Cumulative Distribution Function Calculator

A cumulative distribution function (CDF) calculator works out the probability that a random variable takes a value less than or equal to a given number. You enter the distribution type, its parameters, and the value of interest; it returns the cumulative probability, often shown as a percentage between 0% and 100%.

Formula

The CDF is defined as F(x) = P(X ≤ x). For a continuous distribution, it is the integral of the probability density function (PDF) from −� to x:

F(x) = ∫_{−�}^x f(t) dt

where f(t) is the PDF and x is the value you supply. For discrete distributions, the CDF is a sum: F(x) = Σ_t≤x P(X = t). The calculator above implements the CDF for four common continuous distributions:

• Normal(μ, σ): F(x) = Φ((x − μ)/σ), where Φ is the standard normal CDF.
• Student t(ν): F(x) = 1 − ½ I_{ν/(ν+x²)}(ν/2, 1/2) for x ≥ 0, with symmetry for negative x.
• Chi-square(k): F(x) = γ(k/2, x/2) / Γ(k/2), the regularized lower incomplete gamma function.
• Exponential(λ): F(x) = 1 − e^−λx for x ≥ 0.

Here μ is the mean, σ the standard deviation, ν the degrees of freedom, k the degrees of freedom for chi-square, and λ the rate (1/mean) for the exponential. All inputs must be real numbers; for the t and chi-square, the degrees of freedom must be positive integers.

Worked Example

Using the default settings: Normal distribution with mean μ = 0, standard deviation σ = 1, and x = 1.5.

1. Standardize: z = (x − μ)/σ = (1.5 − 0)/1 = 1.5.
2. Find Φ(1.5) using a standard normal table or calculator: Φ(1.5) ≈ 0.9332.
3. The CDF value is 0.9332, meaning P(X ≤ 1.5) ≈ 93.32%.

The calculator returns 0.9332 for the default inputs, matching the table value.

How to Choose the Right Distribution

The CDF depends entirely on the distribution you select. Use these guidelines:

• Normal: for measurements that cluster around a mean (heights, test scores, measurement errors).
• Student t: for small-sample inference when the population standard deviation is unknown; the shape is similar to normal but with heavier tails.
• Chi-square: for sums of squared standard normals; used in variance estimation and goodness-of-fit tests.
• Exponential: for time between independent events at a constant rate (e.g., radioactive decay, waiting times).

If you are unsure, check whether your data are continuous, bounded (e.g., non-negative), or symmetric. The normal is symmetric and unbounded; the exponential is right-skewed and non-negative; the chi-square is non-negative and right-skewed.

Reading the Result

The output is a probability between 0 and 1. Multiply by 100 to get a percentage. For example, 0.9332 means 93.32% of the distribution lies at or below x = 1.5. This is useful for:

• Finding percentiles: if F(x) = 0.90, then x is the 90th percentile.
• Computing p-values: for a one-sided test, the p-value is often 1 − F(observed).
• Setting confidence intervals: the CDF inverse gives critical values.

Remember that the CDF is non-decreasing and approaches 0 as x → −� and 1 as x → +�. A value near 0.5 for a symmetric distribution indicates x is near the median.

Common Mistakes and Misconceptions

• Confusing CDF with PDF: the PDF gives the height of the density curve; the CDF gives the area under it. The PDF can exceed 1, but the CDF never does.
• Using the wrong parameterization: the exponential distribution is sometimes written with mean β instead of rate λ. Here λ = 1/β. If you enter a mean of 2 as λ, you get the wrong answer.
• Forgetting the continuity correction: for discrete distributions, the CDF at x includes all values ≤ x. For a normal approximation to a binomial, use x+0.5.
• Ignoring units: if x is in minutes and the mean is in hours, convert first. The CDF is unitless, but the input must match the distribution’s scale.
• Rounding too early: keep at least 4 decimal places in intermediate steps to avoid large errors in the tail.

Limitations of the Calculator

This calculator handles four common continuous distributions, but many others exist (Weibull, gamma, beta, uniform, etc.). For those, you would need a different tool or a statistical package. Also, the calculator uses numerical approximations for the t and chi-square CDFs; these are accurate to about 6 decimal places for typical inputs, but may lose precision for extreme values (e.g., very large x or very small degrees of freedom). The exponential formula is exact, and the normal uses the error function, which is accurate to machine precision.

For discrete distributions like binomial or Poisson, the CDF is a sum, not an integral. This calculator does not implement those, but you can approximate them with a normal CDF if the sample size is large enough (e.g., np and n(1−p) both > 5).

Frequently Asked Questions

What is the difference between CDF and PDF?

The PDF, f(x), describes the relative likelihood of a value at exactly x. The CDF, F(x), gives the probability of observing a value ≤ x. The CDF is the integral of the PDF; conversely, the PDF is the derivative of the CDF. For a normal distribution, the PDF is the bell curve, and the CDF is the S-shaped curve that rises from 0 to 1.

How do I find the percentile from a CDF?

To find the 90th percentile, set F(x) = 0.90 and solve for x. For the normal distribution, use the inverse standard normal: x = μ + σ·Φ�¹(0.90). Many calculators have an inverse CDF function. In this calculator, you would adjust x until the output is 0.90, or use a dedicated inverse CDF tool.

Can the CDF be negative or greater than 1?

No. By definition, a CDF is a probability, so it is always between 0 and 1 inclusive. If you get a result outside that range, you likely made a data-entry error, such as using a negative standard deviation or a negative rate for the exponential.

What is the CDF of a standard normal at 0?

For the standard normal (μ=0, σ=1), F(0) = 0.5. This is because the distribution is symmetric about zero, so exactly half of the probability mass lies below zero. This is a quick sanity check for any normal CDF calculation.

How do I use the CDF for hypothesis testing?

For a one-sided test, the p-value is 1 − F(test statistic) for a right-tailed test, or F(test statistic) for a left-tailed test. For a two-sided test, double the smaller tail. For example, with a t-statistic of 2.1 and 10 degrees of freedom, the one-sided p-value is 1 − F(2.1) ≈ 0.031.

Why does the exponential CDF require x ≥ 0?

The exponential distribution models waiting times, which cannot be negative. Its support is [0, �). For x < 0, the CDF is 0, because the probability of a negative waiting time is zero. The formula 1 − e^(−λx) is only valid for x ≥ 0.

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in x | Value x | - | 1.5 | 0
sel dist | Distribution | Normal=1, Student t=2, Chi-square=3, Exponential=4
in p1 | Parameter 1 (mean or df) | - | 0 | 0
in p2 | Parameter 2 (sd or rate) | - | 1 | 0
out cdf | P(X ≤ x) | - | dist==1 ? 0.5*(1+erf((x-p1)/(p2*sqrt(2)))) : dist==2 ? 1-0.5*reginc(0.5, p1/2, p1/(p1+x*x)) : dist==3 ? gammainc(p1/2, x/2) : 1-exp(-p2*x) | 4