In statistics, it is frequent that we come across these two terms known as covariance and correlation. The two terms are often used interchangeably. These two ideas are similar, but not the same. Both are used to determine the linear relationship and measure the dependency between two random variables. But are they the same? Not really.
Despite the similarities between these mathematical terms, they are different from each other.
Covariance is when two variables vary with each other, whereas Correlation is when the change in one variable results in the change in another variable.
In this article, we will try to define the terms correlation and covariance matrices, talk about covariance vs correlation, and understand the application of both terms.
Introduction To The Concept
Covariance and correlation are key concepts in statistics, helping us understand how two variables interact. Here’s a simple breakdown:
Covariance: This measures how two variables change together.
- Positive covariance: Both variables tend to increase or decrease together.
- Negative covariance: The variables tend to move in opposite directions.
- Zero covariance: The variables do not relate to each other.
Correlation: This describes the strength and direction of a relationship between two variables, ranging from -1 to 1.
- Correlation of -1: Perfect negative correlation; as one variable increases, the other decreases.
- Correlation of 1: Perfect positive correlation; as one variable increases, the other also increases.
- Correlation of 0: No relationship between the variables.
Correlation is calculated as the covariance of the two variables divided by the product of their standard deviations. This helps us understand the degree to which these variables deviate from their expected value or how they can deviate together.
So, now you can understand the difference between Covariance vs Correlation.
Difference between Covariance vs Correlation
Both the Covariance and Correlation metrics evaluate two variables throughout the entire domain and not on a single value. The differences between them are summarized in a tabular form for quick reference. Let us look at Covariance vs Correlation:
| Covariance | Correlation |
| Covariance is a measure to indicate the extent to which two random variables change in tandem. | Correlation is a measure used to represent how strongly two random variables are related to each other. |
| Covariance is nothing but a measure of correlation. | Correlation refers to the scaled form of covariance. |
| Covariance indicates the direction of the linear relationship between variables. | Correlation on the other hand measures both the strength and direction of the linear relationship between two variables. |
| Covariance can vary between -∞ and +∞ | Correlation ranges between -1 and +1 |
| Covariance is affected by the change in scale. If all the values of one variable are multiplied by a constant and all the values of another variable are multiplied by a similar or different constant, then the covariance is changed. | Correlation is not influenced by the change in scale. |
| Covariance assumes the units from the product of the units of the two variables. | Correlation is dimensionless, i.e. It’s a unit-free measure of the relationship between variables. |
| Covariance of two dependent variables measures how much in real quantity (i.e. cm, kg, liters) on average they co-vary. | Correlation of two dependent variables measures the proportion of how much on average these variables vary w.r.t one another. |
| Covariance is zero in case of independent variables (if one variable moves and the other doesn’t) because then the variables do not necessarily move together. | Independent movements do not contribute to the total correlation. Therefore, completely independent variables have a zero correlation. |
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What is covariance?
Covariance signifies the direction of the linear relationship between the two variables. By direction we mean if the variables are directly proportional or inversely proportional to each other. (Increasing the value of one variable might have a positive or a negative impact on the value of the other variable).
The values of covariance can be any number between the two opposite infinities. Also, it’s important to mention that covariance only measures how two variables change together, not the dependency of one variable on another one.
The value of covariance between 2 variables is achieved by taking the summation of the product of the differences from the means of the variables as follows:
The upper and lower limits for the covariance depend on the variances of the variables involved. These variances, in turn, can vary with the scaling of the variables. Even a change in the units of measurement can change the covariance. Thus, covariance is only useful to find the direction of the relationship between two variables and not the magnitude. Below are the plots which help us understand how the covariance between two variables would look in different directions.
Example:
| X | Y |
| 10 | 40 |
| 12 | 48 |
| 14 | 56 |
| 8 | 32 |
Step 1: Calculate Mean of X and Y
Mean of X ( μx ) : 10+12+14+8 / 4 = 11
Mean of Y(μy) = 40+48+56+32 = 44
Step 2: Substitute the values in the formula
| xi –x̅ | yi – ȳ |
| 10 – 11 = -1 | 40 – 44 = – 4 |
| 12 – 11 = 1 | 48 – 44 = 4 |
| 14 – 11 = 3 | 56 – 44 = 12 |
| 8 – 11 = -3 | 32 – 44 = 12 |
Substitute the above values in the formula
Cov(x,y) = (-1) (-4) +(1)(4)+(3)(12)+(-3)(12)
___________________________
4
Cov(x,y) = 8/2 = 4
Hence, Co-variance for the above data is 4
Types of Covariance
Covariance can be classified under two types positive or negative:
- Positive Covariance: Indicates that two variables move in the same direction. If one variable increases, the other also increases, and vice versa.
- Negative Covariance: Indicates that two variables move in opposite directions. If one variable increases, the other decreases, and vice versa.
Applications of Covariance
- Covariance is used in Biology such as in Genetics and Molecular Biology to measure certain DNAs.
- Covariance is used in the prediction of amount investment on different assets in financial markets
- Covariance is widely used to collate data obtained from astronomical /oceanographic studies to arrive at final conclusions
- In Statistics to analyze a set of data with logical implications of principal component we can use covariance matrix
- It is also used to study signals obtained in various forms.
Quick check – Introduction to Data Science
What is correlation?
Correlation analysis is a method of statistical evaluation used to study the strength of a relationship between two, numerically measured, continuous variables.
It not only shows the kind of relation (in terms of direction) but also how strong the relationship is. Thus, we can say the correlation values have standardized notions, whereas the covariance values are not standardized and cannot be used to compare how strong or weak the relationship is because the magnitude has no direct significance. It can assume values from -1 to +1.
To determine whether the covariance of the two variables is large or small, we need to assess it relative to the standard deviations of the two variables.
To do so we have to normalize the covariance by dividing it with the product of the standard deviations of the two variables, thus providing a correlation between the two variables.
The main result of a correlation is called the correlation coefficient.
The correlation coefficient is a dimensionless metric and its value ranges from -1 to +1.
The closer it is to +1 or -1, the more closely the two variables are related.
If there is no relationship at all between two variables, then the correlation coefficient will certainly be 0. However, if it is 0 then we can only say that there is no linear relationship. There could exist other functional relationships between the variables.
When the correlation coefficient is positive, an increase in one variable also increases the other. When the correlation coefficient is negative, the changes in the two variables are in opposite directions.
Example:
| X | Y |
| 10 | 40 |
| 12 | 48 |
| 14 | 56 |
| 8 | 32 |
Step 1: Calculate Mean of X and Y
Mean of X ( μx ) : 10+12+14+8 / 4 = 11
Mean of Y(μy) = 40+48+56+32/4 = 44
Step 2: Substitute the values in the formula
| xi –x̅ | yi – ȳ |
| 10 – 11 = -1 | 40 – 44 = – 4 |
| 12 – 11 = 1 | 48 – 44 = 4 |
| 14 – 11 = 3 | 56 – 44 = 12 |
| 8 – 11 = -3 | 32 – 44 = 12 |
Substitute the above values in the formula
Cov(x,y) = (-1) (-4) +(1)(4)+(3)(12)+(-3)(12)
___________________________
4
Cov(x,y) = 8/2 = 4
Hence, Co-variance for the above data is 4
Step 3: Now substitute the obtained answer in Correlation formula
Before substitution we have to find standard deviation of x and y
Lets take the data for X as mentioned in the table that is 10,12,14,8
To find standard deviation
Step 1: Find the mean of x that is x̄
10+14+12+8 /4 = 11
Step 2: Find each number deviation: Subtract each score with mean to get mean deviation
| 10 – 11 = -1 |
| 12 – 11 = 1 |
| 14 – 11 = 3 |
| 8 – 11 = -3 |
Step 3: Square the mean deviation obtained
| -1 | 1 |
| 1 | 1 |
| 3 | 9 |
| -3 | 9 |
Step 4: Sum the squares
1+1+9+9 = 20
Step 5: Find the variance
Divide the sum of squares with n-1 that is 4-1 = 3
20 /3 = 6.6
Step 6: Find the square root
Sqrt of 6.6 = 2.581
Therefore, Standard Deviation of x = 2.581
Find for Y using same method
The Standard Deviation of y = 10.29
Correlation = 4 /(2.581 x10.29 )
Correlation = 0.15065
Types of Correlation
- Simple Correlation: In simple correlation, a single number expresses the degree to which two variables are related.
- Partial Correlation: When one variable’s effects are removed, the correlation between two variables is revealed in partial correlation.
- Multiple correlation: A statistical technique that uses two or more variables to predict the value of one variable.
Applications of correlation
- Time vs Money spent by a customer on online e-commerce websites
- Comparison between the previous records of weather forecast to this current year.
- Widely used in pattern recognition
- Raise in temperature during summer v/s water consumption amongst family members is analyzed
- The relationship between population and poverty is gauged
Methods of calculating the correlation
- The graphic method
- The scatter method
- Co-relation Table
- Karl Pearson Coefficient of Correlation
- Coefficient of Concurrent deviation
- Spearman’s rank correlation coefficient
Before going into the details, let us first try to understand variance and standard deviation.
Quick check – Statistical Analysis Course
Variance
Variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of numbers are spread out from their average value.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range. It essentially measures the absolute variability of a random variable.
Covariance and correlation are related to each other, in the sense that covariance determines the type of interaction between two variables, while correlation determines the direction as well as the strength of the relationship between two variables.
Summary
The difference between Covariance and Correlation has been summarized below:
Covariance (Cov(X, Y))
Definition: Measures the direction of the relationship between two variables, X and Y.
Calculation: The expected value of the product of deviations from their respective means.
Significance of Sign:
- Positive covariance: X and Y move in the same direction.
- Negative covariance: X and Y move in opposite directions.
Limitation: Its magnitude is unbounded and influenced by the scale of the variables, making it hard to interpret alone.
Correlation (Pearson’s Correlation Coefficient, r):
Improvement over Covariance: Standardized covariance by dividing it by the product of the variables’ standard deviations.
Range: Confined between -1 and 1.
- 1 indicates a perfect positive linear relationship.
- -1 indicates a perfect negative linear relationship.
- 0 indicates no linear relationship.
Strength of Relationship: The absolute value indicates the strength.
Mathematical Expression: Derived by normalizing covariance.
Key Differences and Usage:
- Both measure linear relationships and do not imply causation.
- Correlation is preferred over covariance as it provides a scaled, interpretable measure and remains consistent across different scales and locations.
- Correlation is useful for comparing pairs of variables across different domains due to its constrained range (-1 to +1).
Limitations:
Both concepts only measure linear relationships and cannot capture more complex associations.
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Covarinca vs Corelation FAQs
Positive covariance indicates that as one variable increases, the other variable tends to increase as well. Conversely, as one variable decreases, the other tends to decrease. This implies a direct relationship between the two variables.
No, correlation alone cannot be used to infer causation. While correlation measures the strength and direction of a relationship between two variables, it does not imply that changes in one variable cause changes in the other. Establishing causation requires further statistical testing and analysis, often through controlled experiments or longitudinal studies.
Correlation is preferred because it is a dimensionless measure that provides a standardized scale from -1 to 1, which describes both the strength and direction of the linear relationship between variables. This standardization allows for comparison across different pairs of variables, regardless of their units of measurement, which is not possible with covariance.
A correlation coefficient of 0 implies that there is no linear relationship between the two variables. However, it’s important to note that there could still be a non-linear relationship between them that the correlation coefficient cannot detect.
Outliers can significantly affect both covariance and correlation. Since these measures rely on the mean values of the variables, an outlier can skew the mean and distort the overall picture of the relationship. A single outlier can have a large effect on the results, leading to overestimation or underestimation of the true relationship.
Yes, it’s possible to have a high covariance but a low correlation if the variables have high variances. Because correlation normalizes covariance by the standard deviations of the variables, if those standard deviations are large, the correlation can still be low even if the covariance is high.
A high correlation means that there is a strong linear relationship between the two variables. If the correlation is positive, the variables tend to move together; if it is negative, they tend to move in opposite directions. However, “high” is a relative term and the threshold for what constitutes a high correlation can vary by field and context.
Correlation is preferred because it is standardized and unit-free, making it easier to compare relationships between variables of different scales. Unlike covariance, it is not affected by changes in the scale of the variables.
No, while both measure relationships between variables, they serve different purposes. Covariance shows the direction of the relationship, whereas correlation shows both the direction and strength of the relationship.
Positive covariance indicates that two variables tend to increase or decrease together. For example, if X and Y both increase simultaneously, their covariance will be positive.
Negative covariance indicates that as one variable increases, the other decreases. This means the variables move in opposite directions.
Correlation is calculated by dividing the covariance of two variables by the product of their standard deviations. This results in a value between -1 and 1, indicating the strength and direction of the linear relationship.
A correlation of 0 means there is no linear relationship between the variables. Changes in one variable do not predict changes in the other.
Covariance is used in fields like genetics to measure relationships between traits, in finance to assess the co-movement of asset returns, and in principal component analysis to reduce data dimensions.
Correlation is widely used to find patterns in large datasets, such as predicting consumer behavior in marketing, analyzing weather patterns, and in various statistical analyses like regression and factor analysis.
