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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Artem Cheprasov*

The central limit theorem provides us with a very powerful approach for solving problems involving large amount of data. In this lesson, we'll explore how this is done as well as conditions that make this theorem valid.

Have you ever wondered how public opinion polls can provide an accurate prediction of a future event, such as the outcome of a presidential election? After all, the number of people sampled is much smaller than the population size. In fact, this approach of taking samples of a large data set to predict something about it is done in many different fields, including the physical sciences, medicine and finance. In this lesson, you will learn about using the Central Limit Theorem to find probabilities about means.

The **Central Limit Theorem** states that if we take enough independent random samples of size 'n' where 'n' is sufficiently large, then the distribution of the means of the samples will approach the normal distribution. This holds even for data sets that are not normally distributed to start out with, as we shall see next. In the **normal distribution**, most of the values of a data set cluster around the middle of the range, and the rest taper off toward one extreme or another, giving the frequency distribution curve a symmetrical bell shape.

Let's examine what the Central Limit Theorem means with a simple example. Suppose a biologist wants to determine the weight distribution of squirrels in a forest. One way to do it would be to find and weigh every squirrel. Unfortunately, that would be too expensive, and impractical. So, the biologist comes up with a better solution, relying on the Central Limit Theorem. He starts out by weighing groups of three squirrels at a time, randomly selected from the population. After finding the mean rates of these groups of three, he plots the results. The distribution of the means does not seem to have any recognizable shape. He then proceeds to weigh groups of ten squirrels at a time, again randomly chosen from the entire population. The histogram now looks more similar to a normal distribution. However, the biologist decides to try an even larger sample size. He now goes out and weighs groups of 100 squirrels at a time. Now the distribution of average weights looks very similar to a normal distribution.

The biologist's results are in good agreement with the Central Limit Theorem. As the sample size was increased, the distribution of the means came closer and closer to a normal distribution. What if the distribution is inherently bi-modal, such as the weight distribution of ants in a colony where there are only two types of ants? There are large ones and very small ones, as shown on the screen. If the biologist were to study these ants, the distribution of the means of their weights, assuming a large enough sample size, would still be normal. As before, making the sample size larger would cause the distribution to become a better approximation of the normal distribution.

Why is the Central Limit Theorem important? It turns out that when the sample size is large enough, the following properties will hold:

1. The mean of the population can be approximated by the mean of the sample means.

2. The standard deviation of the population can be approximated by the standard deviation of the sample means multiplied by the square root of the sample size.

3. The variance of the population can be approximated by the variance of the sample means multiplied by the sample size.

As you can see, these properties give researchers the ability to find out information about an entire population by sampling it in the right way.

And why don't we sample our lesson with a quick summary? In this lesson, you have learned about the **Central Limit Theorem** and its implications. This theorem states that if we take enough independent random samples of size 'n', where 'n' is sufficiently large, then the distribution of the means of the samples will approach the **normal distribution**. In other words, if we take sufficiently large random samples from a population, subsequently computing the means of these samples, then these means will follow a normal distribution. This will remain true, even if the population is not normally distributed to begin with. Also, recall that there are very important implications of this theorem, mainly, the following:

1. The mean of the population can be approximated by the mean of the sample means.

2. The standard deviation of the population can be approximated by the standard deviation of the sample means multiplied by the square root of the sample size.

3. The variance of the population can be approximated by the variance of the sample means multiplied by the sample size.

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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

- Go to Probability

- Simple Random Samples: Definition & Examples 5:10
- What is Random Sampling? - Definition, Conditions & Measures 5:55
- Stratified Random Samples: Definition, Characteristics & Examples 6:25
- Cluster Random Samples: Definition, Selection & Examples 6:44
- Systematic Random Samples: Definition, Formula & Advantages 8:37
- Understanding the Law of Large Numbers 5:14
- Sampling Distributions & the Central Limit Theorem: Definition, Formula & Examples 5:06
- Find the Mean & Standard Error of the Sampling Distribution 5:03
- Finding Probabilities About Means Using the Central Limit Theorem 4:24
- Go to Sampling

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