Measuring statistical dispersion reveals how tightly data points cluster around their central average versus how widely they spread across a distribution. This calculator evaluates numerical datasets in real time, computing both sample statistics (using Bessel's correction) and population statistics along with arithmetic mean and variance.

### Statistical Dispersion Formulas The calculator parses raw data values and computes key descriptive statistics: - **Data Count ($n$) & Mean ($\\bar{x}$)**: Sums all values and divides by count $n$: `mean = sum / n`. - **Sum of Squared Differences ($SS$)**: Calculates the squared distance of each observation from the mean: `sum((x - mean)^2)`. - **Sample Variance ($s^2$) & Sample Standard Deviation ($s$)**: Applies Bessel's correction dividing by $n - 1$: `s^2 = SS / (n - 1)` and `s = sqrt(s^2)`. Use this when analyzing a sample drawn from a larger group. - **Population Variance ($\\sigma^2$) & Population Standard Deviation ($\\sigma$)**: Divides sum of squares by the full population count $N$: `sigma^2 = SS / N` and `sigma = sqrt(sigma^2)`. Use this when measuring an entire complete population.

### Worked Dataset Example Enter the sample dataset: `12, 15, 18, 22, 25, 29, 31, 35, 40` (9 values): - **Sample Count ($n$)**: `9` - **Sum**: `227` - **Arithmetic Mean**: `227 / 9 = 25.22` - **Sum of Squared Diffs**: `661.56` - **Sample Variance ($s^2$)**: `661.56 / 8 = 82.69` - **Sample Standard Deviation ($s$)**: `sqrt(82.69) = 9.09` - **Population Standard Deviation ($\\sigma$)**: `sqrt(661.56 / 9) = sqrt(73.51) = 8.57`

### Private Client-Side Computation Dataset parsing, summation, and variance root calculations occur 100% inside your browser using client-side JavaScript. No financial records, experimental measurements, or personal data are transmitted across external networks.