Calculate population and sample standard deviation, variance, mean, sum of squares, and margin of error with complete step-by-step computation tables.
Standard deviation is a foundational statistical measurement that quantifies the amount of variation, dispersion, or spread of a set of numerical data values relative to their arithmetic mean.
Used when the dataset represents the entire population of interest (every single subject or measurement). Divides the sum of squared deviations by total size N:
Used when the dataset is a sample drawn from a larger population. Applies Bessel's correction (n - 1) to correct for sample variance underestimation:
For normally distributed datasets (symmetric bell curve):
Dividing by (n - 1) is known as Bessel's correction. Because sample data tends to cluster more closely around the sample mean than the true population mean, dividing by n underestimates population variance. Using (n - 1) provides an unbiased estimator of population variance.
Variance is the average of the squared differences from the mean, expressed in squared units (e.g., dollars²). Standard deviation is the square root of variance, returning the measurement back to original units (e.g., dollars), making it easier to interpret.
Standard error estimates the variability of sample means if multiple samples were drawn from the population. It is calculated as SE = s / √n. As sample size n increases, SE decreases.
No. Standard deviation is always greater than or equal to zero (σ ≥ 0). A standard deviation of zero indicates that all data points in the dataset are identical.