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Even though the distribution in the universe is not normal, the distribution of Xbar values tends to be close to normal. The larger the sample size and the more nearly normal the universe, the closer will the frequency distribution of averages approach the normal curve.
However, even if n is as small as 4 and the universe far from normal, the distribution of the averages of the samples will be very close to normal. Shewhart illustrates this by showing the distributions of averages of 1,000 samples of four from each of two bowls of chips, one containing a rectangular and the other a triangular distribution. Neither of these universes even faintly resembles a normal curve. However, the distribution of the samples drawn from these bowls fairly approximates normal.
The main point of Shewhart’s bowl is this. Even with great departures from normality in the universe, the distribution of Xbar values with n=4 are approximately normal. In sampling from most distributions found in nature and industry, the distribution of Xbar values will be even closer to normal.
However, it is of interest to observe that distributions similar to the rectangular and triangular distributions sometimes found in industry. Although they seldom occur as a result of production alone, they may be found as a result of production followed by 100% inspection. For example, if a production operation gives a distribution on a certain dimension, which is roughly normal with a standard deviation of 0.001, and the specified tolerances on the dimension are +/- 0.001 cm, it is obvious that only about 68% of the product will meet the specification. If the production operation accurately centers the dimension at its specified nominal value, the go gage and another 16% will reject about 16% of the product by the no-go gage. The distribution of the accepted product will not be far from rectangular. There will be two distributions something like the triangular, one for the product rejected by the go gage and the other by the no-go gage.
The great practical importance of the normal curve arises even more from its uses in sampling theory than from the fact that some observed distributions are described by it well enough for practical purposes. Of great practical significance is the fact that distributions of averages of samples tend to be approximately normal even though the samples were drawn from non-normal universes.
Grant and Leavenworth, Statistical Quality Control, pp. 60-62.
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Regards,
dWizard
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I was better but I got over it.