The Central Limit Theorem says that as you take more samples, the distribution of their averages becomes bell-shaped, no matter what the original data looks like. This matters because it lets you use more powerful statistical tests. The video clarifies the difference between CLT and the Law of Large Numbers using a neighbor survey example.
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Great work so far. Now let's move to the Central Limit Theorem. In this video, we will go over what the Central Limit Theorem is and why it is important. The Central Limit Theorem states that the sampling distribution of the sample's means approaches a normal distribution as the sample sizes get larger. Sounds complicated, doesn't it? The interviewer might test if you understand the Central Limit Theorem by asking you to explain it situationally. Let's consider the population of your neighbors. You want to know how much your neighbors liked the last Avengers movie. In the picture, each of the rectangles represents one of your neighbors. You draw random and independent samples of three neighbors and collect the data on how much the neighbors liked the movie on a scale from 1 to 10. In the picture, the three orange rectangles represent the first sample, the three green rectangles the second sample, etc. For each of the samples, you calculate the mean of the scores. The statement of the Central Limit Theorem says that the distribution of the samples means will approximately take the shape of a bell curve around the population's mean. If you increase the sample size, the sampling distribution converges to the shape of a normal distribution. The power of the Central Limit Theorem is that it works with any underlying distribution. Whether the distribution is bimodal or skewed, the theorem holds. Why does it matter that the sampling distribution converges to the shape of a normal distribution? There are statistical tests that require a specific distribution of the underlying data. These tests are called parametric tests. Parametric tests are in general more powerful than non parametric tests. The fact that the sampling distribution is normal allows us to perform parametric tests, otherwise we couldn't. We will cover them in the next chapter. The Central Limit Theorem is sometimes confused with the Law of Large Numbers. It is easy to mix them up, especially in a high stress interview setting. Let's go over the Law of Large Numbers. Imagine a discrete uniform distribution ranging from 0 to 1. You draw a sample from this distribution and calculate the mean value of the data points. The Law of Large Numbers states that if you increase the size of the sample, the mean will be getting closer to the true mean of the distribution we are sampling from. In the case of the discrete uniform distribution ranging from 0 to 1, the true mean amounts to 0.5. The central limit Theorem concerns the distribution of the sample's means, whereas the Law of Large Numbers focuses on a single sample's mean. To summarize, in this video we have covered the Central Limit Theorem and the Law of Large Numbers. Let's review the Central Limit Theorem to get you ready for the interview.
