Statistics

$x̄ = (x₁ + x₂ + ⋯ + xₙ)/n = Σxᵢ/n$

Sum of all values divided by the number of values.

$x̄ = Σfᵢxᵢ / Σfᵢ$

$fᵢ =$ frequency of class; $xᵢ =$ class mark (midpoint).

$MD(x̄) = Σ|xᵢ − x̄| / n (ungrouped)$

Average of the absolute deviations of all observations from the mean x̄. Uses the modulus so that positive and negative deviations don't cancel (they always $\sum$ to exactly zero otherwise).

$MD(M) = Σ|xᵢ − M| / n$

Mean deviation about the median is minimised (least compared to any other point).

$σ² = Σ(xᵢ − x̄)² / n$

Average of squared deviations from the mean.

$σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ$

Variance for discrete or continuous frequency distributions; xᵢ is the value (discrete) or class mark (continuous).

$σ = \sqrt{σ²} = \sqrt{Σ(xᵢ−x̄}²/n)$

The positive square root of variance; same units as data.

$CV = (σ/x̄) × 100$

Relative measure of dispersion; used to compare variability of two datasets.

$σ² = (Σxᵢ²/n) − x̄²$

Computationally efficient; avoids computing deviations explicitly.

$σ² = h² [ (Σfᵢuᵢ²/N) − (Σfᵢuᵢ/N)² ], uᵢ = (xᵢ − A)/h$

Used for grouped/continuous data with a common class width h, especially when class marks are large numbers — reduces arithmetic by working with small integers uᵢ.