The central limit theorem is the basis of several types of statistical tests. In this video, learn how to apply this theorem.
- [Instructor] Now we learn about a very important statistical concept, the central limit theorem. We start with this definition, a sampling distribution is a distribution of all possible values of a statistic for a given sample size. Here's a picture that shows what I'm talking about. A huge number of samples coming out of a population, imagine that each one has the same number of people and we measure each person on some attribute, we calculate a mean for each sample, that set of means is the sampling distribution of the mean. Now, bear in mind, that when we're doing a study, we never actually create a sampling distribution, that would take an infinite amount of effort. Here's what we do, from one sample and some additional statistical knowledge, we can find the parameters of a sampling distribution. What does this enable us to do? It enables us to calculate probabilities and thus to estimate parameters and test hypothesis. When the statistic is the mean, the additional statistical knowledge is the central limit theorem. And here's the central limit theorem, if the sample size, N, is large enough, 30 or more, the sampling distribution of the mean is approximately a normal distribution, and this is important, even if the population is not normally distributed. The sampling distribution's mean, mu sub x bar, is equal to the population mean, mu. Its standard deviation, sigma sub x bar, is equal to the population standard deviation divided by the square root of the sample size. The sampling distribution standard deviation, sigma x bar, is also called the Standard Error of the Mean. If the population is normally distributed, the sampling distribution of the mean is normally distributed regardless of the sample size. This is what the sampling distribution of the mean looks like according to the central limit theorem. It's a normal distribution with the mean equal to the mean of the population and the standard errors are laid out across the x-axis. Here's an example of the central limit theorem at work. In a population of IQ scores, with one version having mean of a hundred and standard deviation of 16, if we take a sample of 64 people, what does the sampling distribution of the mean look like? It looks like this, according to the central limit theorem, its mean is the same as the population mean, which is a hundred and its standard deviation or the standard error of the mean is equal to sigma divided by the square root of N, which is 16 divided by the square root of 64, 16 over eight, which is two. And you can see those values laid out across the x-axis. Now, in the two preceding pictures, the population distribution look very much like the sampling distribution, that's because they appeared on different axes. This is what the population distribution and the sampling distribution look like when you plot both of them on the same set of axes. They're both normal distributions but the sampling distribution has a way smaller standard deviation. Summing up, the central limit theorem is the additional knowledge we need to specify the sampling distribution of the mean. With data from just one sample, the central limit theorem can tell us the mean, the standard error, and the shape of the sampling distribution and that's a huge amount of information. We use this in estimation and in hypothesis testing.
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