This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example. The sampling distributions are:. What we are seeing in these examples does not depend on the particular population distributions involved. In general, one may start with any distribution and the sampling distribution of the sample mean will increasingly resemble the bell-shaped normal curve as the sample size increases.
This is the content of the Central Limit Theorem. The larger the sample size, the better the approximation. 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.
Household size in the United States has a mean of 2. It should be clear that this distribution is skewed right as the smallest possible value is a household of 1 person but the largest households can be very large indeed.
A normal approximation should not be used here, because the distribution of household sizes would be considerably skewed to the right. We do not have enough information to solve this problem. The Central Limit Theorem does not guarantee sample mean coming from a skewed population to be approximately normal unless the sample size is large.
Now we may invoke the Central Limit Theorem: even though the distribution of household size X is skewed, the distribution of sample mean household size x-bar is approximately normal for a large sample size such as Its mean is the same as the population mean, 2. Then we can find the probability using the standard normal calculator or table.
Households of more than 3 people are, of course, quite common, but it would be extremely unusual for the mean size of a sample of households to be more than 3. 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.
Home Introduction Metacognition. CO Apply basic concepts of probability, random variation, and commonly used statistical probability distributions. In particular, be able to identify unusual samples from a given population. Based on our intuition and what we have learned about the behavior of sample proportions, we might expect the following about the distribution of sample means: Center : Some sample means will be on the low side — say 3, grams or so — while others will be on the high side — say 4, grams or so.
Video: Simulation 3 x-bar It helps to look at things visually. The image below represents all possible sample means for samples of size 1 individuals , 2, 3, 4, and 5 the population.
Pay particular attention to the standard deviation. If we think about this a bit, this too, is reasonable. The more individuals we have in our sample, the more likely we are to be closer to the true mean. Things brings us to our first major point. We're now ready to investigate the standard deviation of a bit more in-depth. This is very interesting! So it doesn't matter if the distribution shape was left-skewed, right-skewed, uniform, binomial, anything - the distribution of the sample mean will always become normal as the sample size increases.
What an amazing result! To do some exploring yourself, go to the Demonstrations Project from Wolfram Research, and download the Central Limit Theorem demonstration. If you haven't already, download and install the player by clicking on the image to the right.
Once you have the player installed and the Central Limit Theorem demonstration downloaded, move the slider for the sample size to get a sense of its affect on the distribution shape. You can also move the new sample slider to get a different sample. We can even be more specific about the distribution of :. Key fact: If the population is normally distributed, then the sample mean will be normally distributed, regardless of the sample size. In order to find probabilities about a normal random variable, we need to first know its mean and standard deviation.
With the results of the Central Limit Theorem, we now know the distribution of the sample mean, so let's try using that in some examples. Let's consider again the distribution of IQs that we looked at in Example 1 in Section 7. We saw in that example that tests for an individual's intelligence quotient IQ are designed to be normally distributed, with a mean of and a standard deviation of What is the probability that a randomly selected sample of 20 individuals would have a mean IQ of more than ?
Before we can do that, we need to first find the distribution of.
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