Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation is a measure of dispersion that indicates how spread out the data values are relative to their mean. A low standard deviation means the data points cluster tightly around the average, while a high standard deviation indicates greater variability in the dataset.

Population vs. Sample Standard Deviation

Population standard deviation (sigma): Used when you have data for every member of the entire population. The denominator is N. Sample standard deviation (s): Used when your data is a subset of the population. The denominator is N-1 (Bessel's correction) to produce an unbiased estimate.

Formulas

Mean: mu = Sum of x / n. Population variance: sigma squared = Sum of (x - mu) squared / n. Sample variance: s squared = Sum of (x - mu) squared / (n - 1). Standard deviation equals the square root of the variance.

Frequently Asked Questions

When should I use population vs. sample?

If your data includes every member of the group you are studying, use population standard deviation. If you collected a sample from a larger group, use sample standard deviation to account for sampling variability.

What are common applications?

Standard deviation is used in financial risk analysis, quality control in manufacturing, scientific research, grading curves in education, and any field where understanding data variability is important.