Introduction
Variance tells you how spread out a set of numbers is from the average. A small variance means the values are close together. A large variance means they are far apart. It is one of the most useful ideas in statistics and shows up in science, finance, and everyday data analysis.
This variance calculator finds the variance and standard deviation of any dataset. Just type in your numbers, and the tool does the rest. It works in two modes. The Standard Variance mode takes a list of numbers and calculates both sample variance and population variance. The Discrete Random Variable mode lets you enter values along with their probabilities to find the variance of a probability distribution.
For every calculation, you get a full step-by-step solution, a deviation table, and a chart so you can see exactly how the answer was found. You can also copy or download your results. This calculator makes finding variance quick and simple.
How to Use Our Variance Calculator
Enter your numbers and this calculator will find the variance, standard deviation, mean, and sum of squares. It also shows a step-by-step solution and a deviation table so you can see exactly how each result is found.
Choose a tab. Pick "Standard Variance" if you have a list of numbers. Pick "Discrete Random Variable" if each number has its own probability.
Standard Variance
Select the variance type. Choose "Sample" if your data is a subset of a larger group. Choose "Population" if your data includes every value in the group. Sample divides by n−1, and population divides by N.
Enter your data. Type or paste your numbers into the text box. You can separate them with commas, spaces, or new lines. The calculator will count how many valid numbers it finds.
Click "Calculate." The tool will show your variance, standard deviation, mean, a deviation table, a bar chart, and a full step-by-step breakdown. You can switch between sample and population results at any time using the toggle above the results.
Copy or download your results. Use the "Copy Results" button to copy key values to your clipboard, or click "Download Results" to save them as a text file.
Discrete Random Variable
Enter each probability. In the P(X) column, type the probability for each outcome. Each value must be between 0 and 1, and all probabilities should add up to 1.
Enter each data value. In the X column, type the matching outcome value for that probability.
Add or remove rows. Click "Add Row" to enter more outcomes. Click the trash icon to remove a row you no longer need.
Click "Calculate." The tool will display the expected value (μ), variance (σ²), standard deviation (σ), a computation table, and a step-by-step solution.
What Is Variance?
Variance measures how far a set of numbers is spread out from their average (mean). A small variance means the numbers are close together. A large variance means they are spread far apart. It is one of the most important ideas in statistics.
How Variance Is Calculated
To find variance, you follow these steps:
- Find the mean (average) of all your numbers.
- Subtract the mean from each number. This gives you the deviation for each value.
- Square each deviation (multiply it by itself). This removes negative signs and gives more weight to values far from the mean.
- Add up all the squared deviations. This total is called the sum of squares.
- Divide the sum of squares by the count of values. This gives you the variance.
Sample Variance vs. Population Variance
There are two types of variance. Population variance (σ²) is used when your data includes every member of the group you are studying. You divide by N, the total number of values. Sample variance (s²) is used when your data is only a portion of a larger group. You divide by n − 1 instead of n. This adjustment, called Bessel's correction, makes the result more accurate when working with a sample.
Standard Deviation
Standard deviation is the square root of the variance. It tells you the same thing as variance (how spread out the data is), but in the same units as the original data. For example, if your data is in pounds, the standard deviation is also in pounds, while the variance would be in "pounds squared." This makes standard deviation easier to interpret in everyday use.
Variance of a Discrete Random Variable
A discrete random variable is a value that can only take specific outcomes, each with a known probability. For example, the result of rolling a die. To find its variance, you first calculate the expected value (μ) by multiplying each outcome by its probability and adding the results. Then, for each outcome, you find the squared distance from the expected value, multiply it by the probability, and add everything up. This gives you the variance of the distribution.
Why Variance Matters
Variance helps you understand risk, consistency, and reliability. In school, it shows how spread out test scores are. In finance, it measures how much a stock price changes. In science, it tells researchers how much their measurements vary. Any time you need to know whether data points are tightly grouped or widely scattered, variance gives you a clear answer.