Variance Calculator

What is Variance?

As per the variance definition, Variance is defined as one of the measures of dispersion, which means the measure of by how much numbers in data set possibly differ from mean of values.

It shows the average square of deviations taken from their means. By taking the square of deviations it ensures that negative and positive deviations do not cancel each other out. Variance along with covariance is very useful and these concepts are very important for students.

What is Sample Variance?

A set of data as a data sample is collected from the population. Usually, the population is very large and complete counting of all values is impossible.

Mainly sample is taken from a population with manageable size, say 2,000, and that data is used for calculations. Following sample variance formula is used for sample variance equation:

$$σ^2\;\text{(Sample)}\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N-1}$$

What is Population Variance?

How data points in a particular population are spread identified by population variance (σ2). This is calculated as the average of distances in the population from each data point to mean square.

Following variance formula is used for population variance equation:

$$σ^2\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N}$$

Also find this useful variance tutorial for understanding this concept thoroughly.

Can Variance be negative?

A variance equation never gives a negative because squared values are used for taking the mean and therefore results can be either positive or zero. If we get negative variance, it means we have a calculation error.

How to calculate Variance?

A step-by-step guide on how to calculate variance (σ² using coefficient of variation calculator.

• Sample variance calculator uses the following formula to calculate the Variance(σ2).

  $$σ^2\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N}$$

• Step 1: Determine all possible outcomes

  This calculator calculates the variance from set of values. First step it uses is to take square of all the values available in the entire population:

  x	x2
  400	160000
  270	72900
  200	40000
  350	122500
  170	28900

• Step 2: Calculate the Mean

  Then calculate the sum of all values, ∑x

  $$\sum x\;=\;1390$$

• Take the square of answer and divide that value by size of population.

  $$\frac{(\sum x)^2}{N}\;=\;\frac{1390^2}{5}$$

  $$=\;\frac{1932100} {5}\;=\;386420$$

• Then calculate the sum of all the square values, ∑x2

  $$\sum x^2\;=\; 424300$$

• Subtract,

  $$\frac{\sum x^2\;-\;(\sum x)^2}{N}$$

  $$=\;424300–386420$$

  $$=\;37880$$

• Step 3: Calculate Variance

  For Variance, divide the answer with size of population,

  $$σ^2\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N}$$

  $$=\;\frac{37880} {5}=7576$$

  So the Variance is 7576.

Similar steps were taken for calculating Sample Variance, only the last step is varied according to formula.

$$σ^2\;\text{(Sample)}\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N-1}$$

For Variance, divide the answer with one less than the size of the population,

$$σ^2\;\text{(Sample)}\;=\;\frac{\sum x^2\;-\;\frac{(\sum x)^2}{N}}{N-1}$$

$$=\;\frac{37880}{4}\;=\;9470$$

So the Variance is 9470.

Calculating the variance includes square deviations, so the units are not the same as units entered in the input field for the values variance formula calculator calculates.

Use covariance calculator with mean and standard deviation for your learning & practice of covariance.

How to use Variance Calculator?

Variance calculator is very easy to use. Just follow the below steps:

• Enter the values in the white shaded box. You may also copy/paste data. Values must be numeric and separated by commas. A comma must be used to separate the values otherwise sample variance calculator will show the error "Please match the required format".
• After entering values you may click on the "Calculate" button to execute the computation.
• Variance calculator will calculate the resultant variance and display results for both Variance (σ²) and Variance σ² (Sample).

We hope you liked our calculator & its article. Please send us your valuable feedback.

External Resources:

• What is variance?
• Demonstration of variance with examples and formula.
• Different ways to calculate variance.
• Difference between variance and standard deviation.