Combined Mean & Variance

中文辅助说明

從平均數與變異數的定義逐步移項，推導 Σx = n × 平均數及 Σx² = n ×（變異數 + 平均數²），將兩組總和合併後計算平均數、變異數與標準差。Learn 可修改數據並查看推導與列式；Practice 可練習計算，提交後自行展開解法。本工具採用除以 n 的變異數定義，不是除以 n−1 的不偏估計量。

Stella tests 60 light bulbs: mean life 650 hours, standard deviation 8 hours. Sol tests another 80 bulbs: mean life 660 hours, standard deviation 7 hours. Find the overall mean and standard deviation.

Summary of the two data sets
Group	Size $n$	Mean $\bar{x}$	Standard deviation
A
B

Edit either group to explore. All observations are combined into one data set.

Variance uses divisor $n$ , not $n-1$ . If a standard deviation is given, square it first.

Combined mean

Combined variance

Combined standard deviation

1. Recover the totals

Rearrange the definitions of the mean and variance for each group.

From the mean

Multiply both sides by $n$ .

\displaystyle\begin{aligned}\bar{x}&=\frac{\sum x}{n}\\[8pt]n\bar{x}&=\sum x\\[8pt]\therefore\quad\sum x&=n\bar{x}\end{aligned}

From the variance

Add $\bar{x}^2$ to both sides, then multiply by $n$ .

\displaystyle\begin{aligned}\sigma^2&=\frac{\sum x^2}{n}-\bar{x}^2\\[8pt]\sigma^2+\bar{x}^2&=\frac{\sum x^2}{n}\\[8pt]n(\sigma^2+\bar{x}^2)&=\sum x^2\\[8pt]\therefore\quad\sum x^2&=n(\sigma^2+\bar{x}^2)\end{aligned}

Group	$n$	$\sum x$	$\sum x^2$

2. Combine, then calculate

Use the unrounded values at each step. Final answers below are to 3 significant figures.