Correlation Coefficient Calculator
The correlation coefficient calculator is used to find the correlation coefficient (r) between two sets of data. It also provides different calculators based on the correlation coefficient formula in different cases. You can check other calculators by selecting different calculator types. Please note that the value of the correlation coefficient is always between \(-1\) and \(1\).
Table of Contents
- How to Use the Calculator
- What Is the Correlation Coefficient?
- How to find correlation coefficient when coefficient of determination given
- Related Calculators
- Pearsons correlation coefficient formula
- What is correlation
- Pearson product coefficient
- How to calculate correlation coefficient
- Application
How to use correlation coefficient Calculator
- Input Data: Enter your data sets into the calculator.
- Calculate: Instantly generate the correlation coefficient (r).
What Is the Correlation Coefficient?
The correlation coefficient is define as "The correlation coefficient is a statistical measure of the strength of a linear relationship between two variables." The correlation coefficient is ranges from -1 to 1. If the correaltion coefficient is -1 it shows perfect negative correlation and as one variable increases, the other decreases. If the correlation coefficient is 1 it shows perfect positive correlation, in this case both variables increase together. If the correlation coefficient is 0, it shows no linear relationship between the variables.
Perfect Negative Correlation (-1) graph :
As we can see in above graph when one variable increases, the other decreases at a constant rate. This forms a straight line with a downward slope.
Example : As temperature decreases, the amount of heating needed increases.
Perfect Positive Correlation (1) graph :
As shown in above graph, both variables increase or decrease together at a constant rate, forming a straight line with an upward slope.
Example : The amount of time studied and exam scores.
No Correlation (0) graph :
As shown in above graph, no consistent relationship between the variables; data points are scattered randomly.
Example : Shoe size and intelligence.
How to find correlation coefficient when coefficient of determination given
To find the correlation coefficient (r) when the coefficient of determination (r^2) is given, we can use the following formula:
\( r = \sqrt{ r^{2}} \)
Example :
Find the correlation coefficient (r) when \(r^2 =0.132 \)
Solution :
To find the correlation coefficient (r) when \(r^2 =0.132 \), we can take the square root:
\( r = \sqrt{ 0.132 } \)
Calculating this gives:
r = 0.363
Note : The correlation coefficient can be positive or negative.
Pearsons correlation coefficient formula
\( \mathrm{r = \frac{SS_{xy}}{\sqrt{ SS_{ x} \times SS_{ y}}}}\)
Here's what each term in the formula represents:
- \( \mathrm{SS_{xy}} :\) Sum of cross-products of deviations of \( \mathrm{X} \) and \( \mathrm{Y} \) from their respective means.
- \( \mathrm{SS_{x}} :\) Sum of squares of deviations of \( \mathrm{X} \) from its mean.
- \( \mathrm{SS_{y}} :\) Sum of squares of deviations of \(\mathrm{ Y} \) from its mean.
The numerator values measures the covariance between \( \mathrm{ X}\) and \(\mathrm{Y}\) and the denominator values are used to normalizes the covariance to produce a correlation coefficient that ranges between \(-1\) and \(1\).
Learn more about correlation coefficient r.
What is correlation
The correlation is a statistical measure which shows the strength and direction of a relationship between two variables. In other words it shown how closely two variables are related.
Strength of Relationship
- If the correlation coefficient is equal to \(1\) (+ve number), it shows that perfect positive correlation, where one variable increases exactly in proportion to the other.
- If the correlation coefficient is equal to \(-1\), it shows that perfect negative correlation, where one variable decreases exactly as the other increases.
- If the correlation coefficient is close to \(0\), it shows that no linear relationship between the variables.
Pearson product coefficient
The Pearson product-moment correlation coefficient is nathing but Pearson correlation coefficient or Pearson's r. It is a measure of the strength and direction of the linear relationship between two variables.
- The closer r is to \(+1\) or \(-1\), the stronger the linear relationship.
- The closer r is to \(0\), the weaker the linear relationship.
How to calculate correlation coefficient
Question : Compute the correlation coefficient.
\(\mathrm{x}\) | 6 | 1 | 2 | 3 | 7 |
\(\mathrm{y}\) | 4 | 5 | 7 | 1 | 3 |
Solution
Calculation table :
\(\mathrm{X}\) | \(\mathrm{Y}\) | \(\mathrm{X \times Y}\) | \(\mathrm{X^{2} }\) | \(\mathrm{Y^{2} }\) |
---|---|---|---|---|
6 | 4 | 24 | 36 | 16 |
1 | 5 | 5 | 1 | 25 |
2 | 7 | 14 | 4 | 49 |
3 | 1 | 3 | 9 | 1 |
7 | 3 | 21 | 49 | 9 |
\(\mathrm{\sum X = 19 }\) | \(\mathrm{\sum Y = 20 }\) | \(\mathrm{\sum X \times Y = 67 }\) | \(\mathrm{\sum X^{2} = 99 }\) | \(\mathrm{\sum Y^{2} = 100 }\) |
Lets first find the mean of X :
\( \begin{align} \mathrm{\bar{x}} &= \mathrm{\frac{\sum X}{n}} \\ &=\frac{\text{19}}{\text{5}} \\ &=\text{3.8} \end{align} \)
similarly, lets find the mean for Y :
\( \begin{align} \mathrm{\bar{y}} &=\mathrm{\frac{\sum Y}{n}} \\ &=\frac{\text{20}}{5} \\ &=\text{4} \end{align} \)
To calculate the sum of squares for a set of values X, follow these steps:
\( \begin{align} \mathrm{SS_{x x}} &=\mathrm{\sum X^{2} -\frac{(\sum X)^{2}}{n}} \\ &=\text{99}-\frac{(\text{19})^{2}}{\text{5}} \\ &= \text{26.8} \end{align} \)
To calculate the sum of squares for a set of values Y, follow these steps:
\( \begin{align} \mathrm{SS_{y y}} &=\mathrm{\sum Y^{2} -\frac{\sum Y^{2}} {n} } \\ &= \text{100}-\frac{\text{20}^2}{\text{5}} \\ &= \text{20} \end{align}\)
Sum of the cross-products for both X and Y :
\( \begin{align} \mathrm{SS_{x y}} &=\mathrm{\sum XY - \frac{\sum X \times \sum Y}{n} } \\ &=\text{67}-\frac{\text{19} \times \text{20}}{\text{5}} \\ &= -\text{9} \end{align} \)
To calculate r, we use the formula:
\( \begin{align} \mathrm{r} &= \mathrm{\frac{SS_{xy}}{\sqrt{ SS_{x x} \times SS_{y y}}}} \\ &= \frac{ -\text{9} } { \sqrt{ \text{26.8}\times\text{20} } } \\ &= -\text{0.3887} \end{align} \)
The correlation coefficient is \(-0.3887\)
Application
Pearson's correlation coefficient is widely used in fields such as statistics, economics, psychology, biology, and many others which serve
- Measure relationships between variables.
- Determine predictive power.
- Assess the reliability of measurements.
- Business: Optimize marketing strategies based on consumer behavior correlations.
- Science Analyze experimental data to refine scientific theories.
- Healthcare: Identify relationships between medical variables for informed decision-making.
Why Use Our Correlation Coefficient Calculator?
Understanding correlations is vital across various fields:
Data Analysis: Uncover relationships between variables effortlessly.
Research: Validate hypotheses with statistical rigor.
Finance: Gauge the relationship between economic indicators.
Education: Enhance learning with practical examples and results.