# Mean & SD of Sampling Distribution

A population consists of $$4$$ numbers $$\{0, 2, 4, 6\}$$. Consider drawing a random sample of size $$n = 2$$ with replacement.

(a) What is the sampling distribution of $$\bar x$$?

Is this a normal distribution ? Since $$\bar x$$~ $$N\left(\mu, \dfrac{\sigma^2}{n}\right)$$?

(b) Calculate the mean & standard deviation of the sampling distribution of $$\bar x$$.

I got the answer of mean $$\mu$$ by $$\frac{0+2+4+6}{4} = 3$$

Thereafter, I proceed to calculate $$\sigma$$

$$\sigma = \frac{(0 - 3)^2 + (2 - 3)^2 + (4 - 3)^2 + (6 - 3)^2}{4} = 5$$

Substituting it back into the sample distribution gives:

$$\bar x \sim N\left(3, \dfrac{5^2}{4}\right)$$

Thus, I derive the standard deviation to be:

$$\sqrt{\frac{\sigma^2}{n}} = \dfrac{5}{2}.$$

However, the answer given was $$\dfrac{\sqrt{5}}{\sqrt{2}}$$.

Can someone explain why is this so? I'm really quite confused with the whole concept of sampling distribution..

Thanks a lot!

• Do you seriously believe that drawing two numbers from 0, 2, 4, 6 can result in anything gaussian?
– Did
Commented Apr 13, 2014 at 11:35
• I know the list size is really small, but it's just how the question was given to me. Commented Apr 13, 2014 at 12:19
• They mentioned the gaussian framework, or you tried to fit the question in it?
– Did
Commented Apr 13, 2014 at 12:47

With $$n=2$$, the possible values of $$\bar{x}$$ are $$\{0,1,2,3,4,5,6\}$$ with respective probabilities $$\frac1{16},\frac2{16},\frac3{16},\frac4{16},\frac3{16},\frac2{16},\frac1{16}$$
Your $$3$$ and $$5$$ are the expectation and variance of a sample with $$n=1$$
For a sample with $$n=2$$ the sum would have expectation $$6$$ and variance $$10$$, while the mean of the sample would have expectation $$3$$ and variance $$\frac{5}{2}$$ and so standard deviation $$\frac{\sqrt 5}{\sqrt 2}$$