Continuous uniform CDF calculator

Enter three finite numbers and receive a probability from zero to one. Points below the interval return zero, points above it return one, and points inside the interval are evaluated by their relative position. The upper bound must be strictly greater than the lower bound.

What the continuous uniform CDF measures

A continuous uniform distribution assigns equal probability density to every location in a fixed interval. Its cumulative distribution function, usually called the CDF, answers a practical question: what is the probability that a random value from that interval is less than or equal to a chosen point x? The result is always between zero and one. A result of zero means the point lies at or below the lower boundary, while a result of one means it lies at or above the upper boundary. Between those boundaries, probability grows at a constant rate because equal-length portions of the interval carry equal probability. For example, a point one quarter of the way from a to b has cumulative probability one quarter. This calculator handles all three regions directly, including exact boundary values. It evaluates a single mathematical distribution rather than estimating probability from samples, so there is no simulation error, random variation, confidence interval, or dependence on a sample size. The output is the exact piecewise CDF value subject only to ordinary floating-point representation.

How to enter bounds and interpret the result

Provide a as the lower bound, b as the upper bound, and x as the point where cumulative probability should be evaluated. All three inputs must be finite numbers, and b must be strictly greater than a. Negative bounds, decimal bounds, and an x outside the interval are valid. If x is inside the interval, the calculator subtracts a from x and divides that distance by the full interval width b minus a. If x equals a, the result is zero; if x equals b, the result is one. This boundary behavior follows the standard continuous uniform CDF and does not create point mass at either endpoint, because a single exact point has probability zero in a continuous distribution. Read the returned value as a proportion. Multiplying it by one hundred gives the corresponding percentage, but the capability deliberately returns the canonical probability on the zero-to-one scale. Reversing the bounds or making them equal does not define a valid positive-width continuous uniform distribution, so that input produces a clear validation error instead of a misleading number.

When this calculation is useful

Use the continuous uniform CDF when every value within a known bounded range is modeled as equally likely. It is useful for checking textbook probability exercises, building deterministic test fixtures, validating statistical software, and calculating threshold probabilities in simple uncertainty models. Suppose an arrival time is modeled uniformly over a twenty-minute window: the CDF at a point within that window gives the chance that arrival has occurred by then. In quality assurance, a uniformly generated test value may need to remain below a limit; this result gives the theoretical fraction expected to pass. The calculation is also a convenient building block for quantile checks and probability transformations, although this capability evaluates the forward CDF only and does not solve for x. Be careful about the modeling assumption. Real measurements are often clustered, skewed, rounded, or truncated in ways that make a uniform model inappropriate. Equal density across the entire interval must come from the problem definition or a defensible assumption. The calculator performs the stated mathematics accurately, but it cannot determine whether uniformity is a good description of the underlying process.