# Variance - Definition, Formula, Examples, Properties (2023)

Variance is a statistical measurement that is used to determine the spread of numbers in a data set with respect to the average value or the mean. The standard deviation squared will give us the variance. Using variance we can evaluate how stretched or squeezed a distribution is.

There can be two types of variances in statistics, namely, sample variance and population variance. The symbol of variance is given by σ2. Variance is widely used in hypothesis testing, checking the goodness of fit, and Monte Carlo sampling. To check how widely individual data points vary with respect to the mean we use variance. In this article, we will take a look at the definition, examples, formulas, applications, and properties of variance.

 1 What is Variance? 2 Variance Formula 3 Variance and Standard Deviation 4 How to Find Variance? 5 Variance and Covariance 6 Variance Properties 7 FAQs on Variance

## What is Variance?

Variance is a measure of dispersion. A measure of dispersion is a quantity that is used to check the variability of data about an average value. Data can be of two types - grouped and ungrouped. When data is expressed in the form of class intervals it is known as grouped data. On the other hand, if data consists of individual data points, it is called ungrouped data. The sample and population variance can be determined for both kinds of data.

### Variance Definition

Population Variance - All the members of a group are known as the population. When we want to find how each data point in a given population varies or is spread out then we use the population variance. It is used to give the squared distance of each data point from the population mean.

Sample Variance - If the size of the population is too large then it is difficult to take each data point into consideration. In such a case, a select number of data points are picked up from the population to form the sample that can describe the entire group. Thus, the sample variance can be defined as the average of the squared distances from the mean. The variance is always calculated with respect to the sample mean.

A general definition of variance is that it is the expected value of the squared differences from the mean.

### Variance Example

Suppose we have the data set {3, 5, 8, 1} and we want to find the population variance. The mean is given as (3 + 5 + 8 + 1) / 4 = 4.25. Then by using the definition of variance we get [(3 - 4.25)2 + (5 - 4.25)2 + (8 - 4.25)2 + (1 - 4.25)2] / 4 = 6.68. Thus, variance = 6.68.

(Video) What is Variance in Statistics? Learn the Variance Formula and Calculating Statistical Variance!

## Variance Formula

Depending upon the type of data available and what needs to be determined, the variance formula can be given as follows:

Grouped Data Sample Variance = $$\sum \frac{f\left ( M_{i}-\overline{X} \right )^{2}}{N - 1}$$

Grouped Data Population Variance = $$\sum \frac{f\left ( M_{i}-\overline{X} \right )^{2}}{N}$$

Ungrouped Data Sample Variance = $$\sum \frac{\left ( X_{i}-\overline{X} \right )^{2}}{n - 1}$$

Ungrouped Data Population Variance = $$\sum \frac{\left ( X_{i}-\overline{X} \right )^{2}}{n}$$

where, $$\overline{X}$$ stands for mean, $$M_{i}$$ is the midpoint of the ith interval, $$X_{i}$$ is the ith data point, N is the summation of all frequencies and n is the number of observations.

Mean for grouped data = $$\frac{\sum M_{i}f_{i}}{\sum f_{i}}$$

The general formula for variance is given as,

(Video) How To Calculate Variance

Var (X) = E[( X – μ)2]

## Variance and Standard Deviation

When we take the square of the standard deviation we get the variance of the given data. Intuitively we can think of the variance as a numerical value that is used to evaluate the variability of data about the mean. This implies that the variance shows how far each individual data point is from the average as well as from each other. When we want to find the dispersion of the data points relative to the mean we use the standard deviation. In other words, when we want to see how the observations in a data set differ from the mean, standard deviation is used. σ2 is the symbol to denote variance and σ represents the standard deviation. Variance is expressed in square units while the standard deviation has the same unit as the population or the sample.

## How to Find Variance?

The following steps can be used to find the variance of ungrouped data:

• Find the mean of the observations. This can be done by dividing the sum of all observations by the number of observations.
• Subtract the mean from each observation.
• Square each of these values.
• Add all the values obtained in the previous step.
• Divide the value from step 4 by n (for population variance) or n - 1 (for sample variance).

### Variance of Binomial Distribution

A binomial distribution is defined as a discrete probability distribution that details the number of successes when a binomial experiment is conducted n number of times. Each time the outcome of the experiment can only be either 0 or 1. Say we have a binomial experiment that consists of n number of trials and the probability of success in each trial is given by p, then the variance of the binomial distribution is given as:

σ2 = np (1 - p).

Here, np is also equal to the mean.

### Variance of Poisson Distribution

Poisson distribution is another type of discrete probability distribution that gives the probability of a certain number of events taking place within a specific time frame. The parameter of a Poisson distribution is given by λ. In this distribution the mean and the variance are equal. The variance of the Poisson distribution is given by:

(Video) Variance & Standard Deviation | properties,formulas,examples

σ2 = λ

### Variance of Uniform Distribution

Uniform distribution is a type of continuous probability distribution. It is also known as a rectangular distribution as the outcome of the experiment will lie between a minimum and maximum bound. If a is the minimum bound and b is the maximum bound, then the variance of uniform distribution is as follows:

σ2 = (1/12)(b - a)2

The mean is given by (b + a) / 2.

## Variance and Covariance

Variance is used to describe the spread of the data set and identify how far each data point lies from the mean. Covariance shows us how two random variables will be related to each other. It measures how one variable will get affected due to a change in the other random variable. If we have a positive covariance, it implies that both the variables are moving in the same direction. However, if we have a negative covariance, it means that both variables are moving in opposite directions. Suppose we have two random variables x and y. Here, x is the dependent variable and y is the independent variable. Let n be the number of data points in the sample, $$\overline{x}$$ is the mean of x and $$\overline{y}$$ is the mean of y, then the formula for covariance is given below:

cov (x, y) = $$\frac{\sum_{i = 1}^{n}(x_{i} - \overline{x})(y_{i} - \overline{y})}{n - 1}$$

## Variance Properties

Some of the properties of variance are given below that can help in solving both simple and complicated problem sums.

(Video) Properties of the sample variance

• If the value of the variance is 0, it indicates that all the data points in the data set are of equal value.
• A large variance implies that the data is more vastly spread out from the mean. Similarly, a small variance shows that the values of the data points are closer together and are clustered 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.

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Important Notes on Variance

• Variance is a measure of the variability of data and describes how the data points are spread out with respect to the mean.
• There can be two types of variance - sample variance and population variance.
• There can be two kinds of data - grouped and ungrouped. Thus, we can have grouped sample variance, ungrouped sample variance, grouped population variance, and ungrouped population variance.
• The variance is the standard deviation squared.
• Covariance describes how a dependent and an independent random variable are related to each other.

## FAQs on Variance

### What is Variance in Statistics?

Variance in Statistics is a measure of dispersion that indicates the variability of the data points with respect to the mean. Sample Variance and Population Variance are the two types of variance.

### How to Calculate Variance?

The variance of ungrouped data can be calculated by using the following steps:

• Find the mean and subtract it from each data point.
• Take the summation of the squares of the values obtained in step 1.
• Divide this value by n(number of observations) or n - 1 if the population or sample variance needs to be calculated respectively.

### What Does Variance Tell You About Data?

Variance tells us how spread out the data is with respect to the mean. If the data is more widely spread out with reference to the mean then the variance will be higher. If the data is clustered near the mean then the variance will be lower.

### What is Variance and Standard Deviation?

The square of the standard deviation gives us the variance. The standard deviation will have the same unit as the data while the unit of the variance will differ as it is a squared value.

### Is Variance a Measure of Dispersion?

Variance and standard deviation are the most commonly used measures of dispersion. Standard deviation is the square root of the variance. These measures help to determine the dispersion of the data points with respect to the mean.

(Video) Properties of Variance - BSc Statistics

### Is Variance a Measure of Central Tendency?

Variance is not a measure of central tendency. There are three measures of central tendency, namely, mean, median, and mode. Variance is a measure of dispersion.

### What are the Advantages of Variance?

One of the major advantages of variance is that regardless of the direction of data points, the variance will always treat deviations from the mean like the same. Moreover, variance can be used to check the variability within the data set.

## FAQs

### What are the properties of variance? ›

Variance is the average of the square of the distance from the mean. For this reason, variance is sometimes called the “mean square deviation.” Then we take its square root to get the standard deviation—which in turn is called “root mean square deviation.”

What is variance formula with example? ›

Variance Example

Suppose we have the data set {3, 5, 8, 1} and we want to find the population variance. The mean is given as (3 + 5 + 8 + 1) / 4 = 4.25. Then by using the definition of variance we get [(3 - 4.25)2 + (5 - 4.25)2 + (8 - 4.25)2 + (1 - 4.25)2] / 4 = 6.68. Thus, variance = 6.68.

What is variance and its properties in statistics? ›

Variance is a measure of how data points differ from the mean. According to Layman, a variance is a measure of how far a set of data (numbers) are spread out from their mean (average) value. Variance means to find the expected difference of deviation from actual value.

What is the formula for calculating variance? ›

The variance (σ2), is defined as the sum of the squared distances of each term in the distribution from the mean (μ), divided by the number of terms in the distribution (N). You take the sum of the squares of the terms in the distribution, and divide by the number of terms in the distribution (N).

What are the properties of variance or standard deviation? ›

Standard deviation is the spread of a group of numbers from the mean. The variance measures the average degree to which each point differs from the mean. While standard deviation is the square root of the variance, variance is the average of all data points within a group.

What is the definition of variance? ›

The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean. Variance tells you the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean.

How do you prove property of variance? ›

Here is a useful formula for computing the variance. To prove it note that Var(X)=E[(X−μX)2]=E[X2−2μXX+μ2X]=E[X2]−2E[μXX]+E[μ2X] by linearity of expectation.

What are properties of standard deviation? ›

Properties of Standard Deviation

The square root of the means of all the squares of all values in a data set is described by the standard deviation. In other terms, the standard deviation is also called the root mean square deviation. The smallest value of the standard deviation can only be number zero.

What are the properties of mean in statistics? ›

Properties of Arithmetic Mean

The sum of deviations of the items from their arithmetic mean is always zero, i.e. ∑(x – X) = 0. The sum of the squared deviations of the items from Arithmetic Mean (A.M) is minimum, which is less than the sum of the squared deviations of the items from any other values.

How do you find variance and deviation? ›

To calculate the variance, you first subtract the mean from each number and then square the results to find the squared differences. You then find the average of those squared differences. The result is the variance. The standard deviation is a measure of how spread out the numbers in a distribution are.

### What are the 3 variances? ›

The three main types of variance analysis are material variance, labor variance and fixed overhead variance.

Why do we calculate variance? ›

Variance is a calculation that considers random variables in terms of their relationship to the mean of its data set. You can use variance to determine how far each variable is from the mean and how far each variable is from one another.

What is variation in statistics with example? ›

It is the difference between the smallest data item in the set and the largest. For example, the range of 73, 79, 84, 87, 88, 91, and 94 is 21, because 94 – 73 is 21.

What are the properties of mean and variance of sampling distribution? ›

That is, the variance of the sampling distribution of the mean is the population variance divided by N, the sample size (the number of scores used to compute a mean). Thus, the larger the sample size, the smaller the variance of the sampling distribution of the mean. For N numbers, the variance would be Nσ2.

What are the properties of standard error? ›

The standard deviation of a sampling distribution is called as standard error. In sampling, the three most important characteristics are: accuracy, bias and precision. It can be said that: The estimate derived from any one sample is accurate to the extent that it differs from the population parameter.

What are the properties of a good measure of variation? ›

Properties of Good Measure of Variation
• It should be simple to understand.
• It should be easy to compute.
• It should be rigidly defined.
• It should be based on each and every item of the distribution.
• It should have sampling stability.
• It should not be affected by the extreme items.

What is the variance of random variable property? ›

The variance Var(x) of a random variable is defined as Var(x) = E((x - E(x) 2). Two random variables x and y are independent if E(xy) = E(x)E(y). The standard deviation of a random variable is defined by σ x = √Var(x). The term standard error is used instead of standard deviation when referring to the sample mean.

What are the properties of covariance? ›

One of the key properties of the covariance is the fact that independent random variables have zero covariance. Covariance of independent variables. If X X X and Y Y Y are independent random variables, then Cov ( X , Y ) = 0. \text{Cov}(X, Y) = 0.

What are the three properties of a standard normal distribution? ›

A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation.

What are the two properties of standard normal distribution? ›

What are the properties of normal distributions? Normal distributions have key characteristics that are easy to spot in graphs: The mean, median and mode are exactly the same. The distribution is symmetric about the mean—half the values fall below the mean and half above the mean.

### What is the one property of the standard distribution? ›

Properties of a normal distribution

The mean, mode and median are all equal. The curve is symmetric at the center (i.e. around the mean, μ). Exactly half of the values are to the left of center and exactly half the values are to the right. The total area under the curve is 1.

What are the four properties of mean? ›

The four statistical properties selected for investigation were as follows: Property A: The mean is located between the extreme values; Property B: The sum of the deviations is zero; Property F: When the mean is calculated, a value of zero, if present, must be taken into account; Property G: The mean value is ...

What are properties of means? ›

Properties of means

Algebraic sum of deviations of a set of values from their arithmetic mean is zero. The sum of the squares of the deviation of a set of values is minimum when taken about mean.

What is the property of a mean value? ›

A very useful property of harmonic functions is the mean value principle, which states that the value of a harmonic function at a point is equal to its average value over spheres or balls centred at that point.

What are the three types of variance? ›

The three main types of variance analysis are material variance, labor variance and fixed overhead variance.
...
Businesses may use this type of analysis to calculate variance in the following categories:
• Purchase variance.
• Sales variance.
• Material variance.
• Labor variance.
• Efficiency variance.
Jul 7, 2021

What are the three measures of variance? ›

Variability is most commonly measured with the following descriptive statistics: Range: the difference between the highest and lowest values. Interquartile range: the range of the middle half of a distribution. Standard deviation: average distance from the mean.

What is variance example? ›

Example 1 – Calculation of variance and standard deviation. Let's calculate the variance of the follow data set: 2, 7, 3, 12, 9. The variance is 13.84. To get the standard deviation, you calculate the square root of the variance, which is 3.72.

What are the names of the 10 variances? ›

Types of Variances which we are going to study in this chapter are:-
• Cost Variances.
• Material Variances.
• Labour Variances.
• Fixed Overhead Variance.
• Sales Variance.
• Profit Variance.
Jan 2, 2018

What is a 4 variance analysis? ›

Definition: Variance analysis is the study of deviations of actual behaviour versus forecasted or planned behaviour in budgeting or management accounting. This is essentially concerned with how the difference of actual and planned behaviours indicates how business performance is being impacted.

What are the two types of variances? ›

When effect of variance is concerned, there are two types of variances:
• When actual results are better than expected results given variance is described as favorable variance. ...
• When actual results are worse than expected results given variance is described as adverse variance, or unfavourable variance.

### What is common variance in statistics? ›

In applied statistics, (e.g., applied to the social sciences and psychometrics), common-method variance (CMV) is the spurious "variance that is attributable to the measurement method rather than to the constructs the measures are assumed to represent" or equivalently as "systematic error variance shared among variables ...

What is the unit for variance? ›

The formula for variance uses squares. Therefore, the variance has different units than the data for which it was calculated. For example, if the data were measured in inches, the variance would be measured in square inches. Interpreting the standard deviation in terms of units is easier because there are no squares.

What are the properties of measurement? ›

Measurement Properties: Validity, Reliability, and Responsiveness.

What are properties that can be measured? ›

Density, colour, hardness, melting and boiling points and electrical conductivity are all physical properties. Any property that can be measured, such as an object's density, colour, mass, volume, length, malleability, melting point, hardness, odour, temperature, and so on, is referred to as a property of matter.

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