Model repeated experiments using probabilities, samples, and tolerance targets. Measure convergence speed and error bounds. See averages stabilize as trial counts rise over time.
Expected value: μ = Σ[xᵢ · pᵢ]
Second moment: E[X²] = Σ[xᵢ² · pᵢ]
Variance: σ² = E[X²] − μ²
Standard error of the sample mean: SE(X̄) = σ / √n
Chebyshev lower bound: P(|X̄ − μ| < ε) ≥ 1 − σ² / (nε²)
Hoeffding lower bound for bounded outcomes: P(|X̄ − μ| < ε) ≥ 1 − 2e−2nε²/(b−a)²
The law of large numbers states that the sample mean converges toward the expected value as the number of independent observations increases.
Scenario: fair coin, p = 0.5, ε = 0.1, and bounded outcomes from 0 to 1.
| n | Expected value μ | Variance σ² | SE(X̄) | Chebyshev within ε | Hoeffding within ε |
|---|---|---|---|---|---|
| 10 | 0.5 | 0.25 | 0.1581 | 0.00% | 0.00% |
| 50 | 0.5 | 0.25 | 0.0707 | 50.00% | 26.42% |
| 100 | 0.5 | 0.25 | 0.0500 | 75.00% | 72.93% |
| 500 | 0.5 | 0.25 | 0.0224 | 95.00% | 99.99% |
It estimates how the sample mean approaches the theoretical mean. It also reports variance, standard error, probability bounds, and sample sizes needed for a chosen tolerance target.
They provide conservative guarantees for how close the sample mean can be to the expected value. Hoeffding is usually tighter when outcomes are bounded inside a known range.
Yes. Enter any discrete outcomes and matching probabilities. The calculator then builds the mean, second moment, variance, and convergence bounds from your supplied distribution.
The calculator automatically normalizes positive probability values to sum exactly to one. It also displays a note so you know your original probability totals were adjusted.
Observed samples let you compare an actual empirical mean with the theoretical expectation. The page also shows checkpoint running means to reveal how the average settles as more observations accumulate.
For independent observations with finite variance, the standard error shrinks as n grows. That usually improves concentration around the true mean, although any single sample can still fluctuate.
It is mainly an analytical calculator. You can paste your own sample observations, but the main outputs come from probability formulas and concentration bounds rather than random simulation.
Use Hoeffding when outcomes stay within a fixed minimum and maximum. Use Chebyshev when you only know the variance and need a broad, assumption-light probability guarantee.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.