Quality Quiz from Professor Cleary
"A" is correct.
Click here for a more complete video explanation
Amazingly, Simsack drops the right name this time. The behavior of averages in this case is indeed related to the central limit theorem, which in turn helps explain how control charts work.
The easiest way that I have found to demonstrate the central limit theorem is with the use of Quality Gamebox. Using Quality Gamebox, and taking sample sizes of 5, the following results ensue.
Distribution of sample means with sample size of 5:
Population Distribution of sample means (n=5)
Examining the distribution of sample means derived from the single mode parent population, the shape is predictable. The base is tighter and the shape is normal looking.
The most interesting result lies in the appearance of the distribution of sample means from the bi-modal parent population. Although the base is wider than the one from the single mode population, the shape is again normal-looking. Now it is possible to describe the central limit theorem.
A typical textbook explanation indicates that when one averages many samples of the size n and places them into a distribution of sample means, that distribution of sample means will have a mean close to the mean of the population. It will have the shape of a normal distribution and the variability will be equal to the standard deviation of the population divided by the square root of the sample size. My students can memorize this, but often do not really comprehend what it means until it is broken down into three pieces, indicated below.
Taking many samples of the size n, the averages of these samples form a distribution in which:
- The mean of the distribution of sample means is close to the mean of the population.
- The shape of the distribution of sample means is normal-looking.
- The variability of the distribution of sample means is less than the parent population.
Note: It is equal to the standard deviation of the population divided by the square root of the size of the sample used to create the distribution of sample means. This is called the standard error.
Understanding the central limit theorem means knowing the basis upon which control charts work. This will be addressed in next month’s quality quiz.
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