Overview
A discrete probability distribution lists possible values and the probability attached to each value. The mean is also called the expected value. It tells you the long run average outcome if the same random process repeats many times. This calculator is made for study, quality checks, risk tables, games, surveys, and business cases where outcomes are separate values.
Why the Mean Matters
The mean does not always equal a possible outcome. It is a balance point for the distribution. A lottery may have a mean prize of 2.40, even when no ticket pays exactly 2.40. That value still helps compare choices because it combines every payout with its chance. When probabilities are correct, the mean gives a clear center for future expectations.
What the Calculator Checks
A valid probability distribution has probabilities that are not negative. Their total should be one. The tool shows the total probability so mistakes are easier to find. You can use strict probabilities, normalize values, or treat entries as weights. Normalizing is useful when data comes from counts or rounded percentages. The contribution table shows each x times p, which explains how the final mean is built.
Beyond the Mean
Advanced analysis often needs more than the expected value. Variance measures spread around the mean. Standard deviation is the square root of variance. A larger deviation means outcomes are more scattered. The calculator also returns E(X²), cumulative probability, the highest probability outcome, and the range. These values help you read both center and risk.
Good Input Practice
Enter one outcome per row. Use decimals for probabilities, such as 0.25. If you have percentages, convert 25% to 0.25 or use weight mode with 25 as the entry. Keep enough decimal places to avoid rounding errors. Review the probability total before using the result in reports. Export the CSV for spreadsheets. Use the PDF option when you need a record.
Common Use Cases
Students can verify textbook tables. Analysts can compare demand scenarios. Teachers can prepare examples for class. Businesses can estimate average claims, defects, returns, or revenue bands. Any case with separate outcomes and known chances can use this method, as long as the probabilities describe the full sample space.