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Covariance vs. Correlation: What is the Difference

Whether you are a newbie or a budding professional in the field of Data Science, having a thorough grasp on your basics is a must. And one of them is the very fundamental concept of the difference between covariance and correlation.

With the enormous amount of information generated, consumed, and stored globally, companies have a gold mine of data at their disposal. However, all that data is practically useless if not analyzed and manipulated to derive actionable insights. Here’s where Data Science comes into the picture – with its invaluable arsenal of statistical methods, data analytics, scientific methods, and artificial intelligence algorithms, Data Science is the ultimate savior. Data Science enables business analysts to discover trends and insights from large datasets, which can be further used to shape business decisions.

With the inarguable importance Data Science has in moulding the course of technology, let’s dive into the fundamentals of covariance vs. correlation and upGrad courses that can help you learn them.

Covariance vs. Correlation: What do they mean?

Covariance and correlation are two prevalent terms that one comes across in statistics and probability theory. While both have very similar connotations and describe the dependency and linear relationship between variables, there are stark differences between the two. Covariance signifies the direction of the linear relationship between two variables, whereas correlation indicates both the direction and strength of the linear relationship between variables. 

Before we get into the detailed explanation of covariance vs. correlation, it is essential to understand two other fundamental terms – variance and standard deviation.

Variance

Variance is the measure of the spread between variables in a dataset. In simpler terms, variance measures how far each variable in the dataset is from the average value and thus from every other variable in the set. The larger the spread, the more the variance with respect to the mean (average). Variance is denoted by the symbol S2 (sample variance).

Mathematically, variance is depicted using the formula:

S2 = Σ(X – x̄)2 / n – 1

where,

S2 = sample variance

Σ = sum of

X = each value

x̄ = sample mean

n = number of data values 

Standard Deviation

Standard Deviation measures the amount of dispersion or variation of a dataset relative to its mean. While a high value of standard deviation indicates that the data points are spread out over a broader range, a low value of standard deviation would mean that the data points are close to the mean of the dataset. Standard deviation is denoted by the symbol ‘s’ (sample standard deviation) or σ (population standard deviation).

Mathematically, the standard deviation is depicted using the formula:

s = √Σ(X – x̄)2 / n – 1

where,

s = sample standard deviation

Σ = sum of

X = each value

x̄ = sample mean

n = number of data values 

Covariance

Covariance is an extension of variance and determines the direction of the relationship between two variables. In other words, covariance indicates whether the two variables are directly proportional or inversely proportional to one other. Therefore, a change in the value of one variable will inevitably affect the other. However, it is pertinent to mention that covariance only measures the change of one variable with respect to another and not their inter-dependency.

  • Covariance can take any value between -∞ and +∞. 
  • A positive covariance value signifies a direct relationship between the variables. So, an increase in the value of one variable would lead to a corresponding increase in the other variable, with other conditions remaining constant. Thus, both the variables move together in the same direction as they change.
  • In contrast, a negative covariance would mean an inverse relationship between the two variables. When the value of one variable increases, the other will decrease. Essentially, these variables are said to be inversely related and move in opposite directions.

Mathematically, the covariance between two variables x and y is represented as follows:

Cov(X,Y) = Σ(Xi – x̄)(Yi – ȳ) / n – 1

where,

Cov(X,Y) = covariance between x and y

Σ = sum of

Xi = data value of X

Yi = data value of Y

x̄ = mean of X

ȳ = mean of Y

n = number of data values 

Source

Correlation

In contrast to covariance that only measures the direction of the relationship between two variables, correlation also measures the relationship’s strength. Thus, correlation quantifies the relationship between the variables and signifies how strong or weak the relationship is. The primary outcome of correlation is the correlation coefficient ( r ).

  • Correlation can only take values between -1 and +1.
  • A correlation of +1 signifies a direct and strong relationship between the variables. The increase in one variable leads to a corresponding rise in the other. On the other hand, a correlation of -1indicates a solid, inverse relationship. An increase in one variable will cause an equal and opposite decrease in the other. A correlation value of 0 means that the variables do not have any linear relationship.
  • A correlation value closer to -1 or +1 would mean a close relationship between the variables.

The mathematical expression of correlation is as follows:

r = Cov(x,y) / σX – σY

where,

Cov(x,y) = covariance between X and Y

σX = standard deviation of X

σY = standard deviation of Y


Source

Difference Between Covariance and Correlation

Now that we have covered the basic concepts related to covariance and correlation, it is time to delve into their differences. No doubt, the two statistical terms seem pretty similar at first glance. However, a more detailed study reveals that covariance and correlation are distinct in several aspects.

So, let us look at the difference between covariance and correlation:

  • Meaning

Covariance is a measure of the extent to which two variables change together. 

On the other hand, correlation is a measurement of the strength of the linear relationship between variables.

  • Values

Covariance can take any value between -∞ and +∞. 

The correlation value can be anywhere between -1 and +1.

  • What do they represent?

Covariance shows the direction of the linear relationship between the variables. While a positive value indicates a direct relationship, a negative covariance value means an inverse relationship.

In contrast, correlation indicates both the direction and strength of the linear relationship between the variables. The closer the value to +1 or -1,  the stronger the relationship.

  • Scalability

A change of scale affects covariance. For instance, if the value of two variables is multiplied by the same or different constants, the calculated covariance of the two variables will change.

In contrast, correlation is immune to the change in scale. Hence, multiplication by constants does not change the initial correlation value. 

  • Units

The unit of covariance is the product of the units of the two variables.

On the other hand, correlation is dimensionless. Therefore, it is a unit-free measure of the relationship between the variables that makes the comparison of calculated correlation values easier across variables.

  • Utility

Covariance can be computed for only two variables.

On the other hand, correlation can be calculated for multiple sets of variables, a quality that makes it a more convenient choice for data analysts. 

  • Applications

Covariance mostly finds its use as an input to other analyses. Typical use cases are in stochastic modelling and principal component analysis.

Common applications of correlation include summarizing large amounts of data, input into other analyses, and as a diagnostic for further analyses.

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To Wrap It Up

Both covariance and correlation measure the linear relationship between variables. Nonetheless, given a choice between the two, correlation is favoured over covariance for two primary reasons. First, the correlation coefficient remains unaffected by the change in scale, and second, it is a unitless measure that simplifies comparisons. 

A strong foundation of mathematical and statistical concepts is crucial to a promising career in Data Science and Artificial Intelligence. However, with the cut-throat competition and the constant need for professional upskilling, the best way to future-proof your resume is by choosing the right program – a step that you can take with upGrad. 

How do covariance and correlation help Data Scientists?

Covariance and correlation are two common statistical concepts used by Data Scientists to measure the linear relationship between two variables in data. While covariance identifies how two variables vary simultaneously, correlation determines how change in one variable affects the change in another variable.

Can I do covariance and correlation in MS Excel?

Yes, covariance and correlation can be calculated using MS Excel. The first step is to enter the data into the Excel sheet in clearly labeled columns. Then, you can choose either of the following options:
1. Use Function Codes: The function code for covariance is =covar(array1,array2) and that for correlation is =correl(array1,array2)
2. Use Toolpak Method: Under the Data tab, click on Data Analysis and choose the desired calculation.

What is the difference between population and sample?

Population and sample are two commonly used statistical terms. Their difference lies in how observations are assigned to the dataset - while a population includes all the elements of a dataset, a sample comprises one or more observations drawn from a population. Based on the sampling method, a sample can have fewer, more, or the same number of observations as the population.

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