Question

Hi

My teacher wants 1 page of my understanding of the central limit theorem with example.

If someone could pls do it for me I would be glad. I dont need a bookish thing. Just a jist of what it is explained with an example as by a student.

Answer 1

Central Limit Theorem

It is advantageous to have normally distributed data

When the data are not normally distributed

It is very useful to know about the central limit theorem

Which states that

the distribution of sample averages will tend towards a normal distribution as sample size n approaches infinity

in which subgroup Size of 30 or more are considered sufficiently large

*CLT allows the use of smaller averages to evaluate any process because distributions of sample means tend to form a normal distribution.

* the normal distribution curve appears when the process is in control(predictable)

*CLT leaves variation from common causes to chance, thus distributing accordin g to the central limit theorem.

*CLT identifies and removes variations from special causes.

*Using +/- 3 sigma control limits, CLT is the basis of the prediction that on average only 0.27% of the time the sample mean falls outside the control limits, if the process has not changed.

Central limit theorem has practical application in inferential statstics

Central limit theorem also applies to use of control chart in Statstical process control

Let us first understand the individual observations and sample means

Individual observations:

Represents the distribution of population

Actual values of all observations in all Subgroups

Sample Means:

Represents the distribution of averages (means)

Averages values of subgroup

For example:

5 parts each hour for 20 hours and measure a dimension

Calculate the average dimension for 5 parts & sampled each hour

Plot the 100 measurements (5X20) and observe the distribution of all 100 individual observations

Plot the 20 averages that were calculated based on each hour of production and observe the distribution of 20 averages (Sample Means) in this case sample means are normally distributed

Results are shown in below diagram

Figure shows the impact of central limit theorem on increasing the sample Size

Significant Points:

Sample means curve is narrower in which extreme values are averaged out

Sample means curve tends to be normal regardless of form of shape of distribution of individulas

We can usually approximate the distribution of means with a normal distribution