Continuous Uniform Distribution Calculator

Calculator Inputs

Example Data Table

a	b	x	Interval	Percentile	Density	CDF	Interval Probability	Mean	Standard Deviation
0	10	4	2 to 7	90%	0.1	0.4	0.5	5	2.886751
-5	5	0	-2 to 3	75%	0.1	0.5	0.5	0	2.886751
10	20	25	12 to 16	25%	0.1	1	0.4	15	2.886751

Formula Used

Let the lower bound be a and the upper bound be b. The calculator requires b greater than a.

Range = b - a
PDF: f(x) = 1 / (b - a), when a ≤ x ≤ b. Otherwise, f(x) = 0.
CDF: F(x) = 0 when x ≤ a.
CDF: F(x) = (x - a) / (b - a), when a < x < b.
CDF: F(x) = 1 when x ≥ b.
Mean = (a + b) / 2
Median = (a + b) / 2
Variance = (b - a)² / 12
Standard deviation = (b - a) / √12
P(c ≤ X ≤ d) = overlap width / (b - a)
Percentile value = a + p(b - a)

How to Use This Calculator

Enter the lower bound a.
Enter the upper bound b. It must be greater than a.
Enter an X value for PDF and CDF checks.
Enter interval start and interval end values.
Enter a percentile from 0 to 100.
Press the calculate button.
Review the result table above the form.
Use CSV or PDF export for records.

Understanding Continuous Uniform Distribution

A continuous uniform distribution describes an even spread over a fixed interval. Every value between the lower limit and upper limit has the same density. Values outside the interval have zero density. This makes the model easy to read and useful for teaching probability, simulation, quality checks, and simple risk estimates.

Why This Calculator Helps

Manual uniform distribution work is not hard, but mistakes happen when limits, intervals, or tail areas are entered quickly. This calculator keeps the process organized. It checks that the upper bound is greater than the lower bound. It then calculates density, cumulative probability, interval probability, mean, variance, standard deviation, median, and a selected percentile. The result helps students, analysts, and site visitors confirm answers quickly.

Common Use Cases

Use this tool when outcomes are equally likely across a known range. Examples include a random arrival time during a service window, a simulated measurement within tolerance, or a generated number between two endpoints. The calculator is also helpful when comparing a single point probability statement with an interval statement. A point has no probability mass in a continuous model. The area over an interval is what matters.

Reading The Outputs

The probability density function shows height, not direct probability at one exact point. The cumulative distribution value shows the chance that X is less than or equal to the selected value. The interval probability shows the chance that X falls between two chosen values. The percentile output shows the value where the selected percent of outcomes are expected to fall at or below that point.

Best Practice

Always confirm the bounds first. Use the same unit for all values. Enter interval endpoints carefully. Keep more decimals when preparing formal answers. Use the CSV export for spreadsheet review. Use the PDF export for a compact report. When teaching, compare the formula section with the result table. This makes the idea of area under a flat density curve easier to understand.

Accuracy Notes

The uniform model assumes equal likelihood across the entire interval. It should not be used when values cluster near one side. It also should not replace real data checks. Treat the output as a model result, not proof of natural behavior.

FAQs

What is a continuous uniform distribution?

It is a probability model where every value inside a fixed interval has equal density. The distribution is flat between the lower and upper bounds and zero outside them.

Why is point probability zero?

In a continuous distribution, probability comes from interval area. A single exact point has no width, so its probability is zero, even though the density may be positive.

What does the PDF value mean?

The PDF value is the height of the flat density curve. It is used with interval width to form probability. It is not the probability of one exact value.

What does the CDF value mean?

The CDF gives the probability that X is less than or equal to the chosen value. It ranges from zero below the interval to one above the interval.

Can the interval go outside the bounds?

Yes. The calculator only counts the part overlapping the distribution range. Any portion outside the lower and upper bounds adds no probability.

What happens if I reverse interval endpoints?

The calculator sorts the interval endpoints before calculating overlap. This avoids negative interval width and keeps the probability result meaningful.

How is the percentile value calculated?

The percentile value is found by moving the selected percentage across the range. For example, the 90th percentile is a plus 90 percent of the interval width.

When should I use this calculator?

Use it when values are equally likely across a known continuous range. It works well for classroom probability, quick simulations, simple timing ranges, and basic tolerance examples.