Calculator Inputs
Formula Used
Bias definition
Bias(T) = E[T] - θ
Absolute bias
|Bias(T)| = |E[T] - θ|
Relative bias percentage
Relative Bias (%) = ((E[T] - θ) / θ) × 100
Mean squared error relationship
MSE(T) = Var(T) + [Bias(T)]²
This calculator assumes you already know the estimator's expected value. It does not derive expectation from raw observations.
How to Use This Calculator
Enter the expected value of your estimator in the E[T] field. Then enter the true parameter value you want to estimate.
Optionally add variance, mean squared error, sample size, units, and custom symbols. These fields unlock extra interpretation and consistency checks.
Press Calculate Bias. The result block appears above the form and below the header, showing bias, direction, squared bias, and MSE diagnostics.
Review the Plotly graph to compare expectation, target, and error size. Use the export buttons to save the results as CSV or PDF.
Example Data Table
| Estimator | Expected Value E[T] | True Parameter θ | Bias | Status |
|---|---|---|---|---|
| Sample Mean | 10.00 | 10.00 | 0.00 | Unbiased |
| Shifted Mean | 10.30 | 10.00 | 0.30 | Biased high |
| Shrinkage Estimator | 9.50 | 10.00 | -0.50 | Biased low |
| Biased Variance Form | 7.20 | 8.00 | -0.80 | Biased low |
These rows illustrate the bias formula with simple numerical values.
Frequently Asked Questions
1. What is estimator bias?
Bias is the difference between an estimator's expected value and the true parameter. Zero bias means the estimator is unbiased on average.
2. Why can a useful estimator be biased?
A biased estimator may still have low variance, stability, or smaller mean squared error. Analysts often balance bias and variance instead of demanding zero bias.
3. What does positive bias mean?
Positive bias means the estimator tends to overestimate the parameter. Negative bias means it tends to underestimate the parameter.
4. Is zero bias always best?
Not always. An unbiased estimator can still be noisy. A slightly biased estimator may outperform it when variance is much smaller.
5. How does bias relate to MSE?
Mean squared error equals variance plus squared bias. This decomposition shows overall estimator quality, not only directional average error.
6. Can sample size affect bias?
Yes. Some estimators are biased in small samples but become nearly unbiased as sample size grows. This is asymptotic unbiasedness.
7. What inputs are required here?
Enter the estimator's expected value and the true parameter. Variance and mean squared error are optional, but they add diagnostics.
8. Does this tool derive E[T] from raw data?
No. This page evaluates bias from values you supply. Derive the estimator expectation first, then enter it here.