Properties of Variance

Statistical variance describes the dispersion of numbers in a dataset compared to the average or means value. It is calculated by multiplying the standard deviation by a square. Variance can determine the degree to which a distribution is stretched or compressed. 

In this article, we will understand the properties of expectation and variance, the Properties of mean and variance, and solve some example problems. The variance has the disadvantage that, unlike the standard deviation, its units differ from the random variable, which is why, once the calculation is complete, the standard deviation is more usually reported as a measure of dispersion.

Definition of Variance

The predicted variation between values in a data collection is variance. It calculates the difference between each figure and the average value. Traders and market analysts frequently utilise variance to forecast market volatility and the consistency of a certain investment return over time. Variance is usually represented by the sign σ², whereas its square root – i.e., the standard deviation – is represented by the symbol σ. 

There are two types of variance

Population Variance 

The population refers to a group’s entire membership. The population variance is used to determine how each data point fluctuates or is spread out in a particular population. It calculates each data point’s squared distance from the population mean.

Sample Variance 

It is difficult to consider each data point if the population is too vast. In this situation, a sample of data points from the population is taken to generate a sample that may be used to characterise the entire group. As a result, the sample variance is equal to the average squared deviations from the mean. The sample mean is always used to determine the variance.

Formula of variance

S²= ∑(Xᵢ – X̄ )²/ n-1

S² = sample of variance

Xᵢ = value of one observation

X̄ = the mean value of all observations

n= the number of observations

How do I find the variance?

Variance can be determined using the following steps:

Calculate the average of the observations. Divide the sum of all observations by the number of observations to arrive at this result.

Then the mean should be subtracted from each observation; after Squaring each of these values, add all the values that came in the previous step after subtraction. Then divide the value by n in the case of population variance and n-1 in sample variance.

Variance Properties

The following are some of the properties of variance that may be used to solve both basic and complex problem sums.

• The zero variance indicates that all of the data collection data points are equally important.
• When the variance is high, the data are widely dispersed from the mean. On the other hand, a small variance indicates that the data points’ values are closer together and grouped around the mean.
• Var(X + C) = Var(X), where X is a random variable and C is a constant.
• Var(aX + b) = a2, here a and b are constants.
• Var(CX) = C2
  Var(X), C is a constant.
• Var(x1 + x2 +……+ xn) = Var(x1) + Var(x2) +……..+Var(xn) where x1, x2,……, xn are independent random variables.

Properties of expectation and variance

If X is a discrete random variable, then X’s expected value (or mean) is a weighted average of all the possible values that X could take, each weighing according to the probability that it would occur. Usually, E(X) or m represents the expected value of X.

E(X) = S x P(X = x)

In other words, the expected value is: [(each possible outcome) * (probability of the outcome occurring)].

More specifically, the expectation is what you would expect the results of an experiment to be on average.

These were a few properties of expectation and variance.

Properties of Mean and Variance

• Lets there is a X and Y are random variables, then E(X + Y) = E(X) + E(Y).
• If X1, X2, … , Xn are random variables , then E(x1 + x2 +……+ xn) = E(X1) + E(X2) + … + E(Xn) = Σi E(Xi).
• For random variables, X and Y, E(XY) = E(X) E(Y). Both X and Y are independent variables here.
• If X is a random variable and a is constant, then E(aX)= a E(X)and E(X + a) = E(X) + a.
• No matter what the random variable is, X > 0, E(X) > 0.
• E(Y) ≥ E(X) when X and Y are such that Y ≥ X.

Conclusion

After remembering all these theories and properties, practice some solved examples and questions. Here are some key points below that you should remember about properties of expectation and variance and Properties of mean and variance.

Variance measures data variability that reflects how data points are distributed around the mean. Sample variance and population variance are two forms of variance. There are two types of data: grouped data and ungrouped data. As a result, grouped sample variance, ungrouped sample variance, grouped population variance, and ungrouped population variance are all possible.

The standard deviation squared is the variance. A dependent random variable and an independent random variable are described by covariance.