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
1. What is a correlation coefficient and what does it measure?
A correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It is represented by the symbol 'r'. Its value indicates how well the data points fit a straight line, telling us if the variables tend to move together, in opposite directions, or not at all.
2. How do you interpret the value of a correlation coefficient?
The value of the correlation coefficient 'r' ranges from -1 to +1 and is interpreted based on its sign and magnitude:
- Positive Correlation (r > 0): Indicates that as one variable increases, the other variable tends to increase. A value close to +1 signifies a strong positive relationship.
- Negative Correlation (r < 0): Indicates that as one variable increases, the other variable tends to decrease. A value close to -1 signifies a strong negative relationship.
- No Correlation (r ≈ 0): A value near 0 suggests a weak or no linear relationship between the variables.
For example, a coefficient of 0.8 shows a strong positive correlation, while -0.2 shows a very weak negative correlation.
3. What are some real-world examples of positive and negative correlation?
Understanding correlation is easier with real-world examples:
- Positive Correlation Examples:
- The number of hours a student studies and their exam scores.
- An individual's height and their weight.
- Increased advertising spending and a rise in product sales.
- Negative Correlation Examples:
- The number of hours spent playing video games and academic performance.
- The outside temperature and the sale of winter coats.
- The age of a car and its resale value.
4. What is the formula for Karl Pearson's correlation coefficient?
As per the CBSE syllabus for the 2025-26 session, Karl Pearson's correlation coefficient (r) is the most common method of calculation. The formula is:
r = Cov(x, y) / (σₓ * σᵧ)
Where:
- Cov(x, y) is the covariance between variables x and y, measuring their joint variability.
- σₓ is the standard deviation of variable x.
- σᵧ is the standard deviation of variable y.
This formula effectively standardises the covariance, resulting in a value between -1 and +1.
5. What is the range of the correlation coefficient, and what do the values -1, 0, and +1 signify?
The correlation coefficient 'r' always lies in the range of -1 to +1, inclusive. The extreme and middle values have specific meanings:
- r = +1: This indicates a perfect positive linear relationship. All data points lie exactly on a straight line with a positive slope.
- r = -1: This indicates a perfect negative linear relationship. All data points lie exactly on a straight line with a negative slope.
- r = 0: This indicates a complete absence of a linear relationship between the two variables. However, it does not rule out the possibility of a non-linear relationship.
6. Why does a strong correlation between two variables not necessarily mean one causes the other?
This is a critical concept known as "correlation does not imply causation." A strong correlation shows that two variables are associated, but it doesn't prove that a change in one variable is the direct cause of a change in the other. Often, a third, unobserved variable (a confounding or lurking variable) may be influencing both. For example, ice cream sales and drowning incidents are positively correlated, but eating ice cream doesn't cause drowning. The lurking variable is hot weather, which increases both activities.
7. Why is the correlation coefficient independent of the change of origin and scale?
The correlation coefficient is a unitless measure because of its formula. When you change the origin (add or subtract a constant from all values of a variable) or change the scale (multiply or divide all values by a constant), both the numerator (covariance) and the denominator (product of standard deviations) of the formula are affected proportionally. These effects cancel each other out, leaving the value of 'r' unchanged. This property makes it a robust measure for comparing relationships between variables measured on different scales.
8. How does the correlation coefficient differ from covariance?
While both measures describe the relationship between two variables, their primary difference lies in their scale and interpretation:
- Covariance: It measures the direction of the linear relationship (positive, negative, or zero). However, its magnitude is not standardized and depends on the units of the variables, making it difficult to interpret the strength of the relationship.
- Correlation Coefficient: It measures both the direction and the strength of the relationship. By dividing the covariance by the product of the standard deviations, the correlation coefficient becomes a standardized, unit-free value between -1 and +1, making it easy to interpret and compare across different datasets.
