Standard Deviation Calculator

Calculate population and sample standard deviation, variance, and mean of a dataset.

sigma = sqrt[sum(x - mu)^2 / n] (population); s = sqrt[sum(x - xbar)^2 / (n-1)] (sample); Variance = SD^2
Data: 10, 12, 23, 23, 16, 23, 21, 16 -> Mean = 18, Population SD = 4.90, Sample SD = 5.24, Variance = 24.00

What does standard deviation tell you?

Standard deviation measures how spread out data is from the mean. Low SD = data clustered near average. High SD = data widely dispersed. Example: Test scores - Mean 75, SD 5: Most scores 70-80 (tight grouping). Mean 75, SD 20: Scores 55-95 (wide spread). Used in: Quality control, finance (volatility), sports stats, science. 68% of data falls within 1 SD of mean in normal distribution.

What is the difference between population and sample standard deviation?

Population (sigma): Divide by n. Use when you have ALL data points (entire population). Sample (s): Divide by n-1. Use when you have subset of data (sample). Sample SD is slightly larger (corrects for sample bias). Example: Testing all 100 employees = population. Testing 30 of 100 = sample. Most real-world scenarios use sample SD. Calculator shows both.

What is variance and how does it relate to standard deviation?

Variance (sigma^2) = average of squared differences from mean. Standard deviation = sqrtvariance. Both measure spread, but variance is in squared units. Example: Heights in inches, variance in inches^2. SD brings it back to original units (inches). Formula: Variance = sum(x - mean)^2 / n; SD = sqrtvariance. SD more intuitive; variance used in advanced statistics (ANOVA, regression).

How do I interpret standard deviation in finance?

In finance, SD = volatility (risk). Low SD = stable, predictable returns. High SD = volatile, risky. Example: Stock A returns 8% +- 5% SD, Stock B returns 8% +- 20% SD. Both average 8%, but B is riskier. Sharpe ratio = (Return - Risk-free rate) / SD measures risk-adjusted return. Bonds have low SD (~3-5%), stocks higher (~15-20%), crypto very high (~50%+).