Pareto Distribution Quantile Percent Point Calculator

Inputs

Formula and Domain

For Type I Pareto with shape α > 0 and scale x_m > 0:

Q(p) = x_m * (1 - p)^(-1/α), 0 ≤ p < 1

At p = 1 the quantile diverges to infinity.
Smaller α implies heavier tails and larger high-quantile values.
x_m is the minimum feasible value and sets the scale.

What Is Pareto Distribution Quantile?

The Pareto distribution quantile also called the percent point maps a probability to a threshold x such that only a chosen tail mass lies above it. For the standard Type I Pareto with scale xm and shape alpha the quantile function expresses x as xm multiplied by one minus p raised to the power negative one over alpha for probabilities between zero and one. This inverse cumulative formula converts intuitive percentile targets into tangible cutoffs used in risk limits capacity planning and extreme value screening. Because Pareto tails decay as a power law small increases in p can create very large jumps in the computed threshold especially when alpha is near one. Practitioners use quantiles to answer questions like the ninety fifth percentile loss the required stock level that protects a service level or the size threshold that classifies rare events. Sound workflows validate p below one keep alpha positive and ensure xm positive. Clear units on inputs preserve meaningful results. Interpret results alongside empirical data tails choose parameters responsibly and document assumptions for transparent communication and reproducible decision making.

FAQs

It uses the Type I Pareto inverse cumulative function Q(p) = x_m(1 − p)^−1/α with α > 0, x_m > 0, and 0 ≤ p < 1.

The shape α must be positive, the scale x_m must be positive, and the probability must be in [0, 1). At p = 1 the quantile diverges.

Smaller α indicates heavier tails and larger extreme quantiles. Larger α produces lighter tails and smaller high-percentile thresholds.

x_m is the minimum feasible value. It sets the overall size of the quantile output and should match the units of your data.

Pareto tails follow a power law. As p approaches one, the inverse power causes Q(p) to grow without bound.

Yes. Use the batch box and separate values with spaces, commas, or semicolons. Each value is validated independently.

This page implements the Type I Pareto percent point function. For shifted or generalized forms you must transform inputs to the Type I scale.

Inputs

Formula and Domain

What Is Pareto Distribution Quantile?

FAQs

1) What formula does the calculator use?

2) What ranges are valid for the inputs?

3) How should I interpret the shape parameter?

4) What does the scale parameter represent?

5) Why does the quantile blow up near p = 1?

6) Can I paste a list of probabilities?

7) Does this handle other Pareto variants?

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