Interactive module guide

Laplace Distribution Calculator & Guide

This Laplace distribution calculator lets you calculate probabilities for the double-exponential curve with a sharp peak and heavier-than-normal tails.

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The Laplace distribution—also called the double exponential—glues two exponential tails back to back at a location μ, producing a density with a sharp peak and tails heavier than the normal distribution.

It is the error model under which the sample median, not the mean, is the optimal location estimate, and its negative-log density |x−μ|/b is exactly the absolute-error loss behind LASSO regularization; it is also the noise added by the Laplace mechanism in differential privacy.

This page provides its closed-form density, CDF, quantiles, and moments, including the fixed excess kurtosis of 3 that quantifies how much heavier than normal its tails are.

Is this the right module for my question?

Use it when

Use it for error distributions with occasional large deviations around a sharp center: forecast errors, financial returns at daily horizons, and sensor noise with impulsive components all show the peaked-with-heavy-tails signature.
Use it as the theoretical bridge to robust methods: minimizing absolute error is maximum likelihood under Laplace noise, so this family explains when and why median-type estimators beat mean-type ones.
Use it in privacy calculations: the Laplace mechanism adds noise with scale b = Δ/ε, and this page turns any chosen scale into exact tail probabilities for the added noise.

Choose another method when

Avoid it when empirical tails decay like a power law rather than exponentially; Laplace tails are heavier than normal but still light compared with Pareto or Cauchy behavior.
Avoid it when the density near the center is genuinely smooth: the Laplace peak has a corner at μ, and data with a rounded peak are often better served by a t or logistic model.
Avoid using it merely because excess kurtosis is positive; several families share that property, and the diagnostic that favors Laplace is a linear decay of the log-density on both sides.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

Sharp-peaked forecast errors

Inputs: X ~ Laplace(μ=0, b=1); evaluate an error beyond ±2.
Representative result: P(|X|>2) = e⁻² ≈ 0.1353, about three times the matching normal tail.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

3:14 min

The player loads only after you press play. You can also watch on YouTube.

Read the complete transcript

The Laplace Distribution. Why can one symmetric curve be sharper than a normal, yet produce substantially more extreme errors? The Laplace family uses finite location mu, positive scale b, and the entire real line. Its density is one over two b times an exponential of negative absolute standardized distance. The cumulative function joins two exponential branches continuously at probability exactly one half. Its quantile uses one logarithmic branch below one half and another above the midpoint. Symmetry makes mean, median, and mode all equal mu, despite the density's sharp corner.

Most memorably, probability beyond distance t from mu is exactly exponential negative t over b. Variance is two b squared, so standard deviation is b root two, never simply b. The interquartile range is two b log two, while median absolute deviation is b log two. Skewness is zero and excess kurtosis is exactly three for every positive scale. At equal variance, Laplace has extra center and far-tail mass but lighter intermediate shoulders. Maximum likelihood chooses a sample median for mu and mean absolute deviation for b. Laplace observation noise turns maximum likelihood into least-absolute-deviation fitting, not squared-error fitting. Separately, a zero-centered Laplace prior on coefficients creates the L1 penalty used by LASSO. In the classical privacy mechanism, calibrated scale equals sensitivity divided by epsilon under stated assumptions.

Model daily forecast errors with mu zero and b one, then compare equal-variance normal errors. A miss beyond plus or minus two has probability exponential negative two, about point one three five three. The ninety-fifth percentile equals log ten, approximately two point three zero two six. Variance equals two and standard deviation equals root two, approximately one point four one four two. At distance two, the equal-variance normal tail is actually slightly larger, about point one five seven three. At distance four, Laplace tail probability is point zero one eight three, nearly four times normal.

Applications include forecast errors, impulsive sensors, robust location models, coefficient priors, and privacy noise. Diagnose Laplace by plotting log density against absolute distance, expecting straight lines on both sides. Power-law tails or a genuinely rounded center call for different distribution families. Reject nonpositive b and nonfinite mu before evaluating any density, probability, or quantile. Asymmetric Laplace models used in quantile regression are different and remain outside this explorer. Report mu, b, endpoints, population, result, and exactly how any normal comparison was matched. Explore Laplace probabilities, quantiles, and tail comparisons free at Distri Scope dot com.

How to read the result

Compare like with like: at equal variance, the Laplace places more mass both near the center and far in the tails than the normal—the deficit is in the shoulders, which is what excess kurtosis measures.

The two-sided tail e^{−t/b} makes privacy and error statements one-liners: noise beyond 2b occurs with probability e^{−2} ≈ 0.135, beyond 4.6b with probability about 0.01.

The interquartile range is 2b·ln 2 ≈ 1.386b, a scale reading that is robust to the tails.

Inputs and parameter meaning

Two parameters: the location μ (mean, median, and mode simultaneously) and the scale b; the standard deviation is b√2, not b.

The plotted range covers the central 99% of probability; the exponential tails continue beyond it in both directions.

Interval and quantile calculators are exact via the closed-form piecewise CDF and its inverse.

Formula and available parameters

Density or probability mass

f(x) = \frac{1}{2b} \exp\!\left(-\frac{|x-\mu|}{b}\right)

Cumulative distribution

F(x) = \begin{cases} \tfrac{1}{2} e^{(x-\mu)/b} & x < \mu \\ 1 - \tfrac{1}{2} e^{-(x-\mu)/b} & x \ge \mu \end{cases}

Mean

\mu

Variance

\sigma^2 = 2b^2

Location (μ): interactive range -10 to 10; default 0.
Scale (b): interactive range 0.1 to 5; default 1.

What the method is doing

The CDF is piecewise exponential: F(x) = ½e^{(x−μ)/b} below μ and 1 − ½e^{−(x−μ)/b} above; the quantile function inverts each branch in closed form.

Moments are exact: mean μ, variance 2b², skewness 0, and excess kurtosis exactly 3 at every parameter choice.

The tail bound is memorably simple: P(|X − μ| > t) = e^{−t/b}, one clean exponential in the distance from the center.

Worked example: forecast errors with a sharp center

Daily demand-forecast errors are centered with a sharp peak and occasional large misses, modeled as Laplace with μ = 0 and b = 1 (in units of typical error). Find the probability of a miss beyond ±2, the 95th percentile of error, the standard deviation, and compare the ±2 tail with a normal of equal variance.

1Two-sided tail: P(|X| > 2) = e^{−2} ≈ 0.1353—about one day in seven produces a miss beyond two units.
2Upper percentile: Q(0.95) = −ln(2·0.05) = ln 10 ≈ 2.303 by the closed-form inverse of the upper branch.
3Spread: the variance is 2b² = 2, so the standard deviation is √2 ≈ 1.414 even though the scale parameter is 1.
4Normal comparison at equal variance (σ = √2): P(|X| > 2) = 2Φ(−1.414) ≈ 0.157… slightly higher near two—but at ±4 the Laplace tail e⁻⁴ ≈ 0.018 is more than three times the normal’s 0.0047, and the gap widens with distance.

Interpretation

Near the center the two models look interchangeable; far out they are not. The Laplace model prices a four-unit miss at triple the normal’s rate, which is exactly the kind of difference that matters for safety stock and risk buffers. When large misses are the costly events, the choice between these two error models is a business decision, not a cosmetic one.

Common mistakes

Do not read the scale b as the standard deviation; the standard deviation is b√2, and confusing the two misprices every tail statement by a √2 factor.
Do not use mean-based fitting habits: the maximum-likelihood location is the sample median, and the ML scale is the mean absolute deviation from it.
Do not assume symmetry-plus-heavy-tails uniquely identifies Laplace; check the linear log-density signature before committing.

Limits and independent validation

The page covers the symmetric two-parameter family; asymmetric Laplace variants used in quantile regression are out of scope.

The density’s corner at μ is a modeling idealization—real error distributions are rarely exactly non-smooth at their center.

Fitting b from data and testing fit belong to the data-fitting tool, not this theoretical page.

Before using the result

Check the signature tail identity P(|X−μ| > t) = e^{−t/b} at one or two values of t with the interval calculator.
Recompute Q(0.95) by hand as μ − b·ln(2·0.05) and compare with the quantile lookup.
On candidate data, plot log-frequency against distance from the median; approximate straight lines on both sides support the model.

exponential distribution →See the one-sided building block whose mirror image forms the Laplace.normal distribution →Compare against the squared-error benchmark at equal variance.

← Pareto Distribution Logistic Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.