Probability Distribution Calculator
Build a discrete probability distribution table with x and p values. Get E of X, variance, and standard deviation once probabilities sum to 1.
Enter the Details
Use our probability distribution calculator to find the mean, standard deviation, and variance of a
distribution by entering each value x and its probability P(x) below. Plus, see the steps to solve below.
| Value | Probability |
|---|---|
| X1: | p(X1): |
| X2: | p(X2): |
| X3: | p(X3): |
| X4: | p(X4): |
| X5: | p(X5): |
| X6: | p(X6): |
| X7: | p(X7): |
| X8: | p(X8): |
| X9: | p(X9): |
| X10: | p(X10): |
Result will appear here...
What the probability distribution calculator does
A probability distribution lists the values a random quantity can take and how likely each one is. This calculator summarises such a distribution with three numbers: its mean, its variance, and its standard deviation. You enter each value with its probability, and it does the rest.
Where the expected value gives the centre of a distribution, these three together describe both its centre and its spread. Below is how they are worked out and what they tell you.
How to use it
- Enter each value in the table with its probability beside it, as a decimal.
- Fill in a row for every possible value, making sure the probabilities add up to 1.
- Press Calculate for the mean, variance, and standard deviation, or Reset to clear it.
What a probability distribution is
A probability distribution is the full picture of a random quantity: every value it might take, paired with the probability of that value. The roll of a die, the number of heads in three coin flips, the payout of a game, each has a distribution that lays out its possibilities and their weights.
One rule always holds: the probabilities must add up to 1, because between them they have to account for everything that can happen. That is why the calculator checks the total, and it is the sign of a properly formed distribution.
How the mean, variance, and SD are worked out
The mean is the expected value: each value times its probability, added up. It is the balance point of the distribution. The variance measures spread, worked out as the probability-weighted average of the squared values minus the square of the mean:
Variance = the sum of (value² × its probability) minus the mean²
The standard deviation is the square root of the variance, which brings the spread back into the same units as the values. Together they say where the distribution sits and how widely it is spread around that centre.
Reading the three numbers
The mean tells you the long-run average value, the figure the outcomes cluster around over many repetitions. The variance and standard deviation tell you how tightly they cluster. A small standard deviation means the values stay close to the mean, and a large one means they range widely.
The standard deviation is usually the easier of the two to interpret, since it is in the same units as the values themselves, while the variance is in squared units. Reading them together gives the same two-part summary, centre and spread, that describes any set of numbers, but here applied to the outcomes of a random quantity.
A worked example
Take a fair six-sided die, with values 1 through 6 each having a probability of one sixth. The mean is the expected value, (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5.
For the variance, average the squared values by probability, (1 + 4 + 9 + 16 + 25 + 36) ÷ 6 = 15.1667, and subtract the square of the mean, 3.5² = 12.25, giving a variance of about 2.92. The standard deviation is its square root, about 1.71. So a die roll averages 3.5, with results typically spread about 1.71 either side of that.
Entering your values
Enter each value with its probability as a decimal in the table, up to ten rows. The probabilities must add up to 1, since they cover every possible value. The mean, variance, and standard deviation are returned together, giving the centre and spread of the distribution in one go.
Questions people ask
What is a probability distribution?
A list of the values a random quantity can take, each paired with its probability. The probabilities always add up to 1.
Is the mean the same as the expected value?
Yes. The mean of a probability distribution is its expected value, each value times its probability, added up.
What do the variance and standard deviation tell me?
How spread out the outcomes are around the mean. A small standard deviation means they stay close to it; a large one means they range widely. The standard deviation is in the same units as the values.
Why must the probabilities sum to 1?
Because they must account for every possible value. If they do not add up to 1, the distribution is incomplete or mismeasured.
References
A quick note on where the methods here come from. The mean, variance, and standard deviation of a discrete probability distribution are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering probability distributions.
- NIST/SEMATECH e-Handbook of Statistical Methods (discrete distributions and their moments). https://www.itl.nist.gov/div898/handbook/
- OpenStax, Introductory Statistics (probability distributions, mean, and standard deviation). https://openstax.org/details/books/introductory-statistics-2e
Ankit Khatiwada is a researcher and graduate student in Computer Science at Saarland University, with strengths in statistics, data analysis, data engineering, and full stack development. His work sits at the intersection of quantitative reasoning and applied technology, making him a strong fit for tools that depend on clear numerical logic. At Eon Tools, he reviews number and statistical tools.