Uniform Distribution Calculator

A continuous uniform distribution is defined by two main boundary parameters: the minimum value a and the maximum value b. The probability density function (PDF) remains constant at 1 / (b - a) for any value of x between a and b. The mean (or expected value) and median are both calculated as the midpoint between the boundaries: Mean = (a + b) / 2. The variance measures the dispersion of the distribution and is determined by the formula Variance = (b - a)² / 12, while the standard deviation is simply the square root of the variance.

Imagine a delivery truck arrives at a warehouse at a completely random time between 10:00 AM and 11:00 AM. Let a = 0 (representing 10:00 AM) and b = 60 (representing 60 minutes past the hour). To find the expected arrival time, we compute the mean: Mean = (0 + 60) / 2 = 30 minutes. Next, to find the variance of the arrival time, we use the variance formula: Variance = (60 - 0)² / 12 = 3600 / 12 = 300. Taking the square root gives us a standard deviation of approximately 17.32 minutes. If we want to find the probability that the truck arrives within the first 20 minutes (x = 20), the cumulative distribution function yields P(X ≤ 20) = (20 - 0) / (60 - 0) = 0.333, or a 33.3% chance.

Always verify that your lower bound (a) is strictly less than your upper bound (b) before running calculations. Remember that continuous uniform distributions assign a probability of zero to any single exact point, meaning probabilities are only meaningful across intervals. Finally, keep track of your units of measurement—if boundaries are entered in hours, your inputs for specific values of x must also be converted to hours.