Calculator
Example Data Table
| Case | Parameters | Formula | Result |
|---|---|---|---|
| Between probability | a = 10, b = 20, x1 = 12, x2 = 16 | (16 - 12) / (20 - 10) | 0.4 |
| Less than probability | a = 10, b = 20, x = 14 | (14 - 10) / (20 - 10) | 0.4 |
| Greater than probability | a = 10, b = 20, x = 14 | 1 - 0.4 | 0.6 |
| Density | a = 10, b = 20, x = 15 | 1 / (20 - 10) | 0.1 |
| 75th percentile | a = 10, b = 20, p = 0.75 | 10 + 0.75 × 10 | 17.5 |
Formula Used
For a continuous uniform random variable X on the interval [a, b], use these rules.
- Density: f(x) = 1 / (b - a), for a ≤ x ≤ b. Otherwise, f(x) = 0.
- Cumulative distribution: F(x) = 0 for x < a, F(x) = (x - a) / (b - a) for a ≤ x ≤ b, and F(x) = 1 for x > b.
- Between probability: P(x1 ≤ X ≤ x2) = overlap length / (b - a).
- Greater than probability: P(X ≥ x) = 1 - F(x).
- Mean: (a + b) / 2.
- Variance: (b - a)2 / 12.
- Standard deviation: √Variance.
- Quantile: Q(p) = a + p(b - a), where 0 ≤ p ≤ 1.
How to Use This Calculator
- Choose the calculation type you need.
- Enter the lower bound a and upper bound b.
- Enter x, x1 and x2, or p based on your selected mode.
- Set the decimal places for the output.
- Press Calculate to show the result above the form.
- Use the export buttons to save CSV or PDF output.
Continuous Uniform Distribution Probability Calculator
A continuous uniform distribution models outcomes that are equally likely across a fixed interval. This calculator helps you measure those probabilities fast. It works well for bounded time, length, cost, angle, and simulation inputs. You can estimate interval probability, density, cumulative probability, upper tail probability, and percentile location from one page.
Why this calculator is useful
Manual work is simple for one case. It becomes slow when you compare many ranges. This tool keeps the process organized. Enter the lower bound, upper bound, and your target value or interval. The calculator returns the main result and also shows mean, variance, standard deviation, median, and range. That makes review easier for students, analysts, and quality teams.
What the distribution means
If a random variable X follows a continuous uniform distribution from a to b, every value inside that interval has the same density. No single point has positive probability by itself. Probability comes from interval length. Longer intervals carry more probability. Values outside the bounds have zero density and fixed cumulative limits.
What you can calculate here
Use the between option for P(x1 ≤ X ≤ x2). Use less than for cumulative probability. Use greater than for the upper tail. Use density when you need the constant height of the distribution. Use percentile to convert a probability level into an x value. The result block appears above the form for quick checking and export.
Practical use cases
This calculator supports process control, scheduling, classroom exercises, reliability studies, and quick model validation. It is also helpful when you build examples for reports. The example table below shows how interval width changes probability. That is the key idea behind the continuous uniform model.
Export and review
After calculation, download the values as CSV for spreadsheet work. You can also save a PDF summary for records or sharing. The included formulas section explains exactly how each result is derived, so the page works as both a calculator and a learning aid.
Because the assumptions are transparent, it is easy to spot misuse. The lower bound must stay below the upper bound. Percentiles must stay between zero and one. Interval probability depends only on overlap with the valid range, which this calculator checks automatically.
FAQs
1. What does a continuous uniform distribution mean?
It means every value inside the interval [a, b] has the same density. Probabilities depend on interval width, not on any single exact point.
2. Can this calculator find probability between two values?
Yes. Choose the between option, enter x1 and x2, and the calculator measures the overlap with the valid support interval before computing probability.
3. Why is the density constant?
The model assumes all values in the interval are equally likely in density terms. That makes the density height fixed at 1 divided by the interval length.
4. What happens when x is outside the interval?
Density becomes zero outside [a, b]. The cumulative value is zero before a and one after b. Tail probability adjusts from that cumulative value.
5. Does a single exact value have probability?
No. For continuous distributions, the probability at one exact point is zero. Only intervals with positive width carry probability mass.
6. Can I use decimals and negative bounds?
Yes. The calculator accepts decimal inputs and negative values, as long as the upper bound b remains greater than the lower bound a.
7. What is the percentile feature for?
It converts a probability level p into the corresponding x value in the interval. For example, p = 0.75 gives the 75th percentile location.
8. What do the CSV and PDF exports include?
They include the selected calculation type, entered values, main result, summary statistics, support interval, and the formula line shown in the result block.