Sampling Distribution of the Sample Mean, x-bar
So far, we’ve discussed the behavior of the statistic p-hat, the sample proportion, relative to the parameter p, the population proportion (when the variable of interest is categorical).
We are now moving on to explore the behavior of the statistic x-bar, the sample mean, relative to the parameter μ (mu), the population mean (when the variable of interest is quantitative).
Let’s begin with an example.
- The distribution of the values of the sample mean (x-bar) in repeated samples is called the sampling distribution of x-bar.
Let’s look at a simulation:
The results we found in our simulations are not surprising. Advanced probability theory confirms that by asserting the following:
If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is μ (mu) and the population standard deviation is σ (sigma) then the mean of all sample means (x-bars) is population mean μ (mu).
As for the spread of all sample means, theory dictates the behavior much more precisely than saying that there is less spread for larger samples. In fact, the standard deviation of all sample means is directly related to the sample size, n as indicated below.
Since the square root of sample size n appears in the denominator, the standard deviation does decrease as sample size increases.
Let’s compare and contrast what we now know about the sampling distributions for sample means and sample proportions.
Now we will investigate the shape of the sampling distribution of sample means. When we were discussing the sampling distribution of sample proportions, we said that this distribution is approximately normal if np ≥ 10 and n(1 – p) ≥ 10. In other words, we had a guideline based on sample size for determining the conditions under which we could use normal probability calculations for sample proportions.
When will the distribution of sample means be approximately normal? Does this depend on the size of the sample?
It seems reasonable that a population with a normal distribution will have sample means that are normally distributed even for very small samples. We saw this illustrated in the previous simulation with samples of size 10.
What happens if the distribution of the variable in the population is heavily skewed? Do sample means have a skewed distribution also? If we take really large samples, will the sample means become more normally distributed?
In the next simulation, we will investigate these questions.
To summarize, the distribution of sample means will be approximately normal as long as the sample size is large enough. This discovery is probably the single most important result presented in introductory statistics courses. It is stated formally as the Central Limit Theorem.
We will depend on the Central Limit Theorem again and again in order to do normal probability calculations when we use sample means to draw conclusions about a population mean. We now know that we can do this even if the population distribution is not normal.
How large a sample size do we need in order to assume that sample means will be normally distributed? Well, it really depends on the population distribution, as we saw in the simulation. The general rule of thumb is that samples of size 30 or greater will have a fairly normal distribution regardless of the shape of the distribution of the variable in the population.
- For categorical variables, our claim that sample proportions are approximately normal for large enough n is actually a special case of the Central Limit Theorem. In this case, we think of the data as 0’s and 1’s and the “average” of these 0’s and 1’s is equal to the proportion we have discussed.
Before we work some examples, let’s compare and contrast what we now know about the sampling distributions for sample means and sample proportions.
The purpose of the next activity is to give guided practice in finding the sampling distribution of the sample mean (x-bar), and use it to learn about the likelihood of getting certain values of x-bar.