Descriptive Statistics

Sample Variance

Compute sample variance using Bessel's correction (divide by n-1 instead of n) for an unbiased estimate of population variance from a sample.

Note: Using n-1 (Bessel's correction) gives an unbiased estimate of the population variance.

sample-variance Steps: 1. Find the sample mean x̄. 2. Compute each squared deviation (xᵢ - x̄)². 3. Sum the squared deviations (SS). 4. Divide by (n - 1) to get sample variance s².

Example

Use n minus 1 in the denominator for sample variance.

highlighted = computed this step

Step 1 — Data

List the sample values.

posx12243648510\begin{array}{cc}\text{pos} & \text{x} \\ 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ 4 & 8 \\ 5 & 10\end{array}

Step 2 — Mean

Compute the sample mean.

xˉ=30/5=6\bar{x}= 30 / 5 = \hl{6}

Step 3 — Deviation table

Fill the deviation table.

posxdevsq12−41624−2436004824510416\begin{array}{cccc}\text{pos} & \text{x} & \text{dev} & \text{sq} \\ 1 & 2 & \hlmath{-4} & \hlmath{16} \\ 2 & 4 & \hlmath{-2} & \hlmath{4} \\ 3 & 6 & \hlmath{0} & \hlmath{0} \\ 4 & 8 & \hlmath{2} & \hlmath{4} \\ 5 & 10 & \hlmath{4} & \hlmath{16}\end{array}

Step 4 — Sum squares

Add the squared deviations.

SS=16+4+0+4+16=40SS= 16 + 4 + 0 + 4 + 16 = \hl{40}

Step 5 — Variance

Divide by n minus 1 for sample variance.

s2=40/4=10s^{2}= 40 / 4 = \hl{10}

Step 6 — Result

State the sample variance.

s2=10s^{2}= \hl{10}