Variance: Definition, Formula, Properties & Examples

Variance is an important measurement in the statistics. It measures how far a data set is different from its mean value. The higher the variance, the more scattered the data would be from its mean and vice–versa. 

• In statistics, the whole process of variance is known as the measure of dispersion. 
• It calculates how distantly the numbers scatter from their mean value.
• Variance determines the expected difference with respect to the actual value.
• It is also known as the covariance of random variables.
• The value of variance is always non-negative.
• Investors use this method to determine the profitability of the company.
• If the value of the system is low, then there will be lower risk and lower return.
• When the value of variance is zero, then all data elements are identical.

Read More: Number System

Keyterms: Variance, Standard derivation, Statistics, Mean Value, Dispersion, Variable, Sigma Square, Population, Square Root

---

What is Variance?

[Click Here for Sample Questions]

Variance is the result of the squared deviation from a random variable from its sample mean. The other tool used to measure this relation is Standard Deviation. Variance is denoted by σ² (pronounced as sigma square).

• The formula of variance helps to measure the spread from the mean of the random variable. 
• Its formula varies depending on the population and sample.
• Variance is also called the measure of the spread of data from the mean.
• It helps compute the value of the mean and other data compared to other methods.

\(\text{s} = \sqrt{\frac{\Sigma(x - \overline{x})^2}{n - 1}}\)

Also Read: Mean and variance of random variable

---

Standard Deviation: Formula

[Click Here for Sample Questions]

Variance is particularly square of standard deviation. In other words, standard deviation is defined as the amount of variation in the set of values. The low value of standard deviation indicates that its value lies close to the mean.

• The smallest value of the standard deviation is zero.
• Standard Deviation specifies how the values are spread across the data samples.
• The general formula which is used to calculate the variance is mentioned below :

σ = √∑(X−μ)²/N∑(X−μ)²/N 

Where, X (or x) = Value of Observations

μ = Mean of all Values

n = Number of observations in the sample set

x = Sample mean

N = Total no. of values in the population

There exist separate formulas for a variance for the ungrouped and the grouped data which are as follows:

For Grouped Data

• Population Variance, for population of size N = Σ [f (Mi −¯X)² / N]
• Sample Variance, for a sample of size N = Σ [f (Mi −¯X)² / (N−1)] 

Read More: Measures of Dispersion

For Ungrouped Data

• Population Variance for population of size N = Σ [(Xi−¯X) / N ]
• Sample Variance for a sample of size N = Σ [(Xi−¯X)² / N−1]

Things that need to be kept in mind while calculating variance for ungrouped data and grouped data is:

• For grouped data, X¯= Σ (M_if) / Σf
• For ungrouped data, X¯ = Σ (x_i) / N

Where, X¯ = Mean

M_i = Mid-point of the ith interval

Read More: 

Difference Between Variance and Standard Deviation

---

Properties of Variance

[Click Here for Sample Questions]

The variance measures how far the set of data is dispersed out from its mean or the average value. The list of properties of variance is listed below:

• The variance is always non-negative since every term in the variance sum is squared. Hence, the result can either be positive or zero.
• The variance always has squared units.

There are other properties possessed by a variance. The variance, var(X) of a random variable, X has the following properties those of which are mentioned below:

1. Var (X+C) = Var (X).
2. Var (CX) = C². Var (X)
3. Var (aX+b)= a².Var(X),

Where, a & b are constants

If X1, X2,…….,Xn are n independent randomly picked variables

Hence, Var(X₁₊ X₂₊…….,+X_n) = Var(X₁)+ Var(X₂)+….+ Var(X_n). where, the value of C remains constant in the above equation.

---

How to calculate Variance?

[Click Here for Sample Questions]

The variance can be calculated using the steps mentioned below:

1. First, calculate the mean of the provided dataset.
2. Then calculate the average of the provided set of values.
3. Now you have to subtract the mean from each of the values and square them
4. Finally, calculate the average of the squared values.
5. This result obtained will be the variance.

Also Read:

---

---

Things to Remember

• The higher the value of variance, the more scattered is the data from its mean.
• Variance is the squared deviation of a random set of data from its mean value.
• If the provided data values are identical in a set, the value of variance will be 0.
• In some situations, the covariance of a random variable is treated as the variance of that variable.
• The value of variance can either be positive or zero i.e. non-negative but it can never have a negative value.

---