Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion in a data set. This calculator helps you compute both sample and population standard deviation with detailed step-by-step explanations, making it perfect for students, researchers, and professionals working with statistical data.
Separate with commas, spaces, or new lines
Sample: when data is a subset. Uses n-1 (Bessel's correction)
Statistics for Test Scores:
The 68-95-99.7 rule: In a normal distribution, 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD.
Type or paste your numbers into the data field. Separate them with commas, spaces, or new lines.
Select sample standard deviation (most common) or population standard deviation based on your data.
Get your standard deviation, mean, variance, and a step-by-step breakdown of the calculation.
Standard deviation measures how spread out data points are from the mean. For samples, we use (n-1) as the divisor (Bessel's correction) to get an unbiased estimate.
σ = √[Σ(xi - μ)² / N] or s = √[Σ(xi - x̄)² / (n-1)]
Sample standard deviation (s) uses n-1 in the denominator and is used when your data is a sample from a larger population.
Population standard deviation (σ) uses n in the denominator and is used when your data represents the entire population.
A low standard deviation indicates data points are clustered close to the mean, while a high standard deviation indicates data is spread out.
The variance is simply the standard deviation squared (σ² or s²).
For normally distributed data, about 68% of values fall within 1 standard deviation of the mean, and 95% within 2 standard deviations.
Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion in a data set. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that data points are spread out over a wider range of values.
The key difference lies in the denominator: sample standard deviation divides by (n-1) while population standard deviation divides by n. Use sample standard deviation when your data is a subset of a larger population (which is most common in real-world scenarios). Use population standard deviation only when you have data for every member of the entire population you're studying.
For population: σ = √[Σ(xi - μ)² / N], where μ is the population mean and N is the population size. For sample: s = √[Σ(xi - x̄)² / (n-1)], where x̄ is the sample mean and n is the sample size. The division by (n-1) in the sample formula is called Bessel's correction and provides an unbiased estimate of the population standard deviation.
Standard deviation helps you understand data variability. In a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations (the 68-95-99.7 rule). Compare standard deviations across different data sets to understand which has more variability.
Data sourced from trusted institutions
All formulas verified against official standards.