Explore our collection of 150+ developer tools, PDF editors, and image converters. Everything runs locally in your browser.
Ubify Financial Advisory Board
FINANCIAL ANALYSTS & CPA REVIEWERS
About This Tool
When analyzing summarized research papers, quality audits, or financial reports, raw individual observations are often unavailable. If you already have the total sample size and sum of squared deviations, this calculator determines exact sample variance, sample standard deviation, population dispersion metrics, and standard error of the mean without needing the underlying raw data rows.
### How Summary Statistic Formulas Work The calculator evaluates variance and dispersion from three aggregate summary inputs: - **Sample Size ($N$)**: Total count of observed data points or audit records. - **Sample Mean ($\\bar{x}$)**: Arithmetic average of the dataset. - **Sum of Squared Deviations ($SS$)**: Total squared distance $\\sum(x - \\bar{x})^2$ around the sample mean. - **Sample Variance ($s^2$)**: Applies degrees of freedom $N - 1$: `s^2 = SS / (N - 1)`. - **Sample Standard Deviation ($s$)**: Square root of sample variance: `s = sqrt(sampleVariance)`. - **Population Variance ($\\sigma^2$) & Sigma ($\\sigma$)**: Divides by full sample size $N$: `sigma^2 = SS / N` and `sigma = sqrt(sigma^2)`. - **Standard Error of the Mean ($SE$)**: Quantifies sampling error: `SE = s / sqrt(N)`.
### Worked Statistical Example Given sample size **$N = 10$**, sample mean **$\\bar{x} = 75.40$**, and sum of squared deviations **$SS = 342.80$**: - **Sample Variance**: `342.80 / (10 - 1) = 342.80 / 9 = 38.089` - **Sample Standard Deviation ($s$)**: `sqrt(38.089) = 6.172` - **Population Variance**: `342.80 / 10 = 34.280` - **Population Standard Deviation ($\\sigma$)**: `sqrt(34.280) = 5.855` - **Standard Error ($SE$)**: `6.172 / sqrt(10) = 6.172 / 3.162 = 1.952`
### Raw Dataset vs Summary Stats If you have a list of raw numbers (such as `12, 15, 18, 22...`) rather than precomputed summary figures, use our [Dataset Standard Deviation Calculator](/standard-deviation-calculator).
### In-Browser Computation All degrees of freedom divisions, square roots, and standard error calculations execute locally in your client web browser with zero server data transfer.
How to Use
- 1
Enter your total sample count (N) into the Sample Size field.
- 2
Enter the arithmetic average into the Sample Mean field.
- 3
Enter the precomputed sum of squared deviations (SS).
- 4
Review the immediate sample standard deviation (s), sample variance, population sigma, and standard error of the mean.
Frequently Asked Questions
Why does sample variance divide by N - 1 instead of N?
Dividing by N - 1 (known as Besselโs correction) removes downward bias when estimating population variance from an incomplete sample.
How does this tool differ from the raw dataset standard deviation calculator?
This calculator accepts pre-aggregated summary statistics (N and SS), whereas the raw dataset calculator at /standard-deviation-calculator parses individual numbers from a text field.
What is the purpose of the Standard Error of the Mean (SE)?
Standard error measures how much your sample mean is expected to vary from the true population mean across repeated sampling experiments.