Discrete Probability Distribution Standard Deviation Calculator

Calculator

Distribution label

Input type

Decimal places

Outcome values

Use commas, spaces, or one value per line.

Probabilities, frequencies, or weights

Keep the same order as the outcome values.

Normalize probability inputs when total is not 1

Show cumulative probability column

Example Data Table

Outcome x	Probability p	Meaning
0	0.10	Lowest possible outcome
1	0.20	Below average outcome
2	0.40	Most likely outcome
3	0.20	Above average outcome
4	0.10	Highest possible outcome

Formula Used

The calculator uses the standard population formula for a discrete probability distribution.

Mean: μ = Σ xᵢpᵢ

Expected square: E(X²) = Σ xᵢ²pᵢ

Variance: σ² = Σ pᵢ(xᵢ − μ)²

Alternative variance check: σ² = E(X²) − μ²

Standard deviation: σ = √σ²

How to Use This Calculator

Enter each possible outcome in the outcome box.
Enter the matching probability, frequency, or weight.
Select the correct input type.
Enable normalization if your values need scaling.
Choose decimal places for the final display.
Press Calculate to show the result below the header.
Use CSV or PDF buttons to export the report.

Discrete Distribution Standard Deviation Guide

What It Measures

A discrete probability distribution lists possible outcomes and their probabilities. Each outcome is a number. Each probability shows how likely that number is. The standard deviation describes spread around the mean. A small value means outcomes stay close. A large value means outcomes vary more.

Why This Calculator Helps

This calculator is built for detailed probability work. You can enter exact probabilities, weights, or frequency counts. The tool can normalize totals when needed. It also checks whether probability inputs sum to one. That helps prevent silent mistakes during homework, research, and reporting.

Calculation Method

The calculation starts with the expected value. This is the weighted average of all outcomes. Each value is multiplied by its probability. Those products are then added. Next, the tool finds the expected value of squared outcomes. It also finds each deviation from the mean. Both views help explain the same variance.

Interpreting the Output

Variance measures average squared distance from the mean. Standard deviation is the square root of variance. Because it returns to the original unit, it is easier to interpret. For example, a distribution measured in dollars gives a standard deviation in dollars.

Reviewing Row Details

Use the detailed table to audit every row. It shows probability, contribution to the mean, squared contribution, deviation, and variance contribution. The cumulative probability column can help with distribution checks. The summary also includes range and coefficient of variation when possible.

Common Uses

This page is useful for statistics classes, decision models, risk tables, and quality checks. It supports negative outcomes, decimal outcomes, and uneven probabilities. It also works when outcomes represent scores, returns, claims, defects, or demand levels.

Best Practice

Clean inputs produce reliable results. Keep values and probabilities in the same order. Put one item per line or separate items with commas. Review warnings before exporting. Then download the CSV or PDF report for records, lessons, or client notes.

Final Check

Before using any answer, consider the setting. A game model may need exact probabilities. A survey table may need frequencies. A forecast may use estimated weights. These sources are valid when the assumptions are clear. The calculator does not replace statistical judgment. It gives transparent arithmetic, repeatable output, and organized evidence. That makes review faster and reduces manual spreadsheet errors. Save your inputs with labels when comparing several distributions side by side.

FAQs

What is a discrete probability distribution?

It is a list of possible numeric outcomes and their probabilities. The probabilities describe how likely each outcome is. Their total should be one after any needed normalization.

What does standard deviation show?

Standard deviation shows how far outcomes usually spread from the mean. A lower value means outcomes are closer to the expected value. A higher value means more variation.

Can I use frequencies instead of probabilities?

Yes. Select the frequency or weight option. The calculator divides each entry by the total, then uses those normalized probabilities in the standard deviation formula.

Should probabilities always add to one?

Yes, true probabilities should add to one. If your entries are proportional weights, enable normalization. The calculator will scale them before calculating mean, variance, and standard deviation.

Can outcomes be negative?

Yes. Discrete outcomes can be negative, zero, or positive. Only probabilities, frequencies, and weights must not be negative.

What is variance?

Variance is the weighted average of squared deviations from the mean. Standard deviation is the square root of variance, so it returns the spread to the original unit.

Why is E(X²) shown?

E(X²) is useful for checking variance with the identity σ² = E(X²) − μ². It also helps students see another common probability method.

What do the export buttons do?

The CSV button downloads the full numeric table. The PDF button downloads a compact report with the main summary and row details for review or sharing.