 ## S12 Sampling distributions A sampling distribution is the probability distribution for the means of all samples of size $$n$$ from a given population.

The sampling distribution will be normally distributed with parameters $$\mu_{\overline{x}}$$ and $$\sigma_{\overline{x}}$$, if either:

1. the population from which the samples are drawn is normally distributed, or

2. the samples are large $$(n\geqq30)$$.

The mean of the sampling distribution (i.e. the mean of all the sample means, $$\mu$$ $$_{\overline{x}}$$) and the standard deviation of the distribution ($$\sigma_{\overline{x}}$$) are given by:

$\mu_{\overline{x}}=\mu\quad and\quad\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}$ Note that:

• The sampling distribution has the same centre as the population.

• The measure of variability of a sampling distribution, $$\sigma_{\overline{x}}$$, is called the standard error.

• The distribution of means is not as spread out as the values in the population from which the sample was drawn.

• If we do not know the population standard deviation we approximate with the sample standard deviation: $$s_{\overline{x}}$$ $$\thickapprox$$ $$\sigma_{\overline{x}}$$ and $$\dfrac{s}{\sqrt{n}}$$ $$\thickapprox$$ $$\dfrac{\sigma}{\sqrt{n}}$$ if the sample is large.

### Example of a Sampling Distribution

Consider the little ‘population’ of values P = {1 2 3 4 5}

This population has $$\mu=3$$ and $$\sigma=1.41$$ .

If a sample of size n = 3 was drawn from this population it could be any one of:

(1 2 3) (1 2 4) (1 2 5) (1 3 4) (1 3 5) (1 4 5) (2 3 4) (2 3 5)
(2 4 5) (3 4 5)

The means of each of the samples, and a histogram of the distribution of means, are shown in the table and graph below:

Sample Mean
1 2 3 $$\overline{x}$$ = 2
1 2 4 $$\overline{x}$$ = 2.33
1 2 5 $$\overline{x}$$ = 2.67
1 3 4 $$\overline{x}$$ = 2.67
1 3 5 $$\overline{x}$$ = 3
1 4 5 $$\overline{x}$$ = 3.33
2 3 4 $$\overline{x}$$ = 3
2 3 5 $$\overline{x}$$ = 3.33
2 4 5 $$\overline{x}$$ = 3.67
3 4 5 $$\overline{x}$$ = 4

$$\overline{\overline{x}}$$ = 3 and $$\sigma_{\overline{x}}$$ = 0.61

The sampling distribution of the means for samples of size 3 is:

$$\overline{x}$$ 2 2.33 2.67 3 3.33 3.67 4
P($$\overline{X}=\overline{x}$$) 0.1 0.1 0.2 0.2 0.2 0.1 0.1

Even though this sample is small, and the population is not normally distributed (though it is symmetric) the sampling distribution is reasonably normally distributed: We can see that the mean of the sampling distribution (the mean of all the means) is the same as the population mean, $$\overline{\overline{x}}=\mu=3$$. But the variability in the sampling distribution is less than that of the population: $$\sigma_{\overline{x}}=0.61$$ and $$\sigma=1.41$$. Because larger samples, or those drawn from normally distributed populations, will follow a normal distribution we can use the properties of normal distributions to find probabilities relating to samples: $z_{\overline{x}}=\dfrac{(\overline{x}-\mu)}{\sigma_{\overline{x}}}=\dfrac{(\overline{x}-\mu)}{\sigma/\sqrt{n}}.$

#### Another Example

The shire of Bondara has 1200 preschoolers. The mean weight of pre-schoolers is known to be 18kg with a standard deviation of 3kg. What is the probability that a random sample of 50 preschoolers will have a mean weight more than 19kg?

$$n=50$$, $$\mu=18$$ and $$\sigma=3$$

The sampling distribution of the means for samples of size 50 will have $$\mu$$ $$_{\overline{x}}$$ $$=$$ $$\mu=18$$, and standard error, $\sigma_{\overline{x}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{3}{\sqrt{50}}=0.42$ .

\begin{align*} z_{\overline{x}} & =\dfrac{(\overline{x}-\mu)}{\sigma/\sqrt{n}}\\ & =\dfrac{(19-18)}{3/\sqrt{50}}\\ & =2.38\\ \\ Pr(\overline{x}>19) & =Pr(z_{\overline{x}}>2.38)\\ & =1-0.9913\qquad[from\;tables]\\ & =0.0087 \end{align*}

### Exercise

1. List all samples of size 2 for the population $${1,2,3,4,5,6}$$. What is the probability of obtaining a sample mean of less than $$3$$?

$$4/15$$

2. Samples of size $$40$$ are drawn from a population with $$\mu=50$$ and $$\sigma=5$$. What are the mean and standard error of the sampling distribution? What is the probability that a particular sample has a mean less than $$48.5$$?

(a) $$\mu$$ $$_{\overline{x}}=50$$ and $$\sigma_{\overline{x}}=0.79\quad\quad$$(b) $$0.0288$$
3. If IQ in the general population of secondary students is known to follow a normal distribution with $$\mu=100$$ and $$\sigma=10$$, find the mean and standard error for a random sample of size $$100$$. To test whether a secondary school is representative of the general population a sample of $$100$$ students from that school is chosen. What is the probability of the mean IQ being more than $$105$$? What would be your conclusion?
(a) $$\mu$$ $$_{\overline{x}}=100$$ and $$\sigma_{\overline{x}}=1\qquad$$(b) $$0.00003\thickapprox0$$. This implies that either the sample was not random (perhaps all the smartest students were in the sample) or this school has a higher average IQ than the general population.