What is variance in a discrete probability distribution?

Variance measures the spread of outcomes around the expected value. It is computed as Var(X) = sum of (x - E(X))^2 * P(x), where E(X) is the mean. A higher variance means outcomes are more spread out.

Why must probabilities sum to 1?

A valid probability distribution must cover all possible outcomes with total probability equal to 1. If your probabilities do not sum to 1, the distribution is incomplete or incorrectly specified.

What is the difference between variance and standard deviation?

Standard deviation is the square root of variance. Variance is in squared units of the original values, while standard deviation is in the same units, making it easier to interpret alongside the mean.

Can I enter negative values?

Yes. Outcome values (x) can be any real number, including negatives. Only probabilities must be between 0 and 1.

How many rows can I add?

You can add up to 20 rows, which is sufficient for most discrete distributions used in statistics courses and practical applications.

Variance Discrete

Name: Variance Discrete
Availability: InStock
Author: Nham Vu

Enter values and probabilities for a discrete distribution to instantly compute mean, variance, and standard deviation.

Distribution Table

Enter each outcome value and its probability. Probabilities must sum to 1.

Value (x) Probability P(x)

Sum of P(x): 0.0000

Fill in the table and click Calculate Variance to see results.

Results

Expected Value E(X)

—

Variance Var(X)

—

Std Deviation σ

—

Step-by-step Breakdown

x	P(x)	x·P(x)	(x−μ)²·P(x)
Total	—	—	—

E(X) = Σ x · P(x)

Var(X) = Σ (x − E(X))² · P(x)

σ = √Var(X)

Summary

Enter values and probabilities for a discrete distribution to instantly compute mean, variance, and standard deviation.

How it works

Enter each outcome value (x) and its probability P(x) in the table.
Add or remove rows as needed to match your distribution.
The tool verifies that all probabilities sum to 1.
Expected value E(X) is computed as the sum of x * P(x) for all rows.
Variance Var(X) is computed as the sum of (x - E(X))^2 * P(x).
Standard deviation is the square root of the variance.

Use cases

Solving statistics homework involving discrete probability distributions.
Checking variance calculations for dice rolls, card draws, or game outcomes.
Analyzing risk in financial models with discrete payoff scenarios.
Teaching or learning the relationship between mean, variance, and standard deviation.
Verifying manually computed distribution statistics quickly.
Exploring how changing probabilities affects spread of a distribution.

Frequently Asked Questions

Last updated: 2026-06-09 · Reviewed by Nham Vu