Variance Calculator Online | Calculate Now

A Variance Calculator is an online tool that can be used to calculate the variance of a set of numerical data. Variance refers to the number of ways that individual data points deviate from the mean. It is an important statistical measure that can be used in many areas of mathematics, education, finance, business analytics, research and data analysis.

This online tool can be used by students, teachers, researchers, financial analysts and business professionals in the USA to easily understand how data is distributed.

Variance shows how close or dispersed data points are from the average. If the variance is low then most data points will be close to the mean. If the variance is high then there will be a larger difference between observations.
There are two types of Variances:

• Population Variance – This measures how much the values in an entire population vary.
• Sample Variance – This measures how much the values in a sample vary.

A business owner can calculate the variance of their sales per month to see how consistent their business is. Variance can also be used by students to analyse exam scores, and by financial analysts to analyse how variable their investments are.

Our online Variance Calculator can be used to quickly calculate Population Variance, Sample Variance and other statistical measures.

How to Use the Variance Calculator

Using our online Variance Calculator is easy and does not require any advanced maths skills or statistical software.

Enter your data – Enter your numerical data into the calculator. Ensure your individual numbers are separated by commas or spaces.

Enter valid data – Ensure all data entered is valid numerical data.
Select the type of variance required – Choose between Population Variance or Sample Variance.

Click calculate – Once all your data has been entered click the calculate button.

View mean – See what the average is of your data.

View variance – See the population or Sample Variance.

View other results – If available view the standard deviation, number of observations and sum of squares.

For example, if you enter the sales per month of a business you can see how much the sales values deviate from the average.

Variance Calculator Formula

Variance can be calculated by finding the average of the squared differences from the mean.

There are two common formulas for calculating variance.

Population Variance

\\[
\\sigma^2=\\frac{\\sum(x_i-\\mu)^2}{N}
\\]
Where:
σ 2 = Population Variance
xi = Individual data points.
μ = Population mean.
N = Total number of values.
Σ = Sum of squared differences.
Sample Variance
\\[
s^2=\\frac{\\sum(x_i-\\bar{x})^2}{n-1}
\\]
Where:
s^2 = Sample Variance
xi = Individual data points.
x̄ = Sample mean.
n = Number of observations.
n-1 = Degrees of freedom.

Population Variance vs Sample Variance

The Population Variance takes the sum of squared differences and divides it by the total number of observations (N). Whereas the Sample Variance takes the sum of squared differences and divides it by (n−1) which is known as Bessel’s correction.

The Sample Variance formula helps to give an unbiased estimate of the Population Variance.

Var(X) = E(X^2) – [E(X)] 2
In this case of the normal curve with sigma = 1.4 the equation becomes:
Var(X)=σ 2=(1.4) 2=1.96
σ
μ-σ+σX ValueRelative Density
Feedback
Example
Let’s look at an example of how variance works.
A student gets the following test scores:
Dataset: 10, 12, 14, 16, 18

Step 1 – Work out the mean

Add up all the values and divide by the total number of observations.
\\[
Mean=\\frac{10+12+14+16+18}{5}
\\]
\\[
Mean=\\frac{70}{5}=14
\\]

Step 2 – Work out the differences

Minus the mean from each individual value.
Value Deviation (x-14)
10 -4
12 -2
14 0
16 2
18 4

Step 3 – Square the differences

Value Squared
10 16
12 4
14 0
16 4
18 16
Total 40

Step 4 – Work out the Population Variance

Take the sum of squares and divide by the total number of observations.
\\[
\\sigma^2=\\frac{40}{5}
\\]
\\[
\\sigma^2=8
\\]
Example Results
Mean
14
Population Variance
8
Sample Variance
10
Population Standard Deviation
2.83

If these 5 numbers represent the population then the variance would be 8. If these numbers represent a sample then the Sample Variance would be 10. This is because the sum of squares (40) is divided by 4 rather than 5.

Understanding Results

Understanding your results will help you see how your data is distributed.

• Mean – This shows the average of your data. Variance shows how much individual values deviate from this number.
• Population Variance – This shows how much the data deviates from the mean in terms of squared differences. This is useful when you have all the observations in your population.
• Sample Variance – This is used to estimate the Population Variance using a smaller set of observations. This uses n-1 in the denominator.
• Standard Deviation – This is the square root of the variance. It shows how variable the data is but in the same units.
• Degrees of Freedom – In Sample Variance the degrees of freedom are calculated by taking n-1. This accounts for estimating the mean.

A low variance means observations will be close to the average. A high variance means the values will be more spread out.

For example, if you analysed the monthly revenue of two businesses in the

USA, the business with the lowest variance in sales will have more consistent monthly revenues.

Factors that Affect the Variance

There are a few factors that will affect the outcome of your variance.

• Data – If your data is spread out more from the mean then you will get a larger variance. Whereas if your data is close together you will get a small variance.
• Outliers – If you have very large or very small values then your variance will be increased a lot.
• Number of observations – The number of observations will affect your results. Population Variance and Sample Variance also use different denominators.
• Mean – Variance depends on how far away individual observations are from the mean.
• Incorrect Data – If there are incorrect values, missing data or typos then your results will be misleading.
• Units – Variance will be in squared units. For example if your data was in dollars then your variance will be in dollars squared.
• Type of variance – If you choose Population Variance rather than Sample Variance this will affect your results. Different denominators are used in the formulas.

Advantages of Variance Calculator

There are many benefits to using a Variance Calculator online. This can be used by individuals, students, researchers and businesses.

Saves time – Quickly calculate Population Variance and Sample Variance.

Useful for statistics – Students and teachers can use this to understand averages, squared differences, variance and standard deviation.

Financial Analysis – Financial analysts use variance to analyse investment returns, portfolio performances and market variability.

Business – Businesses can analyse their monthly sales, revenue, costs, customer spend and performances.

Quality Control – Manufacturing companies can analyse their product measurements, output and quality.

Research – Researchers can analyse results from experiments, surveys and scientific measurements.

Data Analysis – Data analysts can use variance to understand how their numerical data is distributed.

Minimise human error – Calculate automatically to minimise any mathematical errors. As long as your data is accurate.

For example, a business owner could use this to see how much the sales per month varies between different locations.

Accuracy, Assumptions & Limitations

As long as your data is accurate and you use the right formula then this calculator will give you accurate results. However there are a few assumptions and limitations.

• Assumes data is accurate – This calculator assumes that your data contains valid numerical data. If there are missing values, incorrect data and formatting issues this will affect your results.
• Use the right calculation – If analysing an entire population use Population Variance. If analysing a sample use Sample Variance.
• Outliers – Variance can be affected by outliers as it uses squared differences. Having one very high or low observation will increase the variance.
• Squared Units – Variance will be in squared units which can be hard to understand. Standard deviation is usually easier to understand as it is in the same units.

Doesn’t explain the full distribution – Variance will not explain how your data is fully distributed. Two sets of data can have the same variance but be very different.

To get a full picture consider using variance alongside standard deviation, mean, median, range and other statistical measurements.

FAQs

1. What is a Variance Calculator?

Variance Calculator is an online statistical tool that calculates how much your data varies from the mean. This can be done by calculating the Population Variance or Sample Variance using squared differences.

2. What is the Difference Between Population Variance and Sample Variance?

Population Variance calculates how variable the entire population is. This is done by dividing the sum of squares by N. Sample Variance calculates how variable the population is using a sample. This is done by dividing the sum of squares by n-1.

3. Can variance be negative?

No. Variance can never be negative. This is because variance uses squared differences. This means it will always be 0 or positive. If you get a variance of 0 then this means all your observations are the same.

4. How is variance related to standard deviation?

Both variance and standard deviation show how variable the data is. Standard deviation is the square root of variance. Variance is the square of standard deviation. Standard deviation is usually easier to understand as it is in the original units.

5. When should I use a Variance Calculator?

You can use this tool to analyse student scores, business sales, financial returns, scientific measurements, manufacturing quality, survey results or any numerical data where it is important to know how variable your data is.