Advanced Mean Variance Probability Calculator

Calculator Inputs

Analysis mode

Variance method

Decimal precision

Probability query

Query value k

Lower bound a

Upper bound b

Probability normalization

Normalize probabilities when they do not total 1

Values or observations

Examples: 1 2 3 4 5 or one value per line.

Probabilities

Use this field only for distribution mode. It must align with the values list.

Input ideas

Distribution mode example:

Values: 1 2 3 4 5

Probabilities: 0.10 0.20 0.30 0.25 0.15

Sample mode example:

Values: 12 14 14 15 15 15 16 17 18

Example Data Table

This example uses a weighted discrete distribution. It demonstrates how outcome values combine with probabilities to produce the expected value and spread metrics.

Outcome x	Probability p(x)	x · p(x)	x² · p(x)
1	0.10	0.10	0.10
2	0.20	0.40	0.80
3	0.30	0.90	2.70
4	0.25	1.00	4.00
5	0.15	0.75	3.75
Total	1.00	3.15	11.35

From this table, the mean is 3.15. The variance is 11.35 − (3.15²), which equals 1.4275.

Formula Used

For discrete probability distributions

Mean: μ = Σ[xᵢ · pᵢ] Second moment: E[X²] = Σ[xᵢ² · pᵢ] Variance: Var(X) = E[X²] − μ² Standard deviation: σ = √Var(X) Event probability: P(event) = sum of probabilities for all outcomes inside the chosen event

For sample data

Mean: x̄ = Σxᵢ / n Population variance: σ² = Σ(xᵢ − x̄)² / n Sample variance: s² = Σ(xᵢ − x̄)² / (n − 1) Empirical probability: count of matching observations / n

How to Use This Calculator

Choose Discrete probability distribution when each value has a matching probability.
Choose Sample data when you only have observations and want empirical statistics.
Paste values into the values box using spaces, commas, or new lines.
In distribution mode, enter the probability list in the same order as the values.
Pick a probability query such as exact, less than or equal, greater than or equal, or between.
Select the variance method and result precision that fit your analysis.
Press Calculate Now to show results above the form.
Use the export buttons to download a CSV file or a PDF report.

FAQs

1. What does this calculator measure?

It measures central tendency, spread, and event likelihood. You can compute mean, variance, standard deviation, and selected probabilities from either raw observations or a discrete probability distribution.

2. When should I use distribution mode?

Use distribution mode when every outcome already has an assigned probability. This is common in modeled scenarios, risk tables, game outcomes, and any finite random variable with known weights.

3. When should I use sample mode?

Use sample mode when you have observed data points rather than predefined probabilities. The calculator estimates relative probabilities from frequencies and then computes statistics from the observed sample.

4. What is the difference between sample and population variance?

Population variance divides by n and assumes your dataset is the full population. Sample variance divides by n − 1 and is better when your data is a sample drawn from a larger population.

5. Why might normalization be useful?

Normalization rescales the probability list so the total becomes 1. It is useful when rounding or manual entry causes a small mismatch, but it should not replace clearly incorrect data.

6. Can I calculate range probabilities?

Yes. Choose the range query option and enter a lower and upper bound. The tool sums all included probabilities or counts matching observations to estimate the requested event probability.

7. What does the chart show?

The bar trace shows either distribution probabilities or empirical relative frequencies. The line trace shows cumulative probability, and the vertical marker shows the mean for quick visual interpretation.

8. Can I export the results?

Yes. After a successful calculation, use the CSV button for spreadsheet-friendly output or the PDF button for a clean report containing the summary table and detailed probability table.