Understanding Discrete Variance
A discrete random variable lists possible outcomes and their probabilities. Each probability describes how likely one outcome is. Variance measures how far those outcomes usually sit from the expected value. A small variance means the values stay close to the mean. A large variance means the distribution spreads out more.
Why This Calculator Helps
Manual variance work can become repetitive. You must multiply each outcome by its probability. You must also square each outcome, weight it, and compare it with the mean. This calculator organizes those steps in a clear table. It checks probability totals, supports decimals, percentages, and fractions, and can normalize values when needed.
Using Probability Inputs
Enter one outcome and one matching probability on each line. You can also use separate outcome and probability lists. Probabilities should describe the full distribution. Their total should equal one. When the total is slightly different, normalization can scale the probabilities to a valid distribution. This is useful when values are rounded.
Reading The Results
The expected value is the long run average. The second moment is the probability weighted average of squared outcomes. Variance equals the second moment minus the square of the expected value. Standard deviation is the square root of variance. It uses the same unit as the original variable.
Practical Uses
Discrete variance appears in games, surveys, quality checks, insurance, finance, and classroom probability. It helps compare risk between two distributions. Two variables can have the same mean but different spreads. The variable with the larger variance is less predictable.
Best Practices
Use exact probabilities when possible. Keep every outcome listed once. Do not mix percentages and decimals unless the selected format matches your entries. Review the probability sum before trusting the final answer. Export the table when you need a record for homework, reports, or further analysis.
Common Mistakes
A frequent mistake is forgetting an outcome with a small probability. Another mistake is using raw frequencies without converting them. Frequencies can work only after division by the total count. Negative probabilities are not valid. Rounded probabilities may create tiny total errors. The calculator flags these issues, so the distribution stays easier to audit before any final interpretation or export choice and careful review.