Calculator Input
Enter matching values and probabilities, percentages, or frequencies. The form uses a 3-column large-screen layout, 2-column medium layout, and 1-column mobile layout.
Example Data Table
This example uses a fair die distribution. The standard deviation is approximately 1.7078.
| X | P(X) | X × P(X) | P(X)(X - μ)² |
|---|---|---|---|
| 1 | 0.1667 | 0.1667 | 1.0417 |
| 2 | 0.1667 | 0.3333 | 0.3750 |
| 3 | 0.1667 | 0.5000 | 0.0417 |
| 4 | 0.1667 | 0.6667 | 0.0417 |
| 5 | 0.1667 | 0.8333 | 0.3750 |
| 6 | 0.1667 | 1.0000 | 1.0417 |
| Total | 1.0000 | 3.5000 | 2.9168 |
Formula Used
Normalization: pi = wi / Σwi
Mean: μ = Σ[xi × pi]
Expected Value of X²: E[X²] = Σ[xi2 × pi]
Variance: σ² = Σ[pi(xi - μ)2] = E[X²] - μ²
Standard Deviation: σ = √σ²
Mean Absolute Deviation: MAD = Σ[pi|xi - μ|]
Coefficient of Variation: CV = (σ / |μ|) × 100
Skewness: Σ[pi((xi - μ) / σ)3], when σ > 0
How to Use This Calculator
- Enter the distribution label if you want a named output.
- Choose whether your second list contains probabilities, percentages, or frequencies.
- Paste all possible X values in the values box.
- Paste the matching probabilities, percentages, or frequencies in the next box.
- Keep auto-normalize enabled unless your totals already meet the exact rule.
- Select decimal places and your preferred chart view.
- Press the calculate button to show results above the form.
- Use the CSV or PDF buttons to export the result summary and table.
Frequently Asked Questions
1. What does this calculator measure?
It measures the spread of a discrete distribution. It also reports the mean, variance, expected value of X², mean absolute deviation, coefficient of variation, and skewness.
2. Can I enter percentages instead of probabilities?
Yes. Choose the percentages option. You may also keep auto-normalize enabled if your percentages do not total exactly 100 because of rounding.
3. Can I use frequencies from observed outcomes?
Yes. Select frequencies. The calculator converts each frequency into a probability by dividing it by the total frequency.
4. What happens if my probabilities do not sum to 1?
If auto-normalize is on, the calculator rescales them to a valid probability distribution. If it is off, the calculator asks for an exact total of 1.
5. What is the difference between variance and standard deviation?
Variance measures average squared spread around the mean. Standard deviation is the square root of variance, so it returns spread in the same unit as X.
6. Why are z-scores included in the table?
Z-scores show how many standard deviations each X value sits above or below the mean. They help compare outcomes on a standardized scale.
7. When is coefficient of variation useful?
It is useful when you want relative spread rather than absolute spread. It compares variability to the size of the mean as a percentage.
8. Does this work for continuous distributions?
This page is built for discrete distributions listed as separate outcomes. Continuous distributions usually require integration or formula-based parameters instead.