Cauchy Distribution Calculator

Introduction to the Cauchy Distribution Calculator

This Cauchy distribution calculator turns a value $x$, a location $x_0$, and a positive scale $\gamma$ into the quantities most people ask for first: density, cumulative probability, survival probability, median, and, when $p$ is supplied, a matching quantile. It is useful whenever you want to move from a point on the curve to a percentile, or work backward from a probability to the value that creates it.

Cauchy curves look reassuringly bell-shaped until you compare their tails with a normal distribution. The center still sits at $x_0$, but the probability mass fades much more slowly, so observations far from the middle are not nearly as unusual as they would be under a Gaussian model. That slow decay is why the Cauchy distribution has no finite mean or variance.

Those properties make the calculator useful as more than a lookup tool. By varying $x_0$ and $\gamma$, you can see how the median stays fixed at the center while the peak widens or narrows, and you can watch the CDF and survival function remain balanced even as the tails keep pulling probability outward.

How to Use the Cauchy Distribution Calculator

Enter the three core parameters first. Value x is the point where the distribution is evaluated. Location x₀ shifts the entire Cauchy curve left or right and also gives the median. Scale γ must be greater than zero and determines how wide the peak and tails are. If you want the inverse CDF, add a probability p between 0 and 1; otherwise leave that box blank and the quantile line will stay empty.

• PDF f(x): the density height at the chosen value of $x$; it is not a point probability.
• CDF F(x): the probability that a Cauchy random variable is at or below $x$.
• Survival S(x): the probability mass remaining above $x$; this equals $1-F\left(x\right)$.
• Median: always equal to the location parameter $x_0$.
• Quantile at p: the value whose cumulative probability is $p$, when $p$ is entered.

Units should stay consistent across $x$, $x_0$, and $\gamma$. If the inputs represent seconds, volts, meters, or another measurement, all three must use the same unit scale. The calculator returns density in inverse units, while the CDF and survival function are ordinary probabilities from 0 to 1.

If you are exploring the distribution for the first time, start with $x_0=0$ and $\gamma =1$. That standard Cauchy form gives you a clear baseline: shifting $x_0$ translates the curve without changing its shape, while increasing $\gamma$ lowers the peak, spreads the middle, and pushes familiar percentiles farther into the tails.

Cauchy Distribution Formula

Cauchy Probability Density Function

The Cauchy density is the familiar rational curve used whenever a model needs a sharp center and very heavy tails. The calculator evaluates it with the closed form below, which is why the PDF reacts immediately as soon as you change the value, location, or scale inputs.

Formula: f x = 1 / (π γ 1 + (x-x_0)/γ^2)

$f\left(x\right)=\frac{1}{\pi \gamma \left(1+{\frac{x-x_0}{\gamma }}^2\right)}$

This formula explains why the Cauchy distribution behaves so differently from the normal distribution. The density falls off only like $\mathrm{|x|}^\{-2\}$, so far-out values keep a meaningful share of probability. Because the mean and variance do not exist, the usual intuition built from bell-curve models does not apply here.

CDF, Survival, and Quantile Formula

The cumulative distribution function is the calculator's bridge from a point $x$ to the probability accumulated to its left:

Formula: F x = 1 / π ⁢ ⁡ arctan (x - x_0) / γ + 1 / 2

$F\left(x\right)=\frac{1}{\pi }\arctan \left(\frac{x-x_0}{\gamma }\right)+\frac{1}{2}$

Once the CDF is known, the survival function is just $1-F\left(x\right)$, which is why values far in the right tail can still leave a noticeable survival probability. When you enter $p$, the calculator uses the inverse formula below to turn that probability into the corresponding Cauchy quantile:

Formula: Q p = x_0 + γ tan π(p - 1 / 2)

$Q\left(p\right)=x_0+\gamma \tan \left(\pi (p-\frac{1}{2})\right)$

The tangent in the inverse CDF is what makes the tails feel so dramatic. Probabilities close to 0 or 1 map to very large negative or positive values, so even a tiny move in $p$ near the ends can send the quantile far from the center.

Reading the Cauchy Outputs

The CDF tells you how much probability lies to the left of the chosen $x$, and the median is the point where that total reaches one-half. For a Cauchy distribution that center point is always $x_0$, which is why the calculator lists it separately.

The heavy tails are the headline feature to watch. A Cauchy curve can look narrow near the middle yet still assign noticeable probability to very distant values. That is why robust summaries such as the median and interquartile range usually make more sense than averages when Cauchy behavior is involved.

When you compare PDF, CDF, survival, and quantile outputs side by side, a useful habit is to read them as different questions about the same curve. The PDF answers “how high is the density here?” The CDF answers “how much has accumulated by here?” The survival function asks “what remains to the right?” The quantile asks “where do I land if I want this probability?”

Cauchy Parameter Intuition

The location parameter $x_0$ moves the entire distribution without changing its shape. It marks both the median and the mode, so changing $x_0$ simply slides the center line left or right.

The scale $\gamma$ controls how broad the Cauchy peak is and how quickly the middle gives way to the tails. Larger $\gamma$ values flatten the center, widen the interquartile range, and place the same percentiles farther from the median. Even though $\gamma$ is not a standard deviation, it is still the clearest spread parameter for this distribution.

A memorable fact is that the middle 50% of the distribution lies between $x_0-\gamma$ and $x_0+\gamma$. That makes $\gamma$ easy to interpret at a glance: double the scale and you immediately double that central interval, while the tail probabilities at a fixed distance from the center also become larger.

Where the Cauchy Distribution Appears

The Cauchy distribution is one of the classic heavy-tailed models in probability and statistics. It can arise as the ratio of two independent standard normal variables, and it also appears when a uniformly distributed angle is passed through a tangent transform. Those connections are exactly why the tangent shows up in the quantile formula above.

In practice, Cauchy models are used when rare but extreme deviations matter more than tidy averages. They show up in robust Bayesian priors, in resonance-style phenomena, and in any setting where occasional large spikes are plausible enough to shape the analysis. The distribution is less about predicting a typical value and more about acknowledging that very large departures do not disappear quickly.

Simulating Cauchy Random Values

The inverse CDF makes Cauchy sampling straightforward. Draw $U$ from a uniform distribution on (0,1), then transform it with $x_0+\gamma \times \tan \left(\pi ,(,U,-,1,/,2,)\right)$. Because the inverse formula is exact, this gives genuine Cauchy samples rather than an approximation.

If you simulate many draws, the histogram will show why this distribution has such a strong reputation for heavy tails. Most points cluster near the center, but every now and then one sample flies far away. That behavior is not a mistake in the generator; it is the distribution acting exactly as designed.

Simulation is also a practical check on the calculator. You can compare the fraction of samples that land above a threshold with the survival function shown here, or compare simulated medians with $x_0$. When the model is correct, the Monte Carlo results should line up with the formulas.

Median and Quantiles in a Cauchy Model

Because the mean does not exist, the median is the most natural measure of center for a Cauchy distribution. The calculator reports it automatically, and it always matches $x_0$.

Quantiles are equally clean. The 25th percentile sits one scale unit below the median, the 75th percentile sits one scale unit above it, and the interquartile range is therefore $2\gamma$. That simple relationship is one of the best reasons to think about $\gamma$ geometrically instead of comparing it with a standard deviation.

Worked Example: Evaluating a Shifted Cauchy

Suppose you are analyzing resonance measurements with a Cauchy model centered at $x_0=0$ and scale $\gamma =2$. For $x=5$, the density is given by the expression below, which the calculator evaluates directly:

Formula: 1 / (π × 2 × 1 + 5/2^2)

$\frac{1}{\pi \times 2\times \left(1+{\frac{5}{2}}^2\right)}$

With those inputs, the PDF is about 0.02195, the CDF is about 0.87888, and the survival probability is about 0.12112. That combination is a good reminder of what “heavy tails” means in practice: even a point five units from the center can still retain more than a tenth of the probability mass to its right.

If you change the scale to a larger value, the same observation at $x$ becomes less surprising because the curve spreads out. If you move the location instead, the entire set of results shifts with it while the heavy-tailed shape remains the same.

Estimating Cauchy Parameters from Data

Fitting a Cauchy model usually starts with robust summaries rather than ordinary averages. The sample median is a consistent estimator for location, and half of the interquartile range is a natural estimate for scale. Those choices work well because they are not pulled around by extreme observations the way the mean and standard deviation are.

Maximum likelihood methods are also available, but the likelihood surface can be flatter and harder to optimize than it is for lighter-tailed families. In practice, analysts often combine quick robust estimates with plots or diagnostics to decide whether a Cauchy model is a sensible description of the data.

That is why the calculator is useful during exploratory work: it lets you plug in candidate $x_0$ and $\gamma$ values and immediately see how the density and percentiles would behave before you commit to a full fit.

Limitations of Cauchy Modeling and This Calculator

The main limitation of the Cauchy distribution is also its defining feature: the mean and variance do not exist. As a result, sample averages can remain unstable no matter how much data you collect, and methods built around squared error can behave poorly when the noise is Cauchy-like. For that reason, medians, quantiles, and other robust summaries are usually a better fit.

Another common mistake is to treat $\gamma$ as if it were a standard deviation. It is not. The scale parameter controls spread through the interquartile range and the shape of the central peak, so comparisons with Gaussian models need to be made carefully.

The calculator itself has a few straightforward constraints. The scale must be strictly positive, the optional probability $p$ must be between 0 and 1 exclusive, and the answers are floating-point approximations. Quantiles near the extremes can become very large in magnitude, which is exactly what the tangent-based inverse formula predicts.

Further Reading on Heavy-Tailed Distributions

Heavy-tailed distributions appear in robust statistics, signal processing, Bayesian modeling, and other areas where large deviations cannot be ignored. Standard references on probability will show how the Cauchy distribution fits alongside the normal and other families, while software documentation can help when you need built-in PDF, CDF, or quantile functions. Using this calculator first can make those references easier to understand, because you can experiment with $x_0$, $\gamma$, and $p$ before diving deeper.