Sampling Variance Calculator
Choose raw data, a frequency table, or summary statistics. The calculator returns sample variance, population variance, standard error, and sampling variance of the mean.
Example Data Table
This worked example uses raw values 4, 6, 8, 10, 12. The mean is 8, the sum of squared deviations is 40, and the sample variance is 10.
| Observation | Value | Deviation from Mean | Squared Deviation |
|---|---|---|---|
| 1 | 4 | -4 | 16 |
| 2 | 6 | -2 | 4 |
| 3 | 8 | 0 | 0 |
| 4 | 10 | 2 | 4 |
| 5 | 12 | 4 | 16 |
| Sum of Squared Deviations | 40 | ||
Formula Used
s² = Σ(xᵢ − x̄)² / (n − 1)
σ² = Σ(xᵢ − μ)² / n
Var(x̄) = s² / n
SE = s / √n
FPC = √((N − n) / (N − 1))Adjusted Var(x̄) = (s² / n) × ((N − n) / (N − 1))
The calculator uses the unbiased estimator for sample variance when your data represent a sample. It also reports the n-denominator version for direct comparison with population variance.
How to Use This Calculator
- Select the calculation method that matches your available information.
- Enter raw data, a frequency table, or summary inputs.
- Add a population size only when you want finite population correction.
- Choose the number of decimal places for the displayed results.
- Press the submit button to view results above the form.
- Download the output as CSV or PDF for documentation.
Frequently Asked Questions
1. What does sampling variance measure?
Sampling variance measures how much sample observations or sample means vary around their average. Higher values show wider spread and lower precision.
2. Why does the sample variance use n minus 1?
Using n minus 1 corrects bias when estimating population variance from a sample. This adjustment is called Bessel’s correction.
3. When should I use the frequency table option?
Use the frequency option when repeated values are already grouped. It saves time and produces the same variance measures as expanded raw data.
4. What is the variance of the sample mean?
It is the sampling variance of the estimator x̄. The calculator estimates it as sample variance divided by sample size.
5. What does the finite population correction do?
It reduces variance and standard error when sampling without replacement from a limited population. The effect increases as the sample becomes a larger share of the population.
6. Can I calculate variance from summary inputs only?
Yes. Enter sample size, choose whether your value is variance or standard deviation, and the calculator derives related measures.
7. Why is coefficient of variation sometimes blank?
The coefficient of variation requires a nonzero mean. If the mean is missing or zero, the percentage ratio would not be meaningful.
8. Are the CSV and PDF files suitable for reports?
Yes. They summarize the displayed metrics clearly, which makes them useful for homework, analysis notes, audit trails, and research appendices.