Probability Variance Calculator

Enter Discrete Probability Data

Use one row per outcome. Format each row as outcome, probability, optional label.

Outcome and Probability Data 0,0.10 1,0.20 2,0.35 3,0.25 4,0.10

Decimal Precision Target Outcome for Probability Split Probability Review Mode Validate total probability Weighted distribution review

Normalize probabilities when total is not 1

Example row formats:

• 10, 0.25
• 20, 0.40, Medium return
• 30, 0.35, High return

Calculate Probability Variance Reset

Formula Used

Expected value: μ = Σ xp

Second moment: E(X²) = Σ x²p

Probability variance: Var(X) = Σ p(x - μ)²

Shortcut variance: Var(X) = E(X²) - μ²

Standard deviation: σ = √Var(X)

Coefficient of variation: CV = σ / |μ| × 100

The calculator uses the distribution variance formula. It is suitable when each outcome has an assigned probability.

How to Use This Calculator

1. Enter each outcome on a separate line.
2. Add its probability after a comma.
3. Add an optional label after another comma.
4. Enable normalization if probabilities are proportional weights.
5. Set decimal precision for final tables.
6. Enter a target value to compare lower, equal, and higher probability.
7. Press the calculate button.
8. Download the result table as CSV or PDF.

Probability Variance Guide

What Probability Variance Means

Probability variance measures spread in a discrete random variable. It shows how far outcomes tend to move from the expected value. A small variance means outcomes cluster near the mean. A large variance means outcomes are more scattered. This makes variance useful for risk, quality checks, classroom examples, games, surveys, and forecasting.

Why the Expected Value Comes First

The expected value is the weighted average of all possible outcomes. Each outcome is multiplied by its probability. The products are then added together. This mean is not always an actual outcome. It is the long-run center of the distribution. Variance needs this center before spread can be measured.

How This Tool Handles Data

This calculator accepts outcome and probability pairs. You can enter probabilities that already sum to one. You can also enter proportional weights and normalize them. Normalization divides every weight by the total. The adjusted probabilities then sum to one. This is helpful when data comes from counts, scores, or estimated weights.

Advanced Statistical Outputs

The result includes variance, standard deviation, second moment, mean absolute deviation, skewness, kurtosis, entropy, and coefficient of variation. Standard deviation keeps the same unit as the outcome. Skewness describes direction of imbalance. Kurtosis describes tail weight. Entropy summarizes uncertainty in the probability pattern.

Interpreting the Result

Compare variance values only when outcome units are similar. Use standard deviation when you need an easier scale. Use coefficient of variation when means differ and relative spread matters. Check the detailed table for each row contribution. Large weighted squared deviations often explain most of the variance.

Practical Uses

In finance, variance can compare uncertain returns. In education, it can analyze score distributions. In operations, it can review defect counts or demand levels. In probability lessons, it shows how formulas connect. The CSV and PDF exports make the result easier to save, share, and audit.