari
Mba (gen)

Types of correlationTypes of correlation
Methods of studying correlationMethods of studying correlation
a) Scatter diagrama) Scatter diagram
b) Karl pearson’s coefficient of correlationb) Karl pearson’s coefficient of correlation
c) Spearman’s Rank correlation coefficientc) Spearman’s Rank correlation coefficient
d) Method of least squaresd) Method of least squares

• While studying statistics, one comes across the concept of correlation. It is aWhile studying statistics, one comes across the concept of correlation. It is a
statistical method which enables the researcher to find whether twostatistical method which enables the researcher to find whether two
variables are related and to what extent they are related. Correlation isvariables are related and to what extent they are related. Correlation is
considered as the sympathetic movement of two or more variables. We canconsidered as the sympathetic movement of two or more variables. We can
observe this when a change in one particular variable is accompanied byobserve this when a change in one particular variable is accompanied by
changes in other variables as well, and this happens either in the same orchanges in other variables as well, and this happens either in the same or
opposite direction, then the resultant variables are said to be correlated.opposite direction, then the resultant variables are said to be correlated.
Considering a data where we find two or more variables getting valued thenConsidering a data where we find two or more variables getting valued then
we might study the related variation for these variables.we might study the related variation for these variables.
• In correlation, when values of one variable increase with the increase inIn correlation, when values of one variable increase with the increase in
another variable, it is supposed to be a positive correlation. On the otheranother variable, it is supposed to be a positive correlation. On the other
hand, if the values of one variable decrease with the decrease in anotherhand, if the values of one variable decrease with the decrease in another
variable, then it would be a negative correlation. There might be the casevariable, then it would be a negative correlation. There might be the case
when there is no change in a variable with any change in another variable.when there is no change in a variable with any change in another variable.
In this case, it is defined as no correlation between the two.In this case, it is defined as no correlation between the two.

• Correlation can be of three types as follows:Correlation can be of three types as follows:
1. Simple correlation1. Simple correlation
2. Multiple correlation2. Multiple correlation
3. Partial correlation3. Partial correlation

Correlation formula:Correlation formula:
Where,
x and y are the variables.
b = the slope of the regression line is also called as the regression coefficient
a = intercept point of the regression line which is in the yaxis.
N = Number of values or elements
X = First Score
Y = Second Score
= Sum of the product of the first and Second Scores
= Sum of First Scores
= Sum of Second Scores
= Sum of square first scores.
= Sum of square second scores.

Coefficient of correlation, r, called the linear correlationCoefficient of correlation, r, called the linear correlation
coefficient, measures the strength and the direction of a linearcoefficient, measures the strength and the direction of a linear
relationship between two variables. It also called as Pearsonrelationship between two variables. It also called as Pearson
product moment correlation coefficient. The algebraic method ofproduct moment correlation coefficient. The algebraic method of
measuring the correlation is called the coefficient of correlation.measuring the correlation is called the coefficient of correlation.
There are mainly three coefficients of correlationThere are mainly three coefficients of correlation
•1. Karl Pearson’s Coefficient of correlation1. Karl Pearson’s Coefficient of correlation
•2. Pearson’s rank correlation coefficient2. Pearson’s rank correlation coefficient
•3. Concurrent correlation3. Concurrent correlation

• The most important algebraic method of measuring correlationThe most important algebraic method of measuring correlation
is Karl Pearson’s Coefficient of correlation or Pearsonian’sis Karl Pearson’s Coefficient of correlation or Pearsonian’s
coefficient of Correlation. It has widely used application incoefficient of Correlation. It has widely used application in
Statistics. It is denoted by rStatistics. It is denoted by r..

• Interpretation of Karl Pearson’s Coefficient ofInterpretation of Karl Pearson’s Coefficient of
correlationcorrelation
Karl Pearson’s Coefficient of correlation denoted by r is the degree of correlation betweenKarl Pearson’s Coefficient of correlation denoted by r is the degree of correlation between
two variables. r takes values between –1 and 1two variables. r takes values between –1 and 1
– When r is –1, we say there is perfect negative correlation.When r is –1, we say there is perfect negative correlation.
– When r is a value between –1 and 0, we say that there is a negative correlationWhen r is a value between –1 and 0, we say that there is a negative correlation
– When r is 0, we say there is no correlationWhen r is 0, we say there is no correlation
– When r is a value between 0 and 1, we say there is a positive correlationWhen r is a value between 0 and 1, we say there is a positive correlation
– When r is 1, we say there is a perfect positive correlation.When r is 1, we say there is a perfect positive correlation.
• Properties of the Coefficient of correlationProperties of the Coefficient of correlation
1. Coefficient of correlation has a well defined formula1. Coefficient of correlation has a well defined formula
2. Coefficient of correlation is a number and is independent of the unit of2. Coefficient of correlation is a number and is independent of the unit of
measurementmeasurement
3. Coefficient of correlation lies between –1 and 13. Coefficient of correlation lies between –1 and 1
4. Coefficient of correlation between x and y will be same as that between y and4. Coefficient of correlation between x and y will be same as that between y and
x.x.

• There are different types of Correlation. They are listed as follows:There are different types of Correlation. They are listed as follows:
Positive CorrelationPositive Correlation: A positive correlation is a correlation in the same direction.: A positive correlation is a correlation in the same direction.
Negative CorrelationNegative Correlation : A negative correlation is a correlation in the opposite direction.: A negative correlation is a correlation in the opposite direction.
Partial CorrelationPartial Correlation: The correlation is partial if we study the relationship between two: The correlation is partial if we study the relationship between two
variables keeping all other variables constant.variables keeping all other variables constant.
Example:Example:
The Relationship between yield and rainfall at a constant temperature is partial correlation.The Relationship between yield and rainfall at a constant temperature is partial correlation.
Linear Correlation:Linear Correlation: When the change in one variable results in the constant change in theWhen the change in one variable results in the constant change in the
other variable, we say the correlation is linear. When there is a linearother variable, we say the correlation is linear. When there is a linear
correlation, the points plotted will be in a straight linecorrelation, the points plotted will be in a straight line
Example:Example:
Consider the variables with the following values.Consider the variables with the following values.
Here, there is a linear relationship between the variables. There is a ratio 1:2 at all points. Also,Here, there is a linear relationship between the variables. There is a ratio 1:2 at all points. Also,
if we plot them they will be in a straight line.if we plot them they will be in a straight line.

• Zero Order CorrelationZero Order Correlation :One of the most common and basic:One of the most common and basic
techniques for analyzing the relationships between variables is zero ordertechniques for analyzing the relationships between variables is zero order
correlation. The value of a correlation coefficient can vary from 1 to +1. A 1correlation. The value of a correlation coefficient can vary from 1 to +1. A 1
indicates a perfect negative correlation, while a +1 indicates a perfectindicates a perfect negative correlation, while a +1 indicates a perfect
positive correlation. A correlation of zero means there is no relationshippositive correlation. A correlation of zero means there is no relationship
between the two variables.between the two variables.
• Scatter Plot CorrelationScatter Plot Correlation : A scatter plot is a type of mathematical: A scatter plot is a type of mathematical
diagram using Cartesian coordinates to display values for two variables for adiagram using Cartesian coordinates to display values for two variables for a
set of data. Scatter plots will often show at a glance whether a relationshipset of data. Scatter plots will often show at a glance whether a relationship
exists between two sets of data. The data displayed on the graph resemblesexists between two sets of data. The data displayed on the graph resembles
a line rising from left to right. Since the slope of the line is positive, there is aa line rising from left to right. Since the slope of the line is positive, there is a
positive correlation between the two sets of data.positive correlation between the two sets of data.

• Spearman's Correlation :Spearman's Correlation : Spearman's rank correlation coefficient allowsSpearman's rank correlation coefficient allows
us tous to identify easily the strength of correlation within aidentify easily the strength of correlation within a
data set of two variables, and whether the correlationdata set of two variables, and whether the correlation
is positive or negative. The Spearman coefficient isis positive or negative. The Spearman coefficient is
denoted with the Greek letter rho ( ).denoted with the Greek letter rho ( ).
• Non Linear Correlation :Non Linear Correlation : When the amount of change in oneWhen the amount of change in one
variable isvariable is not in the other variable, we say thatnot in the other variable, we say that
thethe correlation is non linear.correlation is non linear.
Example:Example:
Consider the variables with the following values.Consider the variables with the following values.
Here there is a non linear relationship between the variables. The ratioHere there is a non linear relationship between the variables. The ratio betweenbetween
them is not fixed for all points. Also if we plot them on the graph, the points willthem is not fixed for all points. Also if we plot them on the graph, the points will
not be in a straight line. It will be a curve.not be in a straight line. It will be a curve.

• Simple Correlation :Simple Correlation : If there are only two variable under study,If there are only two variable under study,
the correlation is said to be simple.the correlation is said to be simple.
Example:Example:
The correlation between price and demand is simple.The correlation between price and demand is simple.
• Multiple Correlations :Multiple Correlations :When one variable is related to a numberWhen one variable is related to a number
of other variables, the correlation is notof other variables, the correlation is not
simple. It is multiple if there is onesimple. It is multiple if there is one
variable on one side and a set ofvariable on one side and a set of
variables on the other side.variables on the other side.
Example:Example:
Relationship between yield with both rainfall and fertilizerRelationship between yield with both rainfall and fertilizer
together is multiple correlationstogether is multiple correlations
• Weak Correlation:Weak Correlation: The range of the correlation coefficientThe range of the correlation coefficient
between 1 to +1. If the linear correlationbetween 1 to +1. If the linear correlation
coefficient takes values close to 0, thecoefficient takes values close to 0, the
correlation is weak.correlation is weak.

• Positive correlation:Positive correlation: A relationship between two variables in which bothA relationship between two variables in which both
variable move in same directions. A positive correlation exists when as one variable decreases, thevariable move in same directions. A positive correlation exists when as one variable decreases, the
other variable also decreases and vice versa. When the values of two variables x and y move in theother variable also decreases and vice versa. When the values of two variables x and y move in the
same direction, the correlation is said to be positive. That is in positive correlation, when there is ansame direction, the correlation is said to be positive. That is in positive correlation, when there is an
increase in x, there will be and an increase in y also. Similarly when there is a decrease in x, there willincrease in x, there will be and an increase in y also. Similarly when there is a decrease in x, there will
be a decrease in y also. Positive Correlation Example Price and supply are two variables, which arebe a decrease in y also. Positive Correlation Example Price and supply are two variables, which are
positively correlated. When Price increases, supply also increases; when price decreases, supplypositively correlated. When Price increases, supply also increases; when price decreases, supply
decreases.decreases.
Positive Correlation GraphPositive Correlation Graph

• Strong Positive Correlation :Strong Positive Correlation :A strong positive correlation hasA strong positive correlation has
variables that has the same changes, but the point are more close together and formvariables that has the same changes, but the point are more close together and form
a line.a line.

Weak Positive Correlation:Weak Positive Correlation: A weak positive correlation has variablesA weak positive correlation has variables
that has the same changes but the points on the graph are dispersed.that has the same changes but the points on the graph are dispersed.

• Negative correlation:Negative correlation: In a negative correlation, asIn a negative correlation, as
the values of one of the variables increase, the values of the second variablethe values of one of the variables increase, the values of the second variable
decrease or the value of one of the variables decreases, the value of the otherdecrease or the value of one of the variables decreases, the value of the other
variable increases. When the values of two variables x and y move in oppositevariable increases. When the values of two variables x and y move in opposite
direction, we say correlation is negative. That is in negative correlation, when there isdirection, we say correlation is negative. That is in negative correlation, when there is
an increase in x, there will be a decrease in y. Similarly when there is a decrease in x,an increase in x, there will be a decrease in y. Similarly when there is a decrease in x,
there will be an increase in y increase.there will be an increase in y increase.
Negative Correlation ExampleNegative Correlation Example
When price increases, demand also decreases; when price decreases,When price increases, demand also decreases; when price decreases,
demand also increases. So price and demand are negativelydemand also increases. So price and demand are negatively
correlated.correlated.
• Perfect Negative Correlation :Perfect Negative Correlation : The closer the correlationThe closer the correlation
coefficient is either 1 or +1, the stronger the relationship is between the two variables.coefficient is either 1 or +1, the stronger the relationship is between the two variables.
A perfect negative correlation of 1.0 indicated that for every member of the sample,A perfect negative correlation of 1.0 indicated that for every member of the sample,
higher score on one variable is related to a lower score on the other variable.higher score on one variable is related to a lower score on the other variable.

In statistics, some times we will have to study the relationship between two or moreIn statistics, some times we will have to study the relationship between two or more
variables. The statistical technique used to study the relationships between thevariables. The statistical technique used to study the relationships between the
variables is called the correlation technique. Correlation analysis is the analysis ofvariables is called the correlation technique. Correlation analysis is the analysis of
association between two or more variables. The tendency of two or more variables toassociation between two or more variables. The tendency of two or more variables to
vary together directly or inversely is called as correlation.vary together directly or inversely is called as correlation.
Two variables are said to be correlated, if the change in one of the variable results in aTwo variables are said to be correlated, if the change in one of the variable results in a
corresponding change in the other variable. That is, when two variables move together,corresponding change in the other variable. That is, when two variables move together,
they are said to be correlated.they are said to be correlated.
Let us take an example to understand the term correlation. In a given data with heightsLet us take an example to understand the term correlation. In a given data with heights
and weights of students in a school, we can assume that students with a more heightand weights of students in a school, we can assume that students with a more height
would have a more weight. Besides, it is assumed that students who have short heightwould have a more weight. Besides, it is assumed that students who have short height
will have less weight.will have less weight.

Correlation is a term that refers to the strength of a relationship between twoCorrelation is a term that refers to the strength of a relationship between two
variables. Correlation and regression analysis are related in the sense that both dealvariables. Correlation and regression analysis are related in the sense that both deal
with relationships among variables. The correlation coefficient is a measure of linearwith relationships among variables. The correlation coefficient is a measure of linear
association between two variables. Values of the correlation coefficient are alwaysassociation between two variables. Values of the correlation coefficient are always
between 1 and +1. The value of 1 represents a perfect negative correlation while abetween 1 and +1. The value of 1 represents a perfect negative correlation while a
value of +1 represents a perfect positive correlation. A value of 0 means that there isvalue of +1 represents a perfect positive correlation. A value of 0 means that there is
no relationship between the variables being tested.no relationship between the variables being tested.
Interpretation of coefficient of correlation based on the error likelyInterpretation of coefficient of correlation based on the error likely
1. If the coefficient of correlation is less than the error likely, then its not1. If the coefficient of correlation is less than the error likely, then its not
significantsignificant
2. If the coefficient of correlation is more than six times the error likely, it is2. If the coefficient of correlation is more than six times the error likely, it is
significant.significant.
3. If the error is too small and coefficient of correlation is 0.5 or more then the3. If the error is too small and coefficient of correlation is 0.5 or more then the
coefficient of correlation is significant.coefficient of correlation is significant.

• Covariance and correlation are both describe the degree of similarity between twoCovariance and correlation are both describe the degree of similarity between two
random variables. Suppose that X and Y are real valued random variables for therandom variables. Suppose that X and Y are real valued random variables for the
experiment with means E(X), E(Y) and variances var(X), var(Y), respectively. Theexperiment with means E(X), E(Y) and variances var(X), var(Y), respectively. The
covariance of X and Y is defined bycovariance of X and Y is defined by

• The cross correlation function is a measure of the similarity between twoThe cross correlation function is a measure of the similarity between two
data sets. One set is displaced related to the other, correspondingdata sets. One set is displaced related to the other, corresponding
values of the two sets are multiplied together and the product arevalues of the two sets are multiplied together and the product are
summed to give the value of the cross correlation. Whenever two setssummed to give the value of the cross correlation. Whenever two sets
are almost same, the product will be positive and the cross correlation isare almost same, the product will be positive and the cross correlation is
large. When set are unlike, some of the products will be positive andlarge. When set are unlike, some of the products will be positive and
some negative and the sum will be small.some negative and the sum will be small.

Correlation IN STATISTICS

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Correlation IN STATISTICS