Correlation Coefficient, Assumptions of
Correlation Coefficient
The correlation coefficient is a statistical measure that calculates the strength of the
relationship between the relative movements of the two variables. The range of values for the
correlation coefficient bounded by 1.0 on an absolute value basis or between -1.0 to 1.0. If
the correlation coefficient is greater than 1.0 or less than -1.0, the correlation measurement is
incorrect. A correlation of -1.0 shows a perfect negative correlation, while a correlation of 1.0
shows a perfect positive correlation. A correlation of 0.0 shows zero or no relationship
between the movements of the two variables.
While the correlation coefficient measures a degree of relation between two variables, it only
measures the linear relationship between the variables. The correlation coefficient cannot
capture nonlinear relationships between two variables.
A value of exactly 1.0 means there is a perfect positive relationship between the two
variables. For a positive increase in one variable, there is also a positive increase in the
second variable. A value of -1.0 means there is a perfect negative relationship between the
two variables. This shows the variables move in opposite directions — for a positive increase
in one variable, there is a decrease in the second variable. If the correlation is 0, there is no
relationship between the two variables.
The strength of the relationship varies in degree based on the value of the correlation
coefficient. For example, a value of 0.2 shows there is a positive relationship between the two
variables, but it is weak and likely insignificant. Experts do not consider correlations
significant until the value surpasses at least 0.8. However, a correlation coefficient with an
absolute value of 0.9 or greater would represent a very strong relationship.
This statistic is useful in finance. For example, it can be helpful in determining how well a
mutual fund performs relative to its benchmark index, or another fund or asset class. By
adding a low or negatively correlated mutual fund to an existing portfolio, the investor gains
diversification benefits.
Correlation Coefficient Formulas
One of the most commonly used formulas in stats is Pearson’s correlation coefficient formula
r = Pearson correlation coefficient
x = Values in first set of data
y = Values in second set of data
n = Total no of values
, The assumptions of Correlation Coefficient are-
1. Normality means that the data sets to be correlated should approximate the normal
distribution. In such normally distributed data, most data points tend to hover close to the
mean.
2. Homoscedascity comes from the Greek prefix hom, along with the Greek word skedastikos,
which means ‘able to disperse’. Homoscedascity means ‘equal variances’. It means that the
size of the error term is the same for all values of the independent variable. If the error
term, or the variance, is smaller for a particular range of values of independent variable and
larger for another range of values, then there is a violation of homoscedascity. It is quite
easy to check for homoscedascity visually, by looking at a scatter plot. If the points lie
equally on both sides of the line of best fit, then the data is homoscedastic.
3. Linearity simply means that the data follows a linear relationship. Again, this can be
examined by looking at a scatter plot. If the data points have a straight line (and not a curve)
relationship, then the data satisfies the linearity assumption.
4. Continuous variables are those that can take any value within an interval. Ratio variables are
also continuous variables. To compute Karl Pearson’s Coefficient of Correlation, both data
sets must contain continuous variables. If even one of the data sets is ordinal, then
Spearman’s Coefficient of Rank Correlation would be a more appropriate measure.
5. Paired observations mean that every data point must be in pairs. That is, for every
observation of the independent variable, there must be a corresponding observation of the
dependent variable. We cannot compute correlation coefficient if one data set has 12
observations and the other has 10 observations.
6. No outliers must be present in the data. While statistically there’s no harm if the data
contains outliers, they can significantly skew the correlation coefficient and make it
inaccurate. When does a data point become an outlier? In general, a data point thats
beyond +3.29 or -3.29 standard deviations away, it is considered to be an outlier. Outliers
are easy to spot visually from the scatter plot.
Coefficient of Determination and
Correlation
COEFFICIENT OF DETERMINATION
The coefficient of determination (denoted by R2) is a key output of regression analysis. It is
interpreted as the proportion of the variance in the dependent variable that is predictable from
the independent variable.
Correlation Coefficient
The correlation coefficient is a statistical measure that calculates the strength of the
relationship between the relative movements of the two variables. The range of values for the
correlation coefficient bounded by 1.0 on an absolute value basis or between -1.0 to 1.0. If
the correlation coefficient is greater than 1.0 or less than -1.0, the correlation measurement is
incorrect. A correlation of -1.0 shows a perfect negative correlation, while a correlation of 1.0
shows a perfect positive correlation. A correlation of 0.0 shows zero or no relationship
between the movements of the two variables.
While the correlation coefficient measures a degree of relation between two variables, it only
measures the linear relationship between the variables. The correlation coefficient cannot
capture nonlinear relationships between two variables.
A value of exactly 1.0 means there is a perfect positive relationship between the two
variables. For a positive increase in one variable, there is also a positive increase in the
second variable. A value of -1.0 means there is a perfect negative relationship between the
two variables. This shows the variables move in opposite directions — for a positive increase
in one variable, there is a decrease in the second variable. If the correlation is 0, there is no
relationship between the two variables.
The strength of the relationship varies in degree based on the value of the correlation
coefficient. For example, a value of 0.2 shows there is a positive relationship between the two
variables, but it is weak and likely insignificant. Experts do not consider correlations
significant until the value surpasses at least 0.8. However, a correlation coefficient with an
absolute value of 0.9 or greater would represent a very strong relationship.
This statistic is useful in finance. For example, it can be helpful in determining how well a
mutual fund performs relative to its benchmark index, or another fund or asset class. By
adding a low or negatively correlated mutual fund to an existing portfolio, the investor gains
diversification benefits.
Correlation Coefficient Formulas
One of the most commonly used formulas in stats is Pearson’s correlation coefficient formula
r = Pearson correlation coefficient
x = Values in first set of data
y = Values in second set of data
n = Total no of values
, The assumptions of Correlation Coefficient are-
1. Normality means that the data sets to be correlated should approximate the normal
distribution. In such normally distributed data, most data points tend to hover close to the
mean.
2. Homoscedascity comes from the Greek prefix hom, along with the Greek word skedastikos,
which means ‘able to disperse’. Homoscedascity means ‘equal variances’. It means that the
size of the error term is the same for all values of the independent variable. If the error
term, or the variance, is smaller for a particular range of values of independent variable and
larger for another range of values, then there is a violation of homoscedascity. It is quite
easy to check for homoscedascity visually, by looking at a scatter plot. If the points lie
equally on both sides of the line of best fit, then the data is homoscedastic.
3. Linearity simply means that the data follows a linear relationship. Again, this can be
examined by looking at a scatter plot. If the data points have a straight line (and not a curve)
relationship, then the data satisfies the linearity assumption.
4. Continuous variables are those that can take any value within an interval. Ratio variables are
also continuous variables. To compute Karl Pearson’s Coefficient of Correlation, both data
sets must contain continuous variables. If even one of the data sets is ordinal, then
Spearman’s Coefficient of Rank Correlation would be a more appropriate measure.
5. Paired observations mean that every data point must be in pairs. That is, for every
observation of the independent variable, there must be a corresponding observation of the
dependent variable. We cannot compute correlation coefficient if one data set has 12
observations and the other has 10 observations.
6. No outliers must be present in the data. While statistically there’s no harm if the data
contains outliers, they can significantly skew the correlation coefficient and make it
inaccurate. When does a data point become an outlier? In general, a data point thats
beyond +3.29 or -3.29 standard deviations away, it is considered to be an outlier. Outliers
are easy to spot visually from the scatter plot.
Coefficient of Determination and
Correlation
COEFFICIENT OF DETERMINATION
The coefficient of determination (denoted by R2) is a key output of regression analysis. It is
interpreted as the proportion of the variance in the dependent variable that is predictable from
the independent variable.
