Distributions of Sample Means and the Central Limit Theorem

Contributor: Lynn Ellis. Lesson ID: 13771

This lesson will introduce you to distributions of sample means. A short animation of the construction of a distribution of sample means leads to understanding and applying the Central Limit Theorem.

categories

Measurement and Data, Statistics and Probability

subject
Math
learning style
Visual
personality style
Beaver
Grade Level
High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

bell curve

  • What if I told you there is a way to make all distributions of data look like this normal curve?

Well, there is. Welcome to the Central Limit Theorem!

The Central Limit Theorem begins with an understanding of distributions of sample means.

Really cool things happen when we look at a distribution of sample means. You may be asking what a "distribution of sample means" even is.

Watch Distributions of Sample Means, from Parabola Magic, below to learn what it is and to see some of the things that happen. As you watch the video, note that normal distributions are bell-shaped and symmetrical. They are one of the standard distributions of data with which statisticians work.

  • Isn't it interesting how everything starts to become normally distributed?

Here are some things that you hopefully noted from the video:

  The distribution of sample means looks normal no matter what the original population looked like.
     
  The mean of the distribution of sample means was the same as the mean of the population.
     
  The standard deviation of the distribution of sample means was smaller than the standard deviation of the population.

 

Under a certain set of circumstances, these three observations will always hold. Together, they form the basis of the Central Limit Theorem (CLT). This theorem is foundational to much of the work of statisticians.

Here is a statement of the Central Limit Theorem:

The distribution of sample means from samples of size n, taken from a population with mean [mu] and standard deviation [sigma], will be approximately normal with mean [mu] and a standard deviation [sigma] / √(n).

The conditions are:

  1. All samples are of the same size.
  1. The samples are taken randomly.
  1. The sampling process is done an infinite number of times.
  1. The samples are "sufficiently" large. For almost every distribution, samples of size 30 or higher meet this condition.

Write the above information in the first four boxes on The Central Limit Theorem & Distributions of Sample Means Graphic Organizer, found under the Downloadable Resources in the right-hand sidebar. Try to put the information in your own words where possible.

Now, let's look at an example problem that uses this information. Go ahead and write it down in your graphic organizer as the first example.

Suppose that the distribution of the number of text messages sent by teens daily has a mean of 60 and a standard deviation of 12.

  • What is the shape, mean, and standard deviation of the distribution of sample means of size 36?

SHAPE: The distribution of sample means will be approximately normally shaped.

MEAN: The mean of the distribution of sample means will be 60.

STANDARD DEVIATION: The standard deviation of the distribution of sample means will be 12 / √(36) = 2.

To check your understanding, try to find the mean and the standard deviation of the following distributions of sample means:

At this point you might be wondering how we apply your new knowledge of the mean and standard deviation of a distribution of sample means.

Let's look at that next. We'll go back to the sample problem of daily texts by teens for this. Go ahead and add this example to the last box on your graphing organizer.

Suppose that the distribution of the number of text messages sent by teens daily has a mean of 60 and a standard deviation of 12. You take samples of 36 teens from this data.

  • What is the probability that the average number of texts sent by a particular sample of 36 teens is between 59 and 61 text messages per day?

We already found the mean and standard deviation of the distribution of sample means. To find the probability, we need to know the area under the normal curve between 59 and 61 on our distribution of sample means.

Technology is our friend here. To see how to use Desmos Graphing Calculator to get the answer, watch Find the Area Under a Normal Curve Using Desmos from Parabola Magic:

You see from the video that the probability is 0.3829

Try this next problem to see if you understand.

If you feel like you understand this section, move on to the Got It? section to quiz yourself.

If you are not quite ready, feel free to read back through this section and watch one or both videos again.

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