Correlation and Covariance for Standardised Attributes

Understanding statistics is an essential part of learning Physics. All primary measurement techniques in Physics require Statistics in some form. It helps in resolving errors and improving the correctness of measurements.

Correlation and covariance are two important aspects of statistics that help physicists determine relations. Covariance and correlation help us determine the effect of two different variables on each other. For instance, it could have a positive or a negative impact.

In this article, we will discuss the concepts of covariance and correlation. But before jumping into the topic, let’s take a brief look at variance and standard deviation.

Variance and Standard Deviation

The formulas for the two important mathematical aspects, variance and standard deviation, are as follows.

Variance

First, find the deviations from the mean. Then, the squares of them. Now, you have to find the mean of the squares.

σ2 =1ni=1n(xi-x̅)2

Standard Deviation

It is the positive square root of the above formula for variance.

σ = [1ni=1n(xi-x̅)2]1/2

Let’s understand the meaning of correlation and covariance.

Covariance and Correlation Meaning

In simple definitions, covariance and correlation combine two variables and tell us the relation between the variables. So, let’s look at the correlation and covariance meaning separately.

Covariance

It measures the dependency of two random variables on each other; that is, how they change in their presence. Two random variables can correspond to be dependent or independent.

Two random variables are dependent when covariance changes positively or negatively. The change is positive when the two variables differ in the same direction. Negative change occurs when the variables differ in opposite directions.

The two random variables are independent if they don’t vary together. So, it makes their covariance zero.

Defining Covariance Mathematically

Now that we have understood the meaning of covariance, here is the mathematical definition.

Suppose there are two random variables, X and Y. The covariance for the variables is denoted by Cov (X, Y) and given by:

Cov (X, Y) = ∑ (X – X) (Y –Y ) / N= ∑ xy/N

Where X = ∑ X / N

Y   ∑ Y / N

x = X –X

y = Y –Y 

x is the deviation of the ith value of X from its mean value.

y is the deviation of the ith value of Y from its mean value.

Correlation

Correlation is defined in terms of covariance. Let’s take a simple example of an increase in the sale of sweaters and people dying of hypothermia. The root cause of both cases is a fall in temperature. Here, the fall in temperature is the reason for the correlation between the two.

Correlation tells us how intensely the two random variables are related. It also studies the direction in which variables change.

So, correlation can either be positive or negative. The positive and negative signs indicate the direction of the relation between the variables. If the correlation is zero, it infers the two variables do not relate in any case. If the covariance is zero, the correlation is also zero.

Correlation Coefficient

The correlation coefficient or Karl Pearson’s Coefficient of correlation is given by:

r = ∑ xy / N σx σy

r = (X –X)(Y –Y )∑(X -X)2∑(Y -Y )2

r = {[∑XY – (∑X) (∑Y)]/ N} / [√∑X2 – (∑X2)/ N] [ √∑Y2 – (∑Y)2/N]

r = N∑XY – (∑X)(∑Y) / √N∑X2 – (∑X)2 × √N∑Y2 – (∑Y)2

r = ∑ xy / N σx σy

r = ∑ (X –X) (Y –Y )  √∑ (X -X)2 √∑ (Y -Y )2

The value of r has no unit since it’s a coefficient. If r is negative, there is an inverse relationship between the two variables. It means the variables change in two different directions. If r is positive, the two variables vary in the same direction. If r is 0, the two variables are unrelated.

The value of correlation always lies between -1 and 1.

Why Does Correlation Have Preference Over Covariance?

Covariance is not as commonly seen as correlation. The reasons are as follows.

• Covariance varies with unit. But correlation has no unit. So, it is easier to make intuitions in the case of correlation.
• Covariance does not have a sign. So, it is not always possible to tell the strength of the relation between the two variables. However, the positive and negative signs of correlation help determine the strength of their relationship.
• A zero covariance does not always imply no relationship between variables. But a 

Conclusion

Correlation and covariance are useful measurement techniques that help determine the strength and relations between two variables.

We have discussed many aspects of covariance and correlation in this article. First, we discussed the correlation and covariance meaning. We also learned the reasons why correlation finds preference over covariance.

The above-given correlation and covariance notes will help aspiring IIT/JEE students understand several concepts of Physics and Maths.