Standard Deviation Calculator

Variance is the average sum of squared deviations from the mean — it expresses data spread but in squared units. Standard deviation is the square root of variance and is expressed in the same units as the data, making it much easier to interpret. For example, if measuring heights in cm, variance is in cm² but standard deviation is in cm.

Use the divisor n−1 (Bessel's correction) when analysing a random sample — you want to estimate the standard deviation of the whole population from a subset. Use n when you have the complete population (all possible observations). For large n the difference is minimal, but for small samples (e.g. n=5) it matters significantly.

Standard deviation σ tells you how far values typically deviate from the mean. Small σ means data clusters near the mean (low variability). Large σ indicates high variability. Example: the set {5, 5, 5, 5} has σ=0, while {1, 5, 5, 9} has σ≈2.83, even though both have the same mean (5).

For a normal distribution: about 68% of observations fall within (mean ± 1σ), about 95% within (mean ± 2σ), and about 99.7% within (mean ± 3σ). This rule helps quickly assess whether a value is "typical" or "unusual". A value more than 3σ from the mean is considered an outlier.

Step 1: Calculate the arithmetic mean. Step 2: For each value, find the difference from the mean and square it. Step 3: Sum all the squared deviations. Step 4: Divide by n (population) or n−1 (sample) — this is the variance. Step 5: Take the square root of the variance — this is the standard deviation.

Standard deviation is widely used in finance (measuring investment risk, price volatility), medicine (analysis of clinical trial results), manufacturing quality control (checking dimensional tolerances), social sciences (analysing survey and test results), and meteorology (temperature variability). Anywhere data spread needs to be quantified.

Standard deviation is the square root of variance, and variance is a sum of squared deviations from the mean — a squared number is always non-negative. Therefore variance ≥ 0 and standard deviation ≥ 0. A value of zero means all elements are identical (no spread). The minimum value is 0; negative standard deviation does not exist.

Standard deviation (SD) averages squared deviations then takes the square root — it is sensitive to outliers because squaring "penalises" large deviations heavily. Median absolute deviation (MAD) computes the median of absolute deviations from the median — it is robust to outliers. For data with extreme values, MAD is often a better measure of spread.

The coefficient of variation (CV) is the standard deviation divided by the absolute value of the mean, expressed as a percentage: CV = (σ / |x̄|) × 100%. It allows comparing variability of datasets with different units or scales. For example, CV=10% indicates low variability, CV>30% indicates high variability.

Yes — the calculator accepts any real numbers including negatives (e.g. −3, −1.5) and decimals (e.g. 2.5). Numbers should be entered separated by commas or spaces, e.g. "−2, 0, 2, 4". Standard deviation is calculated correctly for all such inputs. A minimum of 2 elements is needed when using the sample formula (n−1).