Result
Formula substitution and steps
Illustrative distribution view
The curve is illustrative. Chebyshev's inequality requires no normal distribution.
Multiple k-value results
| k | Minimum inside | Maximum outside | Interval | Normal estimate |
|---|
Calculator inputs
Choose a calculation mode. Enter the available statistics and interval values.
Formula used
Chebyshev's inequality applies to distributions with finite variance. It makes no normality assumption. Its probability result is a guaranteed bound.
Two-sided form
P(|X − μ| ≥ kσ) ≤ 1/k²
P(|X − μ| < kσ) ≥ 1 − 1/k²
One-sided Cantelli form
P(X − μ ≥ a) ≤ σ²/(σ² + a²)
P(μ − X ≥ a) ≤ σ²/(σ² + a²)
How to use the calculator
- Select the calculation matching your known information.
- Enter the mean and positive dispersion value.
- Add k, confidence, bounds, or one-sided distance.
- Choose precision and optional normal comparison settings.
- Calculate, review steps, then export your result.
Example data
| Mean | Standard deviation | k | Interval | Minimum inside |
|---|---|---|---|---|
| 50 | 5 | 2 | 40 to 60 | 75% |
| 100 | 10 | 3 | 70 to 130 | 88.89% |
| 20 | 4 | 4 | 4 to 36 | 93.75% |
Calculation history
| Time | Mode | Inputs | Result |
|---|
Common k-value reference
| k | Minimum within kσ | Maximum outside kσ |
|---|---|---|
| 1.5 | 55.56% | 44.44% |
| 2 | 75% | 25% |
| 2.5 | 84% | 16% |
| 3 | 88.89% | 11.11% |
| 4 | 93.75% | 6.25% |
Interpretation and limitations
The bound describes the worst guaranteed case. Actual probabilities can be much stronger. Distribution details can provide tighter estimates.
The inequality needs a finite mean and variance. It does not identify exact tail probabilities. Small k-values often create weak bounds.
The empirical rule assumes an approximately normal distribution. Chebyshev's inequality remains distribution-free. Compare them only with that distinction.
Frequently asked questions
What does Chebyshev's inequality calculate?
It calculates guaranteed probability bounds around a distribution's mean. The result depends on standard deviation distance. It does not return exact probabilities.
Must the data follow a normal distribution?
No normal distribution is required. The distribution may be skewed or irregular. It only needs finite mean and variance.
Why should k usually exceed one?
The lower bound equals one minus one over k squared. Values at or below one are non-positive. Such bounds provide little practical information.
Can variance be entered directly?
Yes, select variance as the spread input. The calculator converts variance into standard deviation. Negative variance is rejected automatically.
What is a symmetric interval?
Its endpoints are equally distant from the mean. The mean is the interval midpoint. Chebyshev's standard form uses this symmetry.
What is Cantelli's inequality?
Cantelli's inequality gives a one-sided probability bound. It uses variance and distance from the mean. It can outperform a two-sided conversion.
Is the normal comparison guaranteed?
No, that estimate assumes a normal distribution. It is included only for comparison. Chebyshev's bound remains the guaranteed result.
Can the mean be negative?
Yes, the mean may be any finite number. Interval endpoints may also be negative. Dispersion values must remain positive.
How are results exported?
Copy creates a plain-text summary. CSV downloads structured result data. Printing supports saving the page as PDF.