Descriptive Statistics

Population Variance and Standard Deviation

Compute population variance and standard deviation using all N values in the population: σ² = Σ(x - μ)² / N, σ = √σ²

population-variance-and-std Steps: 1. Find the mean μ. 2. Compute each squared deviation (xᵢ - μ)². 3. Sum the squared deviations (SS). 4. Divide by N to get variance σ². 5. Take the square root for std dev σ.

Example

Use all population values in a deviation table.

highlighted = computed this step

Step 1 — Data

List the population values.

posx1224344455657789\begin{array}{cc}\text{pos} & \text{x} \\ 1 & 2 \\ 2 & 4 \\ 3 & 4 \\ 4 & 4 \\ 5 & 5 \\ 6 & 5 \\ 7 & 7 \\ 8 & 9\end{array}

Step 2 — Mean

Compute the population mean.

μ=40/8=5\mu= 40 / 8 = \hl{5}

Step 3 — Deviation table

Fill the deviation table.

posxdevsq12−3924−1134−1144−1155006500772489416\begin{array}{cccc}\text{pos} & \text{x} & \text{dev} & \text{sq} \\ 1 & 2 & \hlmath{-3} & \hlmath{9} \\ 2 & 4 & \hlmath{-1} & \hlmath{1} \\ 3 & 4 & \hlmath{-1} & \hlmath{1} \\ 4 & 4 & \hlmath{-1} & \hlmath{1} \\ 5 & 5 & \hlmath{0} & \hlmath{0} \\ 6 & 5 & \hlmath{0} & \hlmath{0} \\ 7 & 7 & \hlmath{2} & \hlmath{4} \\ 8 & 9 & \hlmath{4} & \hlmath{16}\end{array}

Step 4 — Sum squares

Add the squared deviations.

SS=9+1+1+1+0+0+4+16=32SS= 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = \hl{32}

Step 5 — Variance

Divide by N for population variance.

σ2=32/8=4\sigma^{2}= 32 / 8 = \hl{4}

Step 6 — Standard deviation

Take the square root for standard deviation.

σ=4=2\sigma=\sqrt{ 4 }= \hl{2}