Central Limit Theorem in Statistics Overview

You understood the third property of sampling distributions, which talks about their shape. Basically, it says that for n > 30, the sampling distributions become normally distributed. Let's recall all the three properties that you have learnt so far for sampling distributions.

So, the central limit theorem says that for any kind of data, provided a high number of samples has been taken, the following properties hold true:

1. Sampling distribution’s mean (${\mu }_{\overline{X}}$) = Population mean (μ),

2. Sampling distribution’s standard deviation (standard error) = $\frac{\sigma }{\sqrt{n}}$, and

3. For n > 30, the sampling distribution becomes a normal distribution.

We made two sampling distributions (the upGrad game and the banking data set) and saw that they follow the aforementioned three properties.

Now, let’s listen to Prof. Tricha as she verifies the central limit theorem for some more population distributions.

Prof. Tricha verified the CLT by performing simulations on different kinds of data. In case you want to try out these simulations yourself, you can go to this link. Press the "Begin" button in the top-right corner to get started.

Recall that in the first lecture on samples, we found the mean commute time of 30,000 employees of an office by taking a small sample of 100 employees and finding their mean commute time. This sample’s mean was  = 36.6 minutes and its standard deviation was S = 10 minutes.

We then said that this sample mean cannot be taken as the population mean, as there might be some errors in the sampling process. However, we can say that the population mean, i.e., the daily commute time of all 30,000 employees  = 36.6 (sample mean) + some margin of error.

Now, you may be thinking that you can use the standard error for the margin of error. However, keep in mind that although the standard error provides a good estimate of this margin of error, you cannot use it in place of the margin of error. To understand why and how you would find the margin of error in that case, let's move on to the next lecture, where we will use the CLT (central limit theorem) to find the aforementioned margin of error.