Vol. 9, No. 6

June 2007

PQ Systems
 
Contents

Keeping track of 2,000 gages

Quality Quiz: With a video!

Data in everyday life

MSA with Jackie Graham

Bytes and pieces

FYI: Current releases

 

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Quality Quiz from Professor Cleary

Congratulations:
"B" is correct.

Click here for a more complete video explanation

While the central limit theorem is not related to Einstein’s theories, it nonetheless lies at the heart of statistical analysis.

The current textbook that I use in classes at Wright State University defines the central limit theorem as follows:

In selecting simple random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal probability distribution as the sample becomes large (Anderson, Sweeney, Williams, Statistics for Business and Economics. Southwestern Publishing, 2006, p. 258).

Of course, one can only speculate what might happen if Hy were to recite this definition to Emily.

A way that I have found to be successful in understanding the central limit theorem (CLT) to university students, managers, line workers, and others follows.

I begin by sharing an article from a newspaper citing the average age of the population of Dayton is 30, the standard deviation is 9, and the shape of the distribution is normal. I ask them to sketch this distribution, recalling the characteristics of a normal distribution;

      • It is symmetrical (bell-shaped);
      • of the area under the curve
      • of the area under the curve
      • of the area under the curve
      • The total area under the curve is equal to 1

Eventually, I have a student sketch out the distribution, below. As one can see, the mean is 30, the upper tail comes down at 57, three standard deviations to the right, and the lower tail comes down at 3, two standard deviations to the left.

3 12 21 30 39 48 57

Next, I ask each student to imagine taking a random sample of 36, and then to write down their estimation of the mean of that sample. Then I collect their estimates and post them:

34
28
30
38
31
26
22
=
=

What’s wrong with these estimates of the average? Note the high of 38 and the low of 22 in this group. I ask students to envision a distribution (histogram) of these averages, then ask: a) What would you expect the mean to be? b) Would this distribution of sample means be wider, tighter, or the same as the population? C) What shape would it take?

Some class hero generally emerges, noting that the mean will be close to 30, the shape will be normal, and the distribution will be tighter, since it is made up of averages rather than individual values.

The inevitable question is how much tighter will the distribution of sample means be, relative to the population. The standard deviation of the population is 9; what would the variability of distribution of sample means be?

This is where the magic comes in. When I say that the variation (standard deviation) of the sample means (known as the standard error) is equal to the standard deviation of the population divided by the square root of the sample size used to create the distribution of sample means. In this case:

This leads to the creation of a diagram of this distribution. To review what is known:

  • The shape is normal.
  • The mean is ( ) 30.
  • The standard deviation (known as the standard error) is equal to

The following chart demonstrates this graphically:

When I draw this on the white board, most students are surprised how much narrower the distribution of sample means is, relative to the population. I remind them that this distribution is made up of means from many samples of the size 36.

Tune in next month when I prove the central limit empirically using the Quality Gamebox. It will be so simple a chimpanzee will be able to understand it.

Click here to register to win a free Quality Gamebox program.

 

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