Mean of Squared Uniform Values
A uniform distribution gives every value in a chosen interval the same density. This calculator focuses on the transformed variable y equals x squared. It helps you find the expected value of that square when x is uniformly spread between two bounds. The result is useful in statistics, physics, risk work, simulation checks, and classroom examples.
Why the Mean Matters
The mean of y is not found by squaring the average of x. Squaring changes the shape of the values. Large positive and negative x values both create large y values. For that reason, the calculator integrates x squared across the full interval. It then divides by the interval width. This gives the true continuous expected value.
What the Tool Calculates
The form accepts a lower bound, an upper bound, a sample count, a decimal setting, a test x value, and a y threshold. It returns the exact mean, the midpoint sample mean, the variance of y, the standard deviation, the range of possible y values, and the probability that y stays under the chosen threshold. These extra outputs make the page more than a simple mean finder.
Practical Use Cases
Use the calculator when checking a simulation. Run a random uniform model, then compare its average squared output with the exact value shown here. You can also use it for quality scoring, distance models, error analysis, and general learning. The sample mean option shows how numerical averaging approaches the analytic result as the sample count grows.
Reading the Results
A small sampling error means the midpoint estimate is close to the formula result. A larger variance means y values spread widely. If the interval crosses zero, the minimum y value becomes zero. If the interval stays positive or negative, the minimum comes from the bound closest to zero. The export buttons save the result for reports, worksheets, or later comparison.
Good Input Choices
Choose bounds that match the real problem. Use a positive sample count for numerical checks. Increase it when you want a closer midpoint estimate. Keep decimal precision high for review work, then lower it for a clean report. Always confirm that the upper bound is greater than the lower bound first.