V7 Sampling Distribution Theory

Hi everyone. Welcome back to our video series on sampling distribution theory. This is part two in the series. I'm Renee Clark from the Swanson School of Engineering.

So, in this video we're going to talk about two topics. The first is that of statistics as random variables, and the second is we're going to define a very special distribution in inferential statistics known as a sampling distribution. Okay, so, let's picture taking different, separate, random samples out of a population. Okay, so, let's say you have a certain population. Let's say the… our population that we're considering is all Pitt undergraduate students, okay?

Okay, and let's say we take separate, different, random samples out of this population. Okay, let's say we take samples of size 50. Okay, so, those samples are going to differ among themselves, or from sample to sample, or another way of saying that is that, for example, the students who comprise sample one are likely going to be different from the students who comprise sample two, right? Because we're taking different random samples out of the population, and likewise for the other samples. Okay, so then, let's say that for each of these samples, we calculate a statistic. Okay, for each sample. So, for… for example, let's say we're going to calculate the average for each sample. Let's say the variable that we're considering is grade point average, or GPA. So, for each different sample we're going to calculate the average GPA. Okay, so, since the contents of each sample differ, or the students in each sample differ, the average GPA associated with that sample is going to differ. So, the average GPA, or average one, is likely going to differ for… from average two, and from average three, all the way on down the line for the remainder of the averages. Okay, because those samples differ, the average GPA, or the average for each sample, is going to differ. Okay, so statistics are going to vary sample to sample with random samples. Okay, so, what this says then is that statistics, such as a sample average are actually random variables. Very important concept in inferential statistics- very important to understand this slide. Okay, so, any random variable, as we know, has a distribution for its value. So, for example, this is a distribution for the random variable weight, and it happens to be weights of four-year-old boys. But, because weight varies among four-year-old boys, it forms a distribution. Okay, so, as we all… as we just said, though previously, statistics are random variables. So, like any random variable, they too have a distribution.

But, the distribution of a statistic, such as the sample average, is known by a very special term, and that is sampling distribution. Okay, so, the distribution of a statistic is known as a sampling distribution. Okay, so, let's show this pictorially. Here, let's say we have a certain population. Let's say this is a population of weights. Okay, so, our random variable X is weight. Okay, let's say we take different random samples out of that population- separate random samples. Okay, and we calculate the average weight for each sample. Okay, so, you can see here the average for sample one is 15.5. The average for sample 2 is 16.0, and so forth. And you can see each of those averages differ, okay, because you're taking different random samples. So, since the… the people comprising each of those samples differ, the averages are going to differ. Okay, because those averages differ, they're random variables. Well, we can plot them, right? We can form a distribution of those sample averages as such. This is a distribution of the various averages that we're calculating based on these different random samples. Okay, this distribution of averages has a very special name known as the sampling distribution. And again, because it's a distribution of averages, it’s actually a samp… sampling distribution of the mean.

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

Thank you for watching.