1. Central limit theorem
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.
2. Central limit theorem
The central limit theorem states that the sampling distribution of the samples' means approaches a normal distribution as the samples' 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.
3. Central limit theorem
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.
4. Central limit theorem
You draw random and independent samples of 3 neighbors and collect the data on how much the neighbors liked the movie on a scale from 1 to 10.
In the picture, the 3 orange rectangles represent the first sample; the 3 green rectangles the second sample, etc.
5. Central limit theorem
For each of the samples, you calculate the mean of the scores.
6. Central limit theorem
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.
7. Central limit theorem
If you increase the samples' size, the sampling distribution converges to the shape of a normal distribution.
8. Central limit theorem
The power of the central limit theorem is that it works with any underlying distribution. Whether the distribution is bimodal
9. Central limit theorem
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.
10. Central limit theorem
These tests are called parametric tests.
Parametric tests are, in general, more powerful than nonparametric 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.
11. Law of large numbers
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.
12. 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.
13. Law of large numbers
The law of large numbers states that if you increase the size of the sample,
14. Law of large numbers
the mean will be getting closer to the true mean of the distribution we're sampling from.
15. Law of large numbers
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 samples' means, whereas the law of large numbers focuses on a single sample's mean.
16. Summary
To summarize, in this video, we have covered the central limit theorem and the law of large numbers.
17. Let's practice!
Let's review the central limit theorem to get you ready for the interview!