Example
Use all population values in a deviation table.
highlighted = computed this step
Step 1 — Data
List the population values.
pos x 1 2 2 4 3 4 4 4 5 5 6 5 7 7 8 9 \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} pos 1 2 3 4 5 6 7 8 x 2 4 4 4 5 5 7 9
Step 2 — Mean
Compute the population mean.
μ = 40 / 8 = 5 \mu= 40 / 8 = \hl{5} μ = 40/8 = 5
Step 3 — Deviation table
Fill the deviation table.
pos x dev sq 1 2 − 3 9 2 4 − 1 1 3 4 − 1 1 4 4 − 1 1 5 5 0 0 6 5 0 0 7 7 2 4 8 9 4 16 \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} pos 1 2 3 4 5 6 7 8 x 2 4 4 4 5 5 7 9 dev − 3 − 1 − 1 − 1 0 0 2 4 sq 9 1 1 1 0 0 4 16
Step 4 — Sum squares
Add the squared deviations.
S S = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32 SS= 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = \hl{32} SS = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
Step 5 — Variance
Divide by N for population variance.
σ 2 = 32 / 8 = 4 \sigma^{2}= 32 / 8 = \hl{4} σ 2 = 32/8 = 4
Step 6 — Standard deviation
Take the square root for standard deviation.
σ = 4 = 2 \sigma=\sqrt{ 4 }= \hl{2} σ = 4 = 2
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.