Unbiased and Biased Estimator Calculator

Test estimators from raw samples with practical comparisons. Measure bias, variance, and squared error instantly. See which estimator fits known parameters more reliably today.

Calculator Form

Enter a numeric sample. Add known population values when available. The calculator compares theoretical and empirical estimator behavior.

Sample Data

Use commas, spaces, semicolons, or new lines.

Known Population Mean (optional)

Used to estimate empirical mean bias and squared error.

Known Population Variance (optional)

Used to compare biased and unbiased variance estimates.

Decimal Precision

Plot Title

Estimator Focus

This version compares the most common estimator pair.

Example Data Table

This sample shows how the corrected variance differs from the uncorrected version.

Case	Sample Values	Known Mean	Known Variance	Biased Variance	Unbiased Variance
Example A	12, 15, 18, 21, 24, 27	20	30	26.2500	31.5000
Example B	8, 9, 10, 11, 12	10	2.5	2.0000	2.5000
Example C	4, 7, 9, 10, 14, 16	10	18	14.8056	17.7667

Formula Used

Sample mean estimator:
x̄ = (Σxᵢ) / n

Biased sample variance:
s²_biased = Σ(xᵢ − x̄)² / n

Unbiased sample variance:
s²_unbiased = Σ(xᵢ − x̄)² / (n − 1)

Empirical bias against a known parameter:
Bias = estimate − true value

Squared error against a known parameter:
Squared Error = (estimate − true value)²

The sample mean is an unbiased estimator of the population mean. The variance formula with divisor n is biased downward. The divisor n−1 corrects that bias for repeated sampling.

How to Use This Calculator

Enter your sample values in the data box.
Optionally enter a known population mean.
Optionally enter a known population variance.
Choose how many decimals you want displayed.
Click Calculate Estimators.
Review the result card above the form.
Use the chart to inspect sample behavior visually.
Download the current result as CSV or PDF.

Frequently Asked Questions

1. What is an unbiased estimator?

An unbiased estimator has an expected value equal to the true parameter. Across many samples, it neither overestimates nor underestimates on average.

2. What is a biased estimator?

A biased estimator systematically misses the true parameter on average. The bias can be positive or negative depending on the formula used.

3. Why does variance use n−1 for correction?

Using n−1 compensates for estimating the mean from the same sample. That adjustment removes the average downward bias in sample variance.

4. Is the sample mean always unbiased?

Under standard random sampling assumptions, yes. The sample mean is unbiased for the population mean, though single samples can still differ from the truth.

5. What if I do not know the population parameter?

You can still compare the formulas theoretically. Empirical bias and squared error require a known benchmark, but the corrected variance remains theoretically unbiased.

6. Does unbiased always mean better?

Not always. A slightly biased estimator can sometimes have lower total error. Analysts often compare both bias and variance before choosing an estimator.

7. What does squared error show?

Squared error measures how far one estimate is from a known target. Larger values indicate poorer accuracy for that single observed sample.

8. Can this tool handle decimals and negatives?

Yes. Enter any valid numeric values, including decimals and negative numbers. Keep separators consistent for reliable parsing and output.