explanation of proof that sample mean is unbiased

William Cochran's book on sampling gives the following proof that a sample mean is unbiased:

Since every unit appears in the same number of samples, it is clear that $E[Y_1 + \cdots + Y_n]$ must be some multiple of $y_1+y_2+\cdots+y_N$. The multiplier must be $n/N$ since the first expression has $n$ terms and the second has $N$ terms.

I am trying to understand the last sentence, about the number of terms. The expectation here is taken over all samples of size $n$ (all of equal probability) out of a population of size $N$.

• You misquoted Cochrane. I see only a small excerpt via Google Books, but it is clear: He wrote $$E(y_1+y_2+\cdots+y_n) \text{ must be some multiple of } (y_1 + y_2 + \cdots + y_N)$$ He had lower-case $y$s throughout. – Michael Hardy Apr 30 '17 at 0:21
• @MichaelHardy That was intentional since his notations is, to my experience, nonstandard. The ys on the lhs are random, those on the rhs are not. (I suppose you could consider both sides as random, but the rhs is still deterministic since the population size is $N$.) – Hasse1987 Apr 30 '17 at 0:57
• Actually, he ought to have expressed it the way you did. – Michael Hardy Apr 30 '17 at 3:34

Let $y_1,\dots,y_N$ denote the values of some attribute of the population units $1,\dots,N$. In randomization theory, the $y_i$'s are not random variables, but fixed quantities. A random sample is determined by (dependent) indicator random variables $Z_1,\dots,Z_N\in\{0,1\}$ such that $Z_1+\dots+Z_N$ is equal to the sample size $n$. Each $Z_i$ determines if the respective population unit is included ($Z_i=1$) or not ($Z_i=0$) in the sample. The sample mean is the random variable $$\bar{Y} = \frac{1}{n}\sum_{i=1}^N y_i Z_i.$$ For a simple random sample, if the $i$-th population unit is included in the sample, then the other $n-1$ sample units must be chosen from the remaining $N-1$ population units. Hence, the probability $\Pr\{Z_i=1\}$ is equal to the number of samples of size $n$ which include $i$, given by $n-1\choose N-1$, divided by the number of size $n$ samples, given by $n\choose N$: $$\Pr\{Z_i=1\} = \frac{n-1\choose N-1}{n\choose N} = \frac{n}{N}.$$ It follows that $$\mathrm{E}[\bar{Y}] = \frac{1}{n}\sum_{i=1}^N y_i \mathrm{E}[Z_i] = \frac{1}{N}\sum_{i=1}^N y_i = \bar{y}.$$ Therefore, the sample mean $\bar{Y}$ is an unbiased estimator of the population mean $\bar{y}$.

• Thanks for the proof. But actually I am trying to understand Cochran's particular proof, specifically his remark about the terms on each side. – Hasse1987 Apr 29 '17 at 23:00
• I probably should have stated in the question, Cochran gives a rigorous proof. But he also gives this alternative proof that I was trying to make sense of. – Hasse1987 Apr 30 '17 at 0:58

"The number $A$ must be some multiple of the number $B$." is true because the number $A/B$ exists, unless $B=0$. So Cochran's meaning may take some effort to discern.

Suppose $N=5$ and $n=3$. Then the possible samples (assuming it's without replacement) are these: \begin{align} & y_1 + y_2 + y_3 \\[3pt] & y_1 + y_2 + y_4 \\[3pt] & y_1 + y_2 + y_5 \\[3pt] & y_1 + y_3 + y_4 \\[3pt] & y_1 + y_3 + y_5 \\[3pt] & y_1 + y_4 + y_5 \\[3pt] & y_2 + y_3 + y_4 \\[3pt] & y_2 + y_3 + y_5 \\[3pt] & y_2 + y_4 + y_5 \\[3pt] & y_3 + y_4 + y_5 \end{align} There are $10$ of these, so the average value of the sum is the average of these $10$ sums. The number $y_1$ appears in six of these sums, and $y_2$ appears in six of them, and so one. Thus "every unit appears in the same number of samples", since in this case, that "same number" is $6$. Thus the sum of the ten numbers is $6y_1+6y_2+6y_3+6y_4+6y_5.$ Dividing by $10$ gives the average, or expected value, of the random sample of size $3$. Hence \begin{gather} \text{expected value of the sample sum} = \frac{6y_1+6y_2+6y_3+6y_4+6y_5}{10} \\[10pt] = \frac 6 {10} (y_1+y_2+y_3+y_4+y_5) = \frac 3 5 (y_1+y_2+y_3+y_4+y_5) \end{gather} This ratio, $\dfrac 3 5,$ is the ratio of the sample size to the population size. That is the "multiplier". I expect there is a short elegant combinatorial argument showing that that will happen in general, i.e. with other numbers than $3$ and $5$, and also making Cochran's argument precise, but I'm not sure how best to express that right now.

1. Every unit appears in the same number $$k$$ of samples

2. If $$M$$ is the total number of samples, $$\Rightarrow M=k \cdot N$$

3. \begin{align}\Rightarrow {} & E\left(\frac{y_1+\cdots+y_n}n\right) \\[8pt] = {} & \left(\bar{y}_1 + \cdots + \bar{y}_M\right)/M \\[8pt] = {} & (k \cdot Y_1+\cdots+ k \cdot Y_N)/M \\[8pt] = {} & \left. \left(\frac{M}{N} \cdot Y_1+\cdots+\frac{M}{N} \cdot Y_N \right)\right/M \\[8pt] = {} & \frac{1}{N} \cdot (Y_1+\cdots+ Y_N) \end{align}

• So $M$ is the total number of $y$s that show up, with repetition, in $E(y_1+\ldots+y_n)$ ? Also could you please clarify what $\bar{y}_i$ refer to in line 2 of number 3? – Hasse1987 Apr 30 '17 at 19:04
• Hasse1987: M is the total number of samples, e.g. if n=3, N=5, M=$\binom{5}{3}=10$. $\bar{y}_{i}$ is the mean over the i th sample. – Augusto Apr 30 '17 at 19:30
• But for the $N=5,n=3$ case, $k=6$ as Michael Hardy's answer shows, and $10\neq 6*5$ ? – Hasse1987 Apr 30 '17 at 19:44