Quantile and Percentile Calculator

Math Probability • Continuous Probability Distributions

Written by STEM Calculators Team Published February 14, 2026 Updated February 24, 2026

Compute the quantile \(x_p\) such that \(P(X \le x_p)=p\) (inverse CDF) for common continuous distributions.

Distribution

\(\mu\) (mean) \(\sigma\) (std dev)

Uses \(z\)-scores: \(x_p=\mu+\sigma z_p\), where \(z_p\) satisfies \(\Phi(z_p)=p\).

p input mode p

Output is the cutoff \(x_p\) such that a fraction \(p\) (or the p-th percentile) lies at or below \(x_p\). Website tip: the graph shows the CDF with the inverse point \((x_p,p)\).

Ready

CDF & inverse point \((x_p, p)\)

Animation ready

The animation moves along the CDF from left to the computed quantile. The badge shows \(p\) and \(x_p\).

Choose a distribution, enter \(p\), then click “Calculate”.

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Theory

What is a quantile / percentile?

For a continuous random variable \(X\) with cumulative distribution function (CDF) \(F(x)\), the p-quantile \(x_p\) is the value that satisfies:

\[ \begin{aligned} F(x_p) &= P(X\le x_p) \\ &= p \end{aligned} \]

When you express \(p\) as a percentage, the result is often called a percentile. For example, \(p=0.95\) is the 95th percentile: about 95% of values lie at or below \(x_{0.95}\).

Quantile = inverse CDF: \(x_p = F^{-1}(p)\). A CDF is monotone increasing, so the inverse is well-defined for \(0<p<1\).

Normal distribution quantiles (z-scores and z-tables)

If \(X\sim N(\mu,\sigma^2)\), you can convert to a standard normal variable \(Z\sim N(0,1)\) using a z-score:

\[ \begin{aligned} Z &= \frac{X-\mu}{\sigma} \\ X &= \mu+\sigma Z \end{aligned} \]

The quantile problem \(P(X\le x_p)=p\) becomes \(P(Z\le z_p)=p\), where \(z_p\) is the standard normal quantile. Then:

\[ \begin{aligned} \Phi(z_p) &= p \\ x_p &= \mu + \sigma z_p \end{aligned} \]

Traditionally, \(z_p\) is read from a z-table (possibly with interpolation). Modern calculators compute \(z_p\) using accurate rational approximations to \(\Phi^{-1}\) (the inverse standard normal CDF).

p	Percentile	\(z_p=\Phi^{-1}(p)\)	Common use
0.900	90th	1.282	one-sided 90%
0.950	95th	1.645	one-sided 95%
0.975	97.5th	1.960	two-sided 95% CI cutoff
0.990	99th	2.326	one-sided 99%

Confidence intervals link: a two-sided 95% normal CI uses \(z_{0.975}\approx 1.96\) because each tail gets 2.5%.

Exponential distribution quantiles (closed-form inverse)

If \(X\sim \mathrm{Exp}(\lambda)\) with \(\lambda>0\) and support \(x\ge 0\), the CDF is:

\[ \begin{aligned} F(x) &= 1-e^{-\lambda x},\quad x\ge 0 \end{aligned} \]

To find the quantile \(x_p\), set \(F(x_p)=p\) and solve for \(x_p\):

\[ \begin{aligned} 1-e^{-\lambda x_p} &= p \\ e^{-\lambda x_p} &= 1-p \\ -\lambda x_p &= \ln(1-p) \\ x_p &= -\frac{\ln(1-p)}{\lambda} \end{aligned} \]

This is one reason the exponential distribution is popular: its quantiles have a simple closed form.

University note: inverse CDF for custom distributions

For many distributions, \(F^{-1}(p)\) is not available in elementary closed form. A standard approach is to solve \(F(x)-p=0\) numerically using the fact that \(F(x)\) is monotone increasing.

Bisection method: bracket a solution \([L,U]\) with \(F(L)\le p\le F(U)\), then repeatedly halve the interval.
Newton’s method: use updates \(x_{k+1}=x_k-\dfrac{F(x_k)-p}{f(x_k)}\) when the PDF \(f(x)=F'(x)\) is available and well-behaved.
Hybrid methods: combine bisection (guaranteed) with Newton (fast) for stability.

Practical edge cases: very small or very large \(p\) values (near 0 or 1) can push quantiles far into the tails, where floating-point rounding and approximation error matter more.

CDF & inverse point \((x_p, p)\)

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