# Sample size of sampling distribution of the sample mean

When people say that the mean of the sampling distribution of the sample mean is equal to the population mean, is this true regardless of how small or large the size of the samples are? I'm having a hard time wrapping my head around how the size of the samples would not make a difference.

Edit: mistake in wording

Yes, it is true that the sampling distribution of the mean is equal to the population mean regardless of how small or large the sample sizes are (In other words, the mean is ubiased).

A good way to think about this is to take a small population and study it. Say for example your population consists of 4 elements having values 2, 4, 6, and 10. The sampling distribution can be thought of as taking samples of a certain size over and over again from this population. So for example you wanted to sample 3 elements. Then, on average, the mean of the means of a lot of samples of size 3 will be unbiased (e.g. the sample means of the mean will be a good approximation of the true population mean). To see this note that the population average of 2, 4, 6, and 10 is 5.5. The following R code shows 100000 samples of size 3 and of size 2 and of size 1. After each sample is selected, the mean is calculated. After all 100000 sample means are calculated, we find the sample mean of those means. Note that this mean is very close to the population mean of 5.5 in every case (the estimates round to 5.5 every time), regardless of how large the sample size is:

x<-c(2,4,6,10)
> #Population Mean
> mean(x)
[1] 5.5
>
> sample.means<-rep(NA, 100000)
>
> set.seed(5)
> #Sample of size n=3
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=3))
+ }
> mean(sample.means)
[1] 5.4982
>
> #Sample of size n=2
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=2))
+ }
> mean(sample.means)
[1] 5.49388
>
> #Sample of size n=1
>
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=2))
+ }
> mean(sample.means)
[1] 5.49419

• @Antoine makes a good point which I did not address in my post. Mar 29 '15 at 3:32

It has to do with there being a lack of bias in your samples. If you were to randomly select samples, even very small ones, from the population, their means will be above and below the population mean an equal number of times. So, on average, the mean of the sample means should be a good approximation of the true population mean.

Sample size will, however, affect variance of your estimates around the true mean. If you have a large n, samples will be closer to the true mean; if you have a small n, they will be dispersed widely around the true mean.

The sample mean is sometimes an unbiased estimatorof the population mean, but it only holds for some sampling designs. Under simple random sampling (SRS), it does work, in which case the sample mean is equal to the Horvitz-Thompson estimator :

\begin{align*} \hat{\bar{Y}}_{HT} &= \dfrac{1}{N} \sum_{k \in s} \dfrac{y_k}{\pi_k} \\ &= \dfrac{1}{N} \sum_{k \in s} \dfrac{N}{n} y_k \\ &= \dfrac{1}{n} \sum_{k \in s} y_k \\ &= \bar{y} \end{align*}

In this case, your estimator is unbiased for any sample size. This property may hold for other sampling schemes (systematic sampling with equal probabilities, for example), but it isn't true in general.

But be careful : using an unbiased estimator does not mean your estimator is consistent. Especially for small samples, you could end up with values really far from the true value. See for example : http://www.johndcook.com/blog/bias_consistency/ . Sometimes, using slightly biased estimators (the Hajek estimator for example) gives better results for small sample sizes (and the biais is smaller for larger samples).

As @Antoine R pointed out the idea is that sample means provide an unbiased estimate of the population mean. If we could sample unlimited samples of one single observation (or subject) (n = 1) we would reproduce the entire population, and its exact mean.

I wanted to contribute some simulations to illustrate the point about consistency: the sample mean converges to the population mean as the number of observations tends to infinity. Conversely, sample means with small number of observations can deviate substantially, perhaps explaining the counterintuitive impression you describe in your question.

If we obtain 10,000 samples out of a standard normal ($$N(0,1)$$) distribution each sample containing one additional subject than the previous, and we plot the results we get the following plot,

set.seed(50)
sample_means <- NULL
for (i in 1:1000) sample_means[i] <- mean(rnorm(i))
plot(sample_means, cex = 0.2, pch = 19, col = 'red')
lines(sample_means, col='orange')


showing the convergence to the true mean (0) with increasing sample size.

Or, another way of play with the concept would be to obtain 10,000 samples with n = 10, and then 10,000 samples with n = 10,000, plotting the histograms without altering the x axis limits,

set.seed(50)
par(mfrow=c(1,2))
s_means_10 <- NULL
for(i in 1:1e4) s_means_10[i] <- mean(rnorm(10))
hist(s_means_10,col="dark red", border = 'white', main = "Distr. Means of Samples of 10 rand. numbers")
s_means_10k <- NULL
for(i in 1:1e4) s_means_10k[i] <- mean(rnorm(1e4))
hist(s_means_10k,xlim=c(-4,4),border = "dark red",col="dark red", main = "Distr. Means of Samples of 10,000 numbers")


showing again the convergence with sample size, without any shift in the mean, which is always centered at zero.