Unit 1
Correlation
MBA (Sem-01)
Session 2022-2023
1
Example of Correlation
Is there an association between:
Children’s IQ and Parents’ IQ
Degree of social trust and number of membership in
voluntary association ?
Urban growth and air quality violations?
GRA funding and number of publication by Ph.D.
students
Number of police patrol and number of crime
Grade on exam and time on exam
Correlation
Correlation coefficient: statistical index of the degree
to which two variables are associated, or related.
We can determine whether one variable is related to
another by seeing whether scores on the two variables
covary---whether they vary together.
Scatterplot
The relationship between any two variables can be
portrayed graphically on an x- and y- axis.
Each subject i1 has (x1, y1). When score s for an entire
sample are plotted, the result is called scatter plot.
Scatterplot
Direction of the relationship
Variables can be positively or negatively correlated.
Positive correlation: A value of one variable increase,
value of other variable increase.
Negative correlation: A value of one variable
increase, value of other variable decrease.
Strength of the relationship
The magnitude of correlation:
Indicated by its numerical value
ignoring the sign
expresses the strength of the linear relationship
between the variables.
r =1.00 r = .42
r =.17
r =.85
Pearson’s correlation coefficient
There are many kinds of correlation coefficients but
the most commonly used measure of correlation is the
Pearson’s correlation coefficient. (r)
The Pearson r range between -1 to +1.
Sign indicate the direction.
The numerical value indicates the strength.
Perfect correlation : -1 or 1
No correlation: 0
A correlation of zero indicates the value are not linearly related.
However, it is possible they are related in curvilinear fashion.
Standardized relationship
The Pearson r can be thought of as a standardized measure of
the association between two variables.
That is, a correlation between two variables equal to .64 is the
same strength of relationship as the correlation of .64 for two
entirely different variables.
The metric by which we gauge associations is a standard
metric.
Also, it turns out that correlation can be thought of as a
relationship between two variables that have first been
standardized or converted to z scores.
Correlation Represents
a Linear Relationship
Correlation involves a linear relationship.
"Linear" refers to the fact that, when we graph our two
variables, and there is a correlation, we get a line of
points.
Correlation tells you how much two variables are linearly
related, not necessarily how much they are related in
general.
There are some cases that two variables may have a
strong, or even perfect, relationship, yet the relationship
is not at all linear. In these cases, the correlation
coefficient might be zero.
Coefficient of Determination r2
The percentage of shared variance is represented by
the square of the correlation coefficient, r2 .
Variance indicates the amount of variability in a set
of data.
If the two variables are correlated, that means that we
can account for some of the variance in one variable
by the other variable.
Coefficient of Determination r2
r2
Statistical significance of r
A correlation coefficient calculated on a sample is
statistically significant if it has a very probability of
being zero in the population.
In other words, to test r for significance, we test the
null hypothesis that, in the population the correlation
is zero by computing a t statistic.
Ho: r = 0
HA: r = 0
Some consideration in interpreting correlation
1. Correlation represents a linear relations.
Correlation tells you how much two variables are
linearly related, not necessarily how much they are
related in general.
There are some cases that two variables may have a
strong perfect relationship but not linear. For
example, there can be a curvilinear relationship.
Some consideration in interpreting correlation
2. Restricted range (Slide: Truncated)
Correlation can be deceiving if the full information
about each of the variable is not available. A
correlation between two variable is smaller if the
range of one or both variables is truncated.
Because the full variation of one variables is not
available, there is not enough information to see the
two variables covary together.
Some consideration in interpreting correlation
3. Outliers
Outliers are scores that are so obviously deviant from
the remainder of the data.
On-line outliers ---- artificially inflate the correlation
coefficient.
Off-line outliers --- artificially deflate the correlation
coefficient
On-line outlier
An outlier which falls near where the regression line
would normally fall would necessarily increase the
size of the correlation coefficient, as seen below.
r = .457
Off-line outliers
An outlier that falls some distance away from the
original regression line would decrease the size of the
correlation coefficient, as seen below:
r = .336
Correlation and Causation
Two things that go together may not necessarily mean
that there is a causation.
One variable can be strongly related to another, yet
not cause it. Correlation does not imply causality.
When there is a correlation between X and Y.
Does X cause Y or Y cause X, or both?
Or is there a third variable Z causing both X and Y ,
and therefore, X and Y are correlated?
