不一致的面包师

如果面包师输出的标准差为 $10$，则方差为 $\sigma_{bakery}^2 = 10^2=100$

由于面包店的总产量是独立随机变量的总和，因此制作的甜甜圈总数的方差就是方差之和： $$\sigma_{bakery}^2 = 25\cdot100=2500$$ 我们对平均值感兴趣，可以将其表示为总产量除以面包师数量。将数字缩放 $n$ 会缩放 $n^2$ 的方差，因此将总输出缩放 $\frac1{25}$ 来求平均值意味着烘焙甜甜圈的平均数量的方差将缩放 $\frac1{625}$。 $$\sigma_{\mu}^2 = \frac{2500}{625} = 4 \rightarrow \sigma_{\mu} = \boxed2$$ （请注意，此过程与将 $\sigma$ 除以 $\sqrt{n}$ 相同，其中 $n$ 是群体大小）

---

Original Explanation

If the standard deviation of a baker's output is $10$, the variance is $\sigma_{bakery}^2 = 10^2=100$

Since the total output of the bakery is the sum of independent random variables, the variance in the total number of donuts made is the sum of the variances:

$$\sigma_{bakery}^2 = 25\cdot100=2500$$

We are interested in the average, which is can be notes as total output over number of bakers. Scaling a number by $n$ scales variance by $n^2$, therefore scaling total output by $\frac1{25}$ to find average means that the variance of the average number of donuts baked will be scaled by $\frac1{625}$.

$$\sigma_{\mu}^2 = \frac{2500}{625} = 4 \rightarrow \sigma_{\mu} = \boxed2$$

(Note that this process is the same as dividing $\sigma$ by $\sqrt{n}$ where $n$ is population size)