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Calculate standard deviation, variance, mean, and other statistical measures for any data set. Supports both population and sample calculations with step-by-step solutions.
Enter numbers separated by commas, spaces, or new lines. Decimal values are supported.
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Use when you have a sample and want to estimate the population standard deviation
Use when you have the entire population data or only care about these specific values
This calculator provides statistical calculations for educational and informational purposes. Always verify important calculations.
Results are rounded for display but calculations use full precision. For sample standard deviation, n-1 (Bessel's correction) is used.
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a data set. A low standard deviation indicates that data points are close to the mean, while a high standard deviation indicates that data points are spread out over a wider range. It's one of the most commonly used statistics in data analysis, research, quality control, and finance.
There are two types of standard deviation depending on your data:
Population Standard Deviation (σ) – Use when you have data for the entire population or when you only care about the specific data set at hand. Divides by N (total count).
Sample Standard Deviation (s) – Use when you have a sample from a larger population and want to estimate the population's standard deviation. Divides by n-1 (Bessel's correction) to reduce bias.
σ = √[Σ(xi - μ)² / N]Where σ is the population standard deviation, μ is the population mean, xi is each value, and N is the total count
s = √[Σ(xi - x̄)² / (n-1)]Where s is the sample standard deviation, x̄ is the sample mean, xi is each value, and n-1 is Bessel's correction
Population standard deviation (σ) is used when you have data for the entire population and divides by N. Sample standard deviation (s) is used when you have a sample from a larger population and divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
No, standard deviation cannot be negative. Since it involves squaring the deviations (which makes them positive) and then taking a square root, the result is always zero or positive. A standard deviation of 0 only occurs when all values in the data set are identical.
A standard deviation of zero means there is no variation in the data—all values are exactly the same. For example, the data set {5, 5, 5, 5, 5} has a standard deviation of 0 because every value equals the mean.
Sample standard deviation uses n-1 (called Bessel's correction) to correct for bias when estimating population standard deviation from a sample. Using n would consistently underestimate the true population standard deviation. The n-1 divisor produces an unbiased estimate.
There's no universal 'good' or 'bad' standard deviation—it depends entirely on context. A better measure is the coefficient of variation (CV = SD/mean × 100%), which expresses variability relative to the mean. In quality control, lower SD is typically desired. In finance, some volatility (SD) may be acceptable for higher returns.
Variance is the square of standard deviation (or conversely, standard deviation is the square root of variance). Variance is measured in squared units, while standard deviation is in the original units of measurement, making it more interpretable.
Standard Error of the Mean (SEM) measures how precisely the sample mean estimates the population mean. It equals the standard deviation divided by the square root of the sample size (SEM = s/√n). Smaller SEM indicates more precise estimation.
For normally distributed data, approximately 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This rule helps interpret what standard deviation values mean in practical terms.