Correlation Coefficient Formula Steps and Solved Examples
Using a scale range of - 1 and + 1, the extent to which 2 different variables are related can be identified using the correlation coefficient. ‘r’ is the symbol to denote a coefficient of correlation between 2 ratio variables or for 2 intervals. So, r denotes the level of relationship which means, if the r’s value is closer to zero (0), then there is a minimal correlation between the intervals. And if the value of r is higher, then it denotes a greater correlation between each variable, regardless of positive or negative direction. From learning a few applications to understanding its features, this module covers all about the important basics you need to know about the correlation coefficient.
Defining What Coefficient Correlation is
Coefficient of the correlation is used to measure the relationship extent between 2 separate intervals or variables. Denoted by the symbol ‘r’, this r value can either be positive or negative. Some of the other names of coefficient correlation are:
Pearson’s r
Pearson product-moment correlation coefficient (PPMCC)
Pearson correlation coefficient (PCC)
Bivariate correlation
Cross-correlation coefficient
The value expressed will tell us the extent to which the 2 entities are interlinked. Sometimes, r value can 0 also, hence symbolizes that there is an absence of a relationship between the 2 given variables.
The Standard Formulas of Coefficient of Correlation
Let us consider 2 different variables ‘x’ and ‘y’ that are related commonly
To find the extent of the link between the given numbers x and y, we will choose the Pearson Coefficient ‘r’ method. In the process, the formula given below is used to identify the extent or range of the 2 variables’ equality.
Pearson Correlation Coefficient
r = \[\frac{n(Σxy) - (Σx)(Σy)}{\sqrt{[nΣx² - (Σx)²] [nΣy² - (Σy)²]}}\]
The Keys:
“Σx” denotes the number of First Variable Value
“Σy” represents the count of Second Variable Value
“Σx2” gives us the addition of Squares for the First Value
“Σy2” mentioned the sum of the Second Value’s square
“n” is the total number of data quantity which is available
“Σxy” symbolizes the addition of the First & Second Value’s products
Check out the Following Formula:
r = \[\frac{\sum_{i=1}^{n} (X_{i} - \overline{X})(Y_{i} -\overline{Y})} {\sqrt{\sum_{i=1}^{n}(X_{i} - \overline{X}})^{2} \sqrt{\sum_{i=1}^{n}(Y_{i} - \overline{Y}})^{2}}\]
The equation which is given above is termed the linear coefficient correlation formula, “xi” and “yi” denote the 2 different variables and “n” is the total number of observations.
2 of the other important formulas include the following ones.
Population Correlation equation: ρxy = σxy/σxσy (the population standard deviations are “σx” and “σy”. “Σxy” is the population variance)
Sample Correlation equation: rxy = Sxy /SxSy (“Sx” and “Sy” and 2 sample standard deviations. Sample covariance is denoted as “Sxy”)
Simple Examples for Coefficient Correlation with Applications
As we read before, the value of coefficient correlation can be evaluated using - 1 and + 1 respectively. Following 3 are scenarios using these 2 ranges.
When r is + 1: With some fixed proportional value, the variable is said to increase positively by 1 and this increases the other as well. When the size of a fabric material increases, together with the growth and height of an individual is the best example.
When r is 0: Zero represents the complete absence of a relationship between 2 variables. This means there is no recorded history for increase or decrease in its value of extent/range.
When r is - 1: In a standard parameter of fixation, the positive increase in 1 variable will lead to a negative decrease in the other variable. When you drive your car faster than usual, then the upcoming distance to be covered gets reduced. This is a classic example of a negative-valued coefficient correlation.
Speaking of its applications, the coefficient of correlation is majorly preferred in the field of finance and insurance sectors. For instance, the correlation between any 2 different quantities is comparable when the price of an oil product increase, giving better advantages to the oil-producing brand and agencies such as ROI and enhancing consumer behaviour.
Conclusion
The correlation coefficient is the method of calculating the level of relationship between 2 different ratios, variables, or intervals. The symbol is ‘r’. The value of r is estimated using the numbers - 1, 0, and/or + 1 respectively. - 1 denotes lesser relation, + 1 gives greater correlation and 0 denotes absence or NIL in the 2 variable’s interlink. Pearson’s r, Bivariate correlation, Cross-correlation coefficient are some of the other names of the correlation coefficient.
FAQs on Correlation Coefficient Explained with Meaning and Uses
1. What is the correlation coefficient in statistics?
The correlation coefficient is a numerical measure that shows the strength and direction of the linear relationship between two variables. It is usually denoted by r (for sample data) and ranges from -1 to +1.
- r = +1: Perfect positive linear correlation
- r = -1: Perfect negative linear correlation
- r = 0: No linear correlation
2. What is the formula for the Pearson correlation coefficient?
The Pearson correlation coefficient formula is r = Cov(X,Y) / (σₓσᵧ). In expanded form for sample data:
r = [n∑xy − (∑x)(∑y)] / √([n∑x² − (∑x)²][n∑y² − (∑y)²]).
- n = number of observations
- ∑xy = sum of product of paired values
- σₓ, σᵧ = standard deviations
3. How do you calculate the correlation coefficient step by step?
To calculate the correlation coefficient (r), compute the standardized covariance between two variables.
- 1. Find the mean of X and Y.
- 2. Subtract the mean to get deviations.
- 3. Multiply corresponding deviations and sum them.
- 4. Compute standard deviations of X and Y.
- 5. Divide covariance by the product of standard deviations.
4. What does a correlation coefficient of 0 mean?
A correlation coefficient of 0 means there is no linear relationship between the two variables. This does not necessarily mean there is no relationship at all.
- The variables may have a non-linear relationship.
- There is no consistent upward or downward linear trend.
5. What is the difference between positive and negative correlation?
A positive correlation means both variables move in the same direction, while a negative correlation means they move in opposite directions.
- Positive (r > 0): As X increases, Y increases.
- Negative (r < 0): As X increases, Y decreases.
6. Why does the correlation coefficient range between -1 and 1?
The correlation coefficient ranges from -1 to +1 because it is a standardized measure of covariance. Dividing covariance by the product of standard deviations limits its value.
- +1: Perfect positive linear relationship
- -1: Perfect negative linear relationship
- 0: No linear relationship
7. What is the difference between covariance and correlation coefficient?
The covariance measures the direction of relationship, while the correlation coefficient measures both direction and strength in a standardized way.
- Covariance can take any real value.
- Correlation is always between -1 and +1.
- Correlation is unit-free, while covariance depends on units.
8. Can the correlation coefficient be greater than 1?
No, the correlation coefficient cannot be greater than 1 or less than -1. Its value is mathematically restricted to the interval [-1, 1].
- If you obtain a value outside this range, there is likely a calculation error.
- This restriction comes from dividing covariance by standard deviations.
9. What are the types of correlation coefficients?
The main types of correlation coefficients measure different kinds of relationships between variables.
- Pearson correlation: Measures linear relationship between continuous variables.
- Spearman rank correlation: Measures monotonic relationship using ranks.
- Kendall’s tau: Measures ordinal association between variables.
10. What is a simple example of calculating correlation coefficient?
A simple example of a correlation coefficient calculation shows how two variables are perfectly related. Consider X = (1,2,3) and Y = (2,4,6).
- Mean of X = 2, Mean of Y = 4
- Compute deviations and products
- Apply the Pearson formula
