Correlation is a statistical measure that expresses the extent to which two variables are linearly related (meaning they change together at a constant rate). It’s a common tool for describing simple relationships between data sets. When two sets of data show high fidelity to change with respect to one another we say they have a high correlation. We use the correlation coefficient, r to quantify the magnitude of the relationship. The correlation coefficient r can have a value between -1 to 1. Here, 1 represents a perfect positive correlation between the two data sets, 0 represents no correlation and -1 represents a perfect negative correlation. In this math article, we will study correlation, its types, properties and different correlation coefficients. There are different types of correlation coefficients that indicate the nature of the relationship. Let's explore these types and their significance in statistical analysis.

Types of correlation is a vital topic to be known for the competitive exams such as the UGC-NET Commerce Examination. 

In this article, the readers will be able to know about the types of correlation in detail, along with certain other vital topics in detail.

Read about Correlation and regression.

Correlation

Correlation is a process to establish a relationship between two variables. In statistics under relation and functions, methods of correlation summarize the relationship between two variables in a single unitless number called the correlation coefficient. The correlation coefficient is usually represented using the symbol r, and it ranges from -1 to +1. If the coefficient is close to 0 then the relation between the relationship between the two numbers is less and when the relationship is far away from 0 then the relationship is strong between the two variables.

The correlation coefficient close to plus 1 means a positive relationship between the two variables, with increases in one of the variables being associated with increases in the other variable. A correlation coefficient close to -1 indicates a negative relationship between two variables, with an increase in one of the variables being associated with a decrease in the other variable.

Find out about Mann Whitney test (Utest).

The correlation coefficient, usually written as r, is a number that tells us how strongly two things are related. These things, or variables, are often shown as X and Y.

• The value of r is always between -1 and +1.

  • If r = 1, the two variables move exactly together in a positive way.
  • If r = -1, they move exactly opposite to each other.
  • If r = 0, there is no connection between them.

In simple terms, the closer r is to 1 or -1, the stronger the relationship. The closer it is to 0, the weaker the connection.

Correlation Coefficient Values and Their Meaning

Scatter Diagram

A scatter diagram is a graph used to show how two sets of numbers relate.

• The values of X (first variable) are shown on the horizontal (x) axis.
• The values of Y (second variable) are shown on the vertical (y) axis.

Each point on the graph represents a pair of X and Y values.

This diagram helps us see patterns or trends between the two variables. For example:

• If the points go upward from left to right, it shows a positive correlation.
• If they go downward, it’s a negative correlation.
• If the points are scattered randomly, there may be no correlation.

Correlation Formula

Correlation tells us how two variables are related to each other. The correlation coefficient measures this relationship and helps us understand how closely the two variables move together.

To compare two sets of data, we use formulas to calculate the correlation coefficient.

Pearson Correlation Coefficient

The most common formula for measuring correlation is called the Pearson correlation coefficient. It shows the strength and direction of a straight-line (linear) relationship between two data sets.

• The value of this coefficient is always between -1 and +1.
• If the value is 0, it means there is no relationship between the data sets.
• If the value is +1, it means the data sets are positively correlated (both move in the same direction).
• If the value is -1, it means the data sets are negatively correlated (they move in opposite directions).

r = [n(Σxy) – (Σx)(Σy)] / √{[n(Σx²) – (Σx)²] × [n(Σy²) – (Σy)²]}

Symbols

• n = Number of data points or pairs of values
• Σx = Sum of all values in the first data set
• Σy = Sum of all values in the second data set
• Σxy = Sum of the product of each pair of values (multiply each x and y, then add all)
• Σx² = Sum of the squares of the values in the first data set
• Σy² = Sum of the squares of the values in the second data set

Types of Correlation

There are three classes of correlation in common.

• Positive, negative and no correlation.
• Linear and non-linear correlation.
• Simple, multiple, and partial correlation.

Positive Correlation

A positive correlation is a relationship between two variables that are directly related to each other. A positive correlation exists when one variable decreases as the other variable decreases, or one variable increases while the other increases.

For example, the more money you save, the more financially secure you feel or when the temperature goes up, the rate at which ice melts also goes up. The graph for a strong positive correlation would look like this:

Fig: Positive Correlation

Negative Correlation

Negative correlation is a relationship between two variables in which one variable increases as the other decreases, and vice versa. A perfect negative correlation means the relationship that exists between two variables is exactly opposite all of the time. For example, as we climb up a mountain (increase in height) it gets colder (decrease in temperature). In statistics, a perfect negative correlation is represented by the value -1.0. The graph for a strong negative correlation would look like this:

Fig: Negative Correlation

No Correlation

There also exists a condition known as no Correlation, where there is no relation or dependence between two variables. A zero value of the correlation coefficient indicates no correlation. The graph for data sets with no correlation would look like this:

Fig: No correlation

Linear Correlation

Linearity of a correlation is a measure of the degree to which two variables vary together, or a measure of the intensity of the association between two variables. In simple words, correlation is said to be linear if the ratio of change of the two variables is constant, i.e if one of them doubles then the other one doubles or is halved i.e. changes by a factor of 2.

For example, the demand of vegetables and the prices of vegetables or the time spent on video games and the marks in exams.

Non-linear Correlation

Non-linear or curvilinear correlation is said to occur when the ratio of change between two variables is not constant. It can happen that as the value of one variable increases linearly with time, the value of another variable increases exponentially.

Simple Correlation

Simple correlation is a measure used to determine the strength and the direction of the relationship between two variables, X and Y. A simple correlation coefficient can range from –1 to 1. However, maximum (or minimum) values of some simple correlations cannot reach unity (i.e., 1 or –1).

Yield of paddy and the use of fertilizers is an example of simple correlation as yield of paddy depends on the use of fertilizers i.e. presence of one variable affects another variable.

Multiple Correlation

In statistics, the 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.

For example, a researcher is interested in computing the correlation between crime rates in a region and multiple factors like unemployment, illiteracy, substance abuse, inflation etc.

Partial Correlation

Partial correlation measures the strength of a relationship between two variables, while controlling for the effect of one or more other variables.

For example, we might want to see if there is a correlation between the amount of food eaten and blood pressure, while controlling for weight or amount of exercise.

It’s possible to control for multiple variables (called control variables or covariates). However, more than one or two is usually not recommended because the more control variables, the less reliable our test.

Learn about Karl Pearson’s Correlation Coefficient

Correlation Coefficient Formulas

Correlation coefficients are used in the statistics for measuring how strong a relationship exists between two variables. There are many types of correlation coefficient like Pearson’s correlation that are used in linear regression analysis. It is very much popular and useful in statistics.

The main types of correlation coefficients are given below.

• Pearson Correlation
• Spearman Correlation
• Population Correlation

Pearson Correlation

It is the most common formula used for linear dependency between the data set. Its value lies between -1 to +1. When the coefficient comes down to zero, then the data will be considered as not related.

The formula for Pearson correlation is,

r={ Σ(xi-x)−Σ(yi-y)}/√{Σ(xi-x)^2*Σ(yi-y)^2}

Spearman Correlation

In statistics, Spearman’s rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter 

ρ or rs, is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function.

The formula for Spearman Correlation is given below.

ρ=1-{(6Σdi^2)/(n*(n^2-1))}

Where,

ρ=Spearman’s rank correlation coefficient

di= Difference between the two ranks of each observation

n = Number of observations.

The above formula is used to find correlation using Spearman Correlation.

Population Correlation

The population correlation coefficient is a measure of linearity between A and B. The usual estimate is the sample correlation coefficient given by the below mentioned formula.

rab=σab/(σa×σb)

Where, rab= population correlation coefficient

σab= population covariance

σa=population standard deviation for variable A

σb=population standard deviation for variable B

Read about Variable.

Degree of Correlation

The degree of correlation measures the strength and direction of the relationship between two variables. It is quantified by the correlation coefficient, which ranges from -1 to +1.

Perfect Correlation:

1. When two variables change in exact proportion to each other, the correlation is said to be perfect. This occurs in two forms:

• Positive Perfect Correlation: Both variables increase or decrease together in the same ratio. The correlation coefficient is +1.
• Negative Perfect Correlation: One variable increases exactly as the other decreases in the same ratio. The correlation coefficient is -1.

2. Zero Correlation:

• If changes in one variable do not affect or relate to changes in the other variable at all, there is no correlation. In this case, the correlation coefficient is 0.

3. Limited Degree of Correlation:

Most real-world relationships fall between perfect correlation and no correlation. Here, the correlation coefficient lies between -1 and +1, but is not exactly these values.

• If the variables tend to move in the same direction but not perfectly, the correlation is positive but less than +1.
• If the variables tend to move in opposite directions but not perfectly, the correlation is negative but greater than -1.

The strength of this limited correlation is classified as:

• Low correlation: Coefficient between 0 and 0.25 (weak relationship)
• Moderate correlation: Coefficient between 0.25 and 0.75 (moderate relationship)
• High correlation: Coefficient between 0.75 and 1 (strong relationship)

Properties of Correlation

The main properties of Correlation are listed below.

• The coefficient of correlation cannot take value less than -1 or more than one +1.

Symbolically, 

−1≤r≤+1 or |r|<1

• Coefficients of Correlation are independent of Change of Origin. This property reveals that if we subtract any constant from all the values of X and Y, it will not affect the coefficient of correlation.
• Coefficients of Correlation possess the property of symmetry.
• Coefficient of Correlation is independent of Change of Scale. This property reveals that if we divide or multiply all the values of X and Y, it will not affect the coefficient of correlation.
• Coefficients of correlation measure only linear correlation between X and Y.
• If two variables X and Y are independent, the coefficient of correlation between them will be zero.

Difference between Correlation and Regression

The difference between Correlation and Regression is listed below.

Difference between Correlation and Covariance

The difference between Correlation and Covariance is listed below.

Correlation Examples

Problem 1:

Calculate the correlation coefficient for the following data, using Pearson's correlation coefficient.

A=4, 8, 12, 16 and B=5, 10, 15, 20.

Solution:

We need to first construct a table as follows to get the required values of the formula.

Now we calculate the sum for each column,

ΣA=4+8+12+16=40

ΣB=5+10+15+20=50

ΣA2=16+64+144+256=480

ΣB2=25+100+225+400=750

ΣAB=20+80+180+320=600, and

n=4

Now we just put the values directly in the formula.

Thus the correlation coefficient is 1.

Problem 2:

Calculate the correlation coefficient for the following data, using Spearman's rank correlation coefficient.

A = 15, 25, 35, 45
B = 20, 10, 30, 40

Solution:

We need to first construct a table as follows to get the required values for the formula.

Use Spearman's rank correlation formula:

ρ = 1 - (6 × Σd²) / (n(n² - 1))

ρ = 1 - (6 × 2) / (4(16 - 1))

ρ = 1 - 12 / 60

ρ = 1 - 0.2

ρ = 0.8

Know about Measures of dispersion.

Conclusion

Understanding the types of correlation is crucial for researchers, statisticians, and anyone involved in data analysis. Whether exploring the relationship between economic indicators, studying the impact of variables in scientific research, or making predictions based on historical data, correlation plays a pivotal role. The nuances of positive, negative, or no correlation provide valuable insights into the dynamics of variables, helping to make informed decisions and predictions.

Types of correlation is a vital topic as per several competitive exams. It would help if you learned other similar topics with the Testbook App. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: