Calculator inputs
This page uses a single-column page layout. The form itself becomes three columns on large screens, two on smaller screens, and one on mobile.
Formula used
For a Pareto Type I distribution, the density is:
f(x) = alpha * x_m^alpha / x^(alpha + 1), for x >= x_m.
Random sample generation
This calculator uses inverse transform sampling: X = x_m / (1 - U)^(1 / alpha), where U is a uniform random number between 0 and 1.
Main summary formulas
Sample mean: x̄ = (Σx_i) / n
Sample variance: s² = Σ(x_i - x̄)² / (n - 1)
Confidence interval: x̄ ± z × (s / √n)
Theoretical mean: alpha * x_m / (alpha - 1), valid when alpha > 1
Theoretical variance: alpha * x_m² / ((alpha - 1)² * (alpha - 2)), valid when alpha > 2
Theoretical median: x_m * 2^(1 / alpha)
Theoretical percentile: Q(p) = x_m / (1 - p)^(1 / alpha)
How to use this calculator
- Enter the Pareto minimum value x_m.
- Enter the shape parameter alpha.
- Set the sample size you want to generate.
- Choose a confidence level for the sample mean interval.
- Set a target percentile, such as 0.90 or 0.95.
- Adjust decimal display and histogram bins if needed.
- Use an integer seed to reproduce the same sample later.
- Press Generate Summary to see the results above the form.
Example data table
Example sample values for a Pareto distribution with x_m = 5 and alpha = 2.8.
| Observation | Value | Observation | Value |
|---|---|---|---|
| 1 | 5.3120 | 6 | 6.7728 |
| 2 | 5.6844 | 7 | 5.1431 |
| 3 | 8.2016 | 8 | 9.4105 |
| 4 | 5.9173 | 9 | 6.0242 |
| 5 | 7.4860 | 10 | 5.5578 |
FAQs
1. What does alpha control in a Pareto distribution?
Alpha controls tail heaviness. Smaller alpha values create heavier tails and more extreme large observations. Larger alpha values reduce extreme outcomes and pull the sample closer to x_m.
2. What is x_m in this calculator?
x_m is the minimum possible value for the distribution. Every generated sample value is at least this large, so it acts like the scale threshold of the Pareto model.
3. Why can the theoretical mean or variance become undefined?
For Pareto Type I, the mean exists only when alpha is greater than 1. The variance exists only when alpha is greater than 2. Heavy tails make these moments diverge below those thresholds.
4. How is the random sample generated here?
The calculator uses inverse transform sampling. It first generates uniform random values, then converts them into Pareto values using the inverse cumulative distribution formula.
5. What does the confidence interval represent?
It gives an estimated range for the sample mean based on the selected confidence level. It is a normal-approximation interval using the sample standard error and a z critical value.
6. Why is the histogram often right-skewed?
Pareto data naturally has a long right tail. Many observations stay close to x_m, while a smaller number become much larger, stretching the graph to the right.
7. Can I reproduce the exact same sample later?
Yes. Enter the same x_m, alpha, sample size, and random seed. The calculator will regenerate the same pseudo-random sample and matching summary values.
8. When should I increase the sample size?
Increase sample size when you want more stable summaries, smoother histograms, and narrower mean intervals. Heavy-tailed distributions especially benefit from larger samples because extremes vary strongly.