Laplace Distribution Quantile / Percent Point Calculator

The Laplace distribution quantile, also called the percent point, maps a probability between zero and one to the corresponding value on a Laplace curve defined by location μ and scale b. Because the Laplace distribution is symmetric with heavier tails than the normal distribution, its quantile function uses two branches. For probabilities below one half, the quantile equals μ plus b times the natural logarithm of twice the probability. For probabilities at or above one half, it equals μ minus b times the natural logarithm of twice one minus the probability. This inverse formulation lets analysts compute thresholds, tolerance limits, prediction intervals, or simulated samples directly from probabilities, enabling robust modeling in reliability engineering, signal processing, and privacy mechanisms applications.

For Laplace(μ, b) with b > 0, the quantile Q(p) for 0 < p < 1 is

Q(p) = {
  μ + b · ln(2p),              for 0 < p < 0.5
  μ − b · ln(2(1 − p)),        for 0.5 ≤ p < 1
}

Derived by inverting the CDF F(x) = 0.5·exp((x−μ)/b) for x < μ and F(x) = 1 − 0.5·exp(−(x−μ)/b) for x ≥ μ.

• Use multiple p values to compute central intervals quickly.
• Set μ to your central tendency and b to your dispersion estimate.
• Choose decimals to control rounding in the table and CSV.