Method of Moments Estimator Uniform Distribution Calculator

Calculator Inputs

Uniform model

Decimal places

Accepted separators

Sample data

Enter raw observations only. The calculator derives sample moments automatically.

Model support

Estimate parameters for U(0, θ), U(-θ, θ), and U(a, b) from raw sample values.

Moment calculations

The page computes the first raw moment, second raw moment, and moment variance automatically.

Export options

Download a CSV summary or create a quick PDF report after each calculation.

Visual output

Plotly shows your sample points together with the estimated bounds or parameter lines.

Formula Used

Uniform U(0, θ)

Population mean: E[X] = θ / 2

Method of moments estimator: θ̂ = 2x̄

Uniform U(-θ, θ)

Second raw moment: E[X²] = θ² / 3

Method of moments estimator: θ̂ = √(3m2)

Uniform U(a, b)

Mean: E[X] = (a + b) / 2

Variance: Var(X) = (b - a)² / 12

Estimators: â = x̄ - √(3v), b̂ = x̄ + √(3v)

Here x̄ is the sample mean, m2 is the sample second raw moment, and v is the moment variance computed with divisor n.

How to Use This Calculator

Choose the uniform distribution form that matches your statistical model.
Paste sample observations using commas, spaces, semicolons, or separate lines.
Set the number of decimal places you want in the output.
Press Calculate Estimator to display the result above the form.
Review the moment values, estimator output, and notes about model fit.
Use the CSV or PDF buttons to export the current result summary.

Example Data Table

Example	Sample Data	Suggested Model	Sample Mean	Example MOM Estimate
Example A	2, 4, 5, 6, 7, 8, 9	U(0, θ)	5.8571	θ̂ = 11.7143
Example B	-3, -2, -1, 0, 1, 2, 3	U(-θ, θ)	0.0000	θ̂ ≈ 3.4641
Example C	3, 5, 6, 7, 8, 10, 11	U(a, b)	7.1429	â ≈ 2.9854, b̂ ≈ 11.3003

Plain HTML FAQs

1) What is the method of moments estimator?

It is a parameter estimate found by matching sample moments to population moments. You replace theoretical expressions such as E[X] or E[X²] with sample counterparts and solve for the unknown parameter values.

2) find the method of moments estimator of θ

For X ~ U(0, θ), the first moment is E[X] = θ/2. Set x̄ = θ/2 and solve. The method of moments estimator is θ̂ = 2x̄.

3) find the method of moments estimator for gamma distribution

For Gamma(shape k, scale θ), use x̄ = kθ and s² = kθ². Then k̂ = x̄² / s² and θ̂ = s² / x̄. Using raw moments gives the same result after simplification.

4) Why does the U(a, b) model use both mean and variance?

The interval U(a, b) has two unknown parameters. One sample moment is not enough. Matching the sample mean and moment variance provides two equations, which lets you solve for both a and b.

5) Can the method of moments estimate fall outside the sample range?

Yes. Unlike maximum likelihood estimates for uniform models, method of moments estimates are based on moment matching, not strict support coverage. That can produce an estimate smaller than the observed maximum or slightly wider than the data range.

6) What is the difference between MOM and maximum likelihood here?

Method of moments matches sample moments. Maximum likelihood chooses values that make the observed sample most probable. For uniform distributions, the two methods often give different estimates, especially for small samples.

7) What sample format should I enter?

Enter raw observations separated by commas, spaces, semicolons, or line breaks. The calculator ignores empty separators and computes the moments automatically from the numeric entries.

8) When should I use the U(-θ, θ) option?

Choose U(-θ, θ) when the variable is symmetric around zero and the support is equally wide on both sides. Then the second raw moment gives a direct method of moments estimate for θ.