Compute sample (s) and population (σ) standard deviation, variance, mean, and sum of squared deviations with full step-by-step arithmetic.
Using Sample (n - 1) with Bessel's correction to remove downward bias when estimating population parameters.
| # | Value (x) | Deviation (x - xฬ) | Squared (x - xฬ)² |
|---|---|---|---|
| Enter numbers above and click "Calculate Results" to inspect every step. | |||
| Total | 0 | Σ = 0.00 | SS = 0.00 |
Step 1: Find the mean by adding all numbers and dividing by the total count.
Step 2: Subtract the mean from each number to find the deviation.
Step 3: Square each deviation to remove negative signs.
Step 4: Sum the squared values to get the Sum of Squares (SS).
Step 5: Divide by degrees of freedom (n - 1) for variance, then take the square root for standard deviation.
In statistical analysis, standard deviation quantifies the dispersion or spread of data points around their arithmetic mean. When choosing between sample and population formulas:
Used when analyzing a subset or random sample drawn from a larger universe. Divides the sum of squared deviations by (n - 1) (Bessel's correction). This corrects the systematic tendency of sample variances to underestimate population variability. To see where a standardized value lands on the bell curve, follow up with our z-score calculator.
Used when the dataset represents the entire group under study (e.g. all 120 employees in an office). Divides by N because every member is known with zero sampling error.
A standard deviation of zero occurs if and only if every single number in the dataset is identical (e.g., 5, 5, 5, 5). It means there is zero variability or dispersion.
For normally distributed data, approximately 68.2% of all observations fall within ±1 standard deviation of the mean, 95.4% fall within ±2 standard deviations, and 99.7% fall within ±3 standard deviations.
Standard deviation measures the variability between individual data points. In contrast, standard error measures how accurately the sample mean estimates the true population mean (SEM = s / √n).
Free, 100% client-side educational solvers, Sarkari image resizers, and academic converters