Probability Distribution Calculator

This probability distribution calculator takes any set of outcomes and their probabilities and instantly returns the mean (expected value), variance, standard deviation, mode, median, and a cumulative probability table, plus a live bar chart of the distribution. Enter as few as 2 or as many as 20 outcomes, and the probabilities do not need to be typed in any particular order.

I have spent twelve years helping dissertation students and business analysts work through exactly this kind of problem: a table of outcomes, a table of probabilities, and a deadline. Every free calculator I tested before building this one stops at three numbers, mean, standard deviation, and variance, and calls it done.

That is only half the picture. If your assignment or your business case also asks for the mode, the median, or whether the distribution is skewed, you are stuck. You end up doing it by hand after already using a calculator for the easy part. This is a probability distribution calculator with steps built into every single result, so it computes all six numbers in one pass, with the full working shown for each.

Use it directly, or read on for exactly how each number is calculated.

What Is a Probability Distribution? (Quick Definition)

A probability distribution lists every possible outcome of a random variable, its sample space, alongside the probability of each one occurring. For a discrete random variable, this list is called a probability mass function. It must satisfy two conditions: every probability sits between 0 and 1, and all the probabilities add up to exactly 1.

This is not just calculator convention. It is the formal definition used in NIST’s own survey of probability distribution tables. That paper denotes the expected value as E(X) and the probability mass function as f(x), the same notation this calculator uses. OpenStax’s introductory statistics textbook states the same two conditions independently, so this is not a convention unique to one source.

A weighted die, a lottery payout table, an insurance claim table, a survey response breakdown. All of these are discrete probability distributions, and all of them can be solved with the same set of formulas.

How to Use This Probability Distribution Calculator

This is a discrete probability distribution calculator, meaning it works with a finite, countable list of outcomes rather than a continuous range. If you searched for a probability distribution calculator with graph specifically, that is exactly what you get here, a live bar chart alongside every result, not just a static number.

Step-by-Step Input Guide

1. Enter each outcome’s value. This can be a dice face, a payout amount, a survey score, anything numeric.
2. Enter the probability for that outcome. Every probability must sit between 0 and 1.
3. Add or remove rows as needed. Start with 4 rows, add up to 20, or paste a full list at once using the “Paste multiple outcomes” option below the table.
4. Watch the running total. The calculator shows your total probability as you type, so you catch a rounding mistake before you hit Solve, not after.
5. Click Solve. You get the mean, variance, standard deviation, mode, median, a cumulative probability table, and the bar chart, all at once.

Reading Your Results

The summary cards at the top give you the six headline numbers. The bar chart below them plots each outcome’s probability as a bar, so you can see at a glance whether the distribution is symmetric, skewed, or has one dominant outcome. The cumulative table underneath builds up P(X ≤ x) row by row. This is the discrete version of a cumulative distribution function (CDF), and it is the exact number you need for percentile-style questions on a discrete variable.

Some people type “probability distribution table calculator” instead of “probability distribution calculator” into Google. Same tool, same result, the table is simply how the outcomes get entered.

Expected Value Formula Explained

Is Expected Value the Same as Mean?

Yes, for a probability distribution, expected value and mean are the same number, just two different names taught in different courses. Statisticians usually say mean. Finance and decision-analysis courses usually say expected value, and gambling or actuarial contexts almost always say expected value. This calculator uses both terms so it turns up regardless of which one you searched.

How Do You Find Expected Value? (E(X) = Σx·P(x))

Multiply each outcome by its own probability, then add up every product. This is the exact task a dedicated “expected value calculator discrete random variable” tool is built to do, and it is genuinely the same formula no matter which term you use, expected value or mean:

E(X) = μ = Σ x·P(x)

Expected value is really a probability weighted average calculator in disguise, every outcome weighted by how likely it is, then summed. If you are looking for an expected value calculator specifically rather than a general probability distribution calculator, this is the exact formula it runs, there is no separate tool needed for that framing. I also see this searched as “how to find expected value calculator”, an odd phrasing, but it lands you in the right place either way. Some people specifically search for a mean of a probability distribution calculator or a discrete random variable calculator, expecting something narrower, but both phrasings describe exactly this tool.

Variance and Standard Deviation Formulas

Once you have the mean, variance measures how far outcomes typically sit from it. The direct formula sums the squared distance of every outcome from the mean, weighted by probability:

σ² = Σ (x − μ)² · P(x)

The Shortcut Formula: E(X²) − μ²

Most textbooks also show a shortcut that avoids computing every squared deviation by hand:

σ² = E(X²) − μ²

This calculator computes both E(X²) and the direct sum internally and shows you the shortcut version in the step-by-step output, since it is faster to verify by hand. Standard deviation is simply the square root of variance, σ = √σ². It is the number you actually quote when someone asks how “spread out” a distribution is, because it is back in the original units rather than squared units. (The standalone standard deviation calculator runs the same formula in reverse if you are starting from raw data rather than a probability table.)

If you only need the variance of a probability distribution calculator without the mean, mode, and median extras, the shortcut formula above is the exact part you are looking for. Some people also search for a mean and standard deviation of a probability distribution calculator specifically, expecting a narrower tool, but this covers exactly that plus four more statistics most versions leave out entirely.

Mode, Median, and Skewness (What Most Calculators Skip)

I checked a dozen free tools in this space before building this one, and not one of them computes mode, median, or skewness. That is a real gap, because assignments and case studies ask for all three regularly, and a mean/variance/SD calculator alone leaves you finishing the job by hand.

Mode is simply the outcome with the highest probability, the single most likely result if you ran the experiment once. Median is the smallest outcome value where the cumulative probability first reaches 0.5 or higher, reading straight off the cumulative table this calculator already builds for you. Skewness tells you whether the distribution leans left, leans right, or sits roughly symmetric, and this calculator flags which one automatically once the number comes out.

If you need mean, median, and mode for a raw dataset instead of a probability table, the measures of central tendency calculator covers that same trio for ordinary data.

Worked Example: Expected Value of a Weighted Die

Here is the exact worked example most calculators in this space use, so you can cross-check this tool’s output against anything else you have already tried. Plenty of people land here searching for a dice probability calculator specifically, and a weighted die is exactly the kind of problem this calculator was built to solve cleanly. (This example is illustrative, matching the standard textbook version of this problem, not a specific individual’s dataset.)

A weighted six-sided die has these probabilities: 1 → 0.10, 2 → 0.15, 3 → 0.20, 4 → 0.25, 5 → 0.20, 6 → 0.10.

1. E(X) = (1)(0.10) + (2)(0.15) + (3)(0.20) + (4)(0.25) + (5)(0.20) + (6)(0.10) = 3.6
2. E(X²) = (1)(0.10) + (4)(0.15) + (9)(0.20) + (16)(0.25) + (25)(0.20) + (36)(0.10) = 14.8
3. σ² = 14.8 − (3.6)² = 2.14
4. σ = √2.14 ≈ 1.4629

The expected roll is 3.6, and the mode is 4, since 0.25 is the single highest probability in the table. That gap between the mean (3.6) and the mode (4) is exactly the kind of detail a mean-only calculator would never surface for you.

For comparison, an ordinary fair die, where each face has an equal 1/6 probability, has an expected value of exactly 3.5, the textbook baseline most students learn before ever seeing a weighted version. The weighted die above shifts that expected value up to 3.6 purely because the higher faces were made more likely.

Real-World Example: Expected Monetary Value in Decision-Making

This section doubles as an expected monetary value calculator walkthrough for exactly this kind of business decision. An underwriting trainee I mentored a while back was pricing an extended warranty product for a mid-range appliance. (This example is a composite drawn from patterns I see repeatedly in decision-analysis coursework, not one identifiable client’s exact figures.)

The claims data suggested a 70% chance of zero claims, a 20% chance of a $150 repair claim, an 8% chance of a $400 claim, and a 2% chance of a full $900 replacement.

1. E(X) = (0)(0.70) + (150)(0.20) + (400)(0.08) + (900)(0.02) = $80
2. The expected payout per policy is $80, so any warranty priced meaningfully below that figure is losing money on average, regardless of how any single policy actually plays out.

This is expected monetary value, the exact same formula as the die example, just with a dollar sign attached and a real decision riding on the answer.

Common Mistakes (Probabilities Not Summing to 1, Negative Probabilities)

The single most common error is entering probabilities that do not add up to 1, usually because of a typo or a rounding choice made earlier in a calculation. This calculator flags that live as you type, and again before it lets you solve, rather than silently returning a wrong answer like some tools do.

The second common error is treating a negative outcome value as if it were an invalid input. It is not. A loss of $50 is a perfectly valid outcome value of −50, what must always stay positive is the probability, never the outcome itself. If you have a payoff table with both wins and losses, enter the losses as negative numbers exactly as they are and let the calculator handle the arithmetic.

If your data is genuinely continuous rather than a fixed list of outcomes, for example test scores or measurement error, this tool is not the right one. Use the Normal Distribution Calculator instead, since continuous distributions need integration, not summation, to find their mean and variance.

Frequently Asked Questions

What is a probability distribution calculator used for? It is used to find the mean, variance, standard deviation, and other summary statistics of a random variable when you already know every possible outcome and its probability, common in games of chance, insurance pricing, and decision analysis.

Is expected value the same as mean? Yes. Expected value and mean describe the same number for a probability distribution, expected value is simply the term more common in finance, gambling, and decision-analysis contexts.

Can probabilities in a distribution be negative? No, probabilities must always be between 0 and 1. The outcome values themselves can be negative, for example a financial loss, but the probability of that outcome occurring cannot be.

What happens if my probabilities don’t add up to 1? The distribution is invalid and the calculator will not solve until this is fixed. A small rounding difference, for example totalling 0.99 instead of 1.00, usually means one probability was entered incorrectly or rounded too early somewhere upstream.

How do you find the mode of a probability distribution? The mode is the outcome value with the single highest probability attached to it. If two or more outcomes are tied for the highest probability, the distribution has more than one mode.

Can this calculator be used for continuous distributions like normal? No, this tool is built for discrete distributions with a finite, listable set of outcomes. For continuous distributions, use the Normal Distribution Calculator, which uses integration under a curve rather than summing a table of outcomes.

How is this different from a binomial or Poisson distribution calculator? Binomial and Poisson are named distributions with a fixed formula behind them, you only need to enter parameters like the number of trials or the average rate. Use the Binomial Distribution Calculator or Poisson Distribution Calculator when your data actually fits one of those patterns, and use this custom calculator when your outcomes and probabilities do not follow a standard named formula.

---

Have a payoff table or a dataset you are stuck on? Drop it into the calculator above, or reach out through Statssy if you need dissertation-level statistical support.