3-4-10. Sampling distributions and the Central Limit Theorem

There are two mathematical theorems that are commonly discussed when looking at sampling distributions, the Law of Large Numbers and the Central Limit Theorem. First, let’s go through the Law of Large Numbers. This theorem makes a lot of sense and it tells us that if we choose the right statistics to estimate a parameter, … Read more

You now have seen how a sampling distribution provides how a statistic varies. As we saw, the proportions change with different samples of students. However, we might also look at the distribution of other statistics, like how the sample standard deviation, the variance, difference in mean or any other statistic varies from one sample to … Read more

There are common ways to notate parameters that are different than the way we notate statistics. In general, parameters are notated with Greek symbols, where statistics are either notated by lower case letters or the same Greek symbol with a hat on it. Here, this symbol called mu represents the mean of a population, while … Read more

It’s important to understand notation. You might not even know it, but you use notation all the time. Consider this example of five plus three. Plus is an English word. This symbol is notation, and it’s universal. Notation is a common math language used to communicate. Regardless of whether you speak English, Spanish, Greek, or … Read more

So you probably notice that though our sample is still five students, our statistic changed, because we chose five different students than were chosen in the first sample. We could select all possible combinations of five cups, and we could recompute the proportion of coffee drinkers for each of these samples. If we were to … Read more

If we wanted to identify the sample and statistic, from this visual, we would only use these cups, which give a sample of five students where four of them don’t drink coffee. This gives us a statistic that 20% of students drink coffee. Remember, a population is our entire group of interest. Therefore, we have … Read more

In this video, we’ll be defining the term Sampling Distribution, and looking at an example of one specific sampling distribution. A sampling distribution is the distribution of a statistic. This could be any statistic, but what does it really mean, to look at the distribution of a statistic? Consider again the coffee drinking habits of … Read more

The topics covered this far have all been aimed at descriptive statistics. That is, describing the data we’ve collected. There’s an entire other field of statistics known as inferential statistics that’s aimed at drawing conclusions about a population of individuals based only on a sample of individuals from that population. Imagine I want to understand … Read more

In this lesson, you’ve looked a lot at sampling distributions. We might still not understand the use of this idea in practice. In the next lessons, you’re going to learn more about inference. Specifically, you’ll be learning about Confidence Intervals and Hypothesis Testing. When looking at these techniques online, you might find a lot of … Read more

If bootstrap sampling seems pretty amazing, that’s because it kind of is. But the application of bootstrap sampling actually goes beyond even the use cases here. Bootstrapping techniques have been used for leading machine learning algorithms. More on this as provided in the instructor notes below. This technique is credited to Bradley Efron in 1979, … Read more

Here’s the idea of using bootstrapping to simulate the sampling distribution for any statistic. This might take more than one time through to fully grasp. So bear with me. We know that in inferential statistics, we want to use a statistic to try and say something about the corresponding population parameter. Imagine we treat our … Read more

So in the last video, we talked about how relying on mathematical theorems, like the central limit theorem leads to gaps in whether we’ve achieved a large enough sample size or, which statistics the theorem applies to, and that instead of relying on theorems we could simulate the sampling distribution. This introduces a technique known … Read more

You now have gained some intuition for how the Central Limit Theorem works. But it doesn’t work for all sampling distributions. Sure, a mean and proportion are normally distributed with large enough sample sizes. But what does it mean for a sample size to be large enough? Is a sample size of 10 large enough? … Read more

The second theorem is one of the most popular theorems in all of statistics. And it pertains specifically to the sample mean and sample proportions statistics. The central limit theorem states, that with a large enough sample size, the sampling distribution of the mean will be normally distributed, since a proportion is like a mean … Read more

In this lesson, you’ll be learning about sampling distributions. In order to gain a firm grasp of how sampling distributions work, it’s important to first have a strong grasp of inferential statistics. We will do a recap in the next concepts to make sure you’re comfortable with the ideas surrounding inferential statistics. This is also … Read more