Calculator Input
Use the responsive grid below. It shows three columns on large screens, two on medium screens, and one on mobile.
Example Data Table
Use this sample dataset to test the variance or mean mode. Paste the values into the sample data field.
| Observation | Value | Observation | Value |
|---|---|---|---|
| 1 | 12.4 | 7 | 14.4 |
| 2 | 15.2 | 8 | 13.9 |
| 3 | 14.8 | 9 | 16.6 |
| 4 | 13.7 | 10 | 15.8 |
| 5 | 16.1 | 11 | 14.9 |
| 6 | 15.0 | 12 | 13.6 |
Proportion example: 42 successes out of 60 trials. Poisson example: 3, 4, 2, 5, 1, 4 with exposure per interval set to 1.
Formula Used
Population Mean
x̄ = Σx / n
The sample mean is an unbiased estimator of the population mean when observations are randomly sampled and identically distributed.
Population Variance
s² = Σ(x - x̄)² / (n - 1)
Dividing by n - 1 removes the downward bias produced by dividing by n. This is the classic Bessel correction.
Population Proportion
p̂ = x / n
The observed proportion of successes is an unbiased estimator for the true Bernoulli or binomial success probability.
Poisson Rate
λ̂ = total counts / total exposure
For count data observed over equal or unequal exposure, this estimator gives the unbiased rate per unit exposure.
How to Use This Calculator
- Select the estimator mode that matches your parameter.
- Enter raw sample values, trial counts, or Poisson interval counts.
- Choose the confidence level and preferred decimal precision.
- Click Calculate Unbiased Estimate to view the result above the form.
- Review the summary cards, the detailed metrics table, and the Plotly chart.
- Download the output as CSV or PDF for reporting or teaching.
Why Unbiased Estimators Matter
An unbiased estimator targets the true parameter on average across repeated samples. That does not mean every sample lands exactly on the truth, but it avoids a consistent upward or downward pull. In practice, this is especially important for variance estimation, because the naive divisor n understates population variability.
Frequently Asked Questions
1) What is an unbiased estimator?
An unbiased estimator has an expected value equal to the true population parameter. Across many repeated samples, it does not systematically overestimate or underestimate the target quantity.
2) Why does the variance formula divide by n - 1?
Using the sample mean consumes one degree of freedom. Dividing by n - 1 corrects the downward bias that appears when variance is computed with n in the denominator.
3) Is the sample mean already unbiased?
Yes. Under standard random sampling assumptions, the sample mean is an unbiased estimator of the population mean. The calculator still adds interval estimates and supporting diagnostics.
4) Does unbiased always mean best?
No. An unbiased estimator can still have high variance. Some biased estimators may produce lower mean squared error, so estimator choice depends on the problem and the loss function.
5) Can I use this for success rates?
Yes. Choose the proportion mode, enter successes and total trials, and the calculator returns the unbiased proportion estimate with a percentage and interval.
6) What does the Poisson rate option estimate?
It estimates event rate per unit exposure. This is useful for defects per hour, arrivals per minute, incidents per day, or counts observed over equal intervals.
7) Are the confidence intervals exact?
They are approximate large-sample intervals. They work well in many routine settings, but very small samples or extreme proportions may require specialized exact methods.
8) What sample issues can weaken the result?
Small samples, strong dependence, selection bias, recording errors, and nonrandom sampling can all reduce reliability. Good input quality matters as much as the formula itself.