V13 Sampling Distribution Theory

Okay, this is part eight of our video series on sampling distribution theory. I'm Renee Clark from the Swanson School of Engineering at Pitt. In this video, we're going to discuss the central limit theorem. Previously, we learned that if the parent population of X is normally distributed, then automatically x-bar will be normally distributed.

Okay, however, what if x, or the parent population of X, is not normally distributed. Possibly, you know, say it's skewed right or skewed left, or we don't know what the… the distribution of X is. Okay, can we say, or can we know, that x-bar is normally distributed? In this case, the answer is possibly… maybe. Okay, come back to this.

But, before we do, why is this even relevant? Why is it relevant to know that x-bar is normally distributed? Okay, the answer to that is that if x-bar is normally distributed, then we can standardize it. Okay, and we can use the Z table in the back of the textbook to determine probabilities that we can use in our statistical techniques.

Okay, so, in order to standardize a normally distributed x-bar, which it needs to be in order to standardize it, okay, we're going to subtract its mean, divide by its standard deviation. And, recall that these are the parameters of the sampling distribution of x-bar. Its expected value, or its mean, is Mu. Its standard deviation is Sigma over the square root of n. Okay, so, going back to our answer given previously about x-bar possibly being normally distributed, even if X is not, here is the… the answer. Fuller answer to that is that x-bar will become normally distributed even if the parent population X is not normally distributed, and this is the key as the sample size n increases.

Okay, so, the central limit theorem says that the sample means, or the x-bars, will be approximately normally distributed for sufficiently large sample sizes, n. Okay, so this is a picture here of a sampling distribution, right? Sampling distribution is the distribution of a statistic. This happens to be a… the sampling distribution of the mean because it's the… a distribution of the x-bars. Okay, but the question is which sample size is this? Okay, it's the sample size that is involved in calculating the average. Okay, recall that, to calculate the average, you add up the X's, divide by the number of x's that you have in the numerator. This is the sample size that must be sufficiently large in order to ensure normality of x-bar, even if X is not normal. Okay, so, it's not… n is not the number of x-bars that you have in the distribution. Okay, it's the number of items that went into calculating each sample average that… that comprises this distribution.

Okay, so, relative to the central limit theorem, what is considered sufficiently large in order for x-bar to be normally distributed even if X is not? Okay, stat… statisticians say an N of about 30 is sufficient for most parent po- populations, X. Okay, or, in most cases, okay, however, the more the parent population of X differs from normality, okay, the larger that n will need to be in order that x-bar be normally distributed. Okay, so, if your… your parent population, X, is, say, strongly skewed or differs greatly from normality, you may need an n greater than 30 in order to ensure normality of x-bar.

Thank you to the National Science Foundation under Grant 233 5802 for supporting our work. Thank you for watching.