Input Distribution
Advanced
When checked, enter counts in the f column. Probabilities will be computed automatically.
Controls rounding for outputs.
| # | Outcome xi | Probability pi (0–1) | Frequency fi | xi·pi | Remove |
|---|
Accepted headers: x,p or x,f. Extra columns ignored.
Results
Sum of probabilities
0
Mean E[X] = Σ xipi
0
Rows considered
0
| # | xi | pi | xi·pi | xi2 | xi2·pi |
|---|---|---|---|---|---|
| Totals | 0 | 0 | — | 0 | |
Tip: If probabilities do not sum to 1, use Normalize to scale them proportionally.
Example Data Table
| x | p | x·p |
|---|---|---|
| 0 | 0.10 | 0.000 |
| 1 | 0.20 | 0.200 |
| 2 | 0.40 | 0.800 |
| 3 | 0.30 | 0.900 |
| — | 1.00 | 1.900 |
Example distribution with mean 1.9. Use Load example to paste these values into the input grid.
Formula Used
For a discrete random variable X taking outcomes xi with probabilities pi, the mean (expected value) is:
E[X] = Σi xi · pi
- Each probability satisfies 0 ≤ pi ≤ 1 and Σ pi = 1.
- If you have frequencies fi, probabilities are pi = fi / Σ fi.
- Only rows with valid numeric entries are included in the sum.
E[X²] = Σ xi2 · pi
Var(X) = E[X²] − (E[X])², σ(X) = √Var(X)
- Equivalently, Var(X) = Σ (xi − μ)²·pi where μ = E[X].
- With frequencies fi, compute pi = fi/Σf and apply the same formulas.
How to Use This Calculator
- Choose whether you will enter probabilities or frequencies.
- Fill the table with outcomes x and either p or f values.
- Click Compute mean to calculate Σ xipi.
- If Σp ≠ 1, click Normalize to scale probabilities.
- Import values from a CSV file or export results as CSV/PDF.
CSV format: header row then values; either x,p or x,f.
Common Discrete Distributions & Mean Formulas
| Distribution | Parameters | Support | Mean E[X] | Example Parameters → Mean |
|---|---|---|---|---|
| Bernoulli | p ∈ [0,1] | {0,1} | p | p = 0.65 → E[X] = 0.65 |
| Binomial | n ∈ ℕ, p ∈ [0,1] | {0,1,…,n} | n·p | n = 20, p = 0.3 → E[X] = 6 |
| Geometric (trials until first success) | p ∈ (0,1] | {1,2,3,…} | 1 / p | p = 0.2 → E[X] = 5 |
| Poisson | λ > 0 | {0,1,2,…} | λ | λ = 3.4 → E[X] = 3.4 |
| Discrete Uniform | a ≤ b, integers | {a, a+1, …, b} | (a + b) / 2 | a = 1, b = 6 → E[X] = 3.5 |
Use these references to sanity‑check computed means for standard models.
Worked Examples (Datasets & Computed Means)
| Case | Outcomes xi | Probabilities pi | Σp | E[X] = Σ xipi |
|---|---|---|---|---|
| A | 0, 1, 2, 3 | 0.10, 0.20, 0.40, 0.30 | 1.00 | 1.90 |
| B | -1, 0, 2 | 0.25, 0.50, 0.25 | 1.00 | 0.25 |
| C | 10, 20, 30 | 0.20, 0.50, 0.30 | 1.00 | 21.00 |
Load Case A via “Load example”. Enter the others manually to verify outcomes.
Real‑World Use Cases & Interpretation
| Domain | Variable X | Example Outcomes | Unit | Mean Interpretation |
|---|---|---|---|---|
| Quality Control | Defects per batch | 0,1,2,3,… | defects | Average defects expected per inspected batch |
| Customer Support | Tickets per hour | 0,1,2,3,… | tickets/hour | Average incoming load for staffing decisions |
| Gaming | Dice payout | -1, 0, +5, … | currency | Average return per play of the game |
| Reliability | Failures per day | 0,1,2,3,… | events/day | Expected failures guiding spare‑parts planning |
| Inventory | Demand per period | 0,1,2,3,… | units | Average demand driving reorder thresholds |
The mean summarizes long‑run average behavior useful in planning and forecasting.
FAQs
It’s a variable that takes countable outcomes, each with an assigned probability, like dice rolls or defect counts.
Yes. If rounding errors occur, use the Normalize button to scale them proportionally so they total 1.
Yes. Toggle “I have frequencies” and provide counts; probabilities are computed as each count divided by the total count.
As many as needed. The grid grows dynamically and only valid numeric rows are used in calculations.
A summary with date, the table of x, p, and x·p, along with totals and the computed mean E[X].