Solved – Sample size of sampling distribution of the sample mean

samplesample-size

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

Best Answer

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