# Dealing with non-normal distributions

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- July 27, 2005 at 10:12 pm #40159
Is the Empirical Rule applicable to non-normal distributions?

Once non-normality is determined, how does one then determine what type of distribution he or she is dealing with?

Thanks.

0July 27, 2005 at 10:23 pm #1237651. No. But, Chebychev applies a similar concept but to any continuous distribution. Research it.

2. Fit a model, usually a probability plot and look for the p value. Easy to do with Minitab.0July 27, 2005 at 10:30 pm #123766Thanks Darth.

0July 27, 2005 at 10:45 pm #123771Can the empirical rule be applied to non-normal distributions? Im also tempted to just say No.

But you need to ask the question could a non-normal distribution plot both symmetrically and bell-shaped? The answer to that is Yes.

Therefore could you conceivably find a non-normal but symmetrical bell-shaped plot that met the empirical rule segmentation ratio? Maybe.

But rather than futz with it why would you want to try to apply the empirical rule, if you knew you were dealing with non-normality? Why potentially compound your margin of error by misapplying tool fitments?

Vinny0July 27, 2005 at 10:45 pm #123772FYI

Chebychev and Empirical Rules

Knowing the mean and standard deviation of a sample or a population gives us a good idea of where most of the data values are because of the following two rules:‘s Rule The proportion of observations within k standard deviations of the mean, where , is at least , i.e., at least 75%, 89%, and 94% of the data are within 2, 3, and 4 standard deviations of the mean, respectively.

Empirical Rule If data follow a bell-shaped curve, then approximately 68%, 95%, and 99.7% of the data are within 1, 2, and 3 standard deviations of the mean, respectively.

EXAMPLE: A pharmaceutical company manufactures vitamin pills which contain an average of 507 grams of vitamin C with a standard deviation of 3 grams. Using Chebychev’s rule, we know that at leastor 75% of the vitamin pills are within k=2 standard deviations of the mean. That is, at least 75% of the vitamin pills will have between 501 and 513 grams of vitamin C, i.e.,

EXAMPLE: If the distribution of vitamin C amounts in the previous example is bell shaped, then we can get even more precise results by using the empirical rule. Under these conditions, approximately 68% of the vitamin pills have a vitamin C content in the interval [507-3,507+3]=[504,510], 95% are in the interval [507-2(3),507+2(3)]=[501,513], and 99.7% are in the interval [507-3(3),507+3(3)]=[498,516].

NOTE: Chebychev’s rule gives only a minimum proportion of observations which lie within k standard deviations of the mean.0July 27, 2005 at 10:52 pm #123774Just looking to broaden my understanding. Can someone indicate where in minitab you would go to deternine the the type of non-normal distribution? Thanks.

0July 27, 2005 at 11:07 pm #123777Disregard. Thanks.

0July 27, 2005 at 11:09 pm #123778Not knowing what you are trying to do, I suppose Id use a goodness of fit test either the Anderson-Darling or the Kolmogorov-Smirnov test depending on sensitivity needed and how much weight needed to be given to the tails of the distribution.

Vinny

0July 27, 2005 at 11:32 pm #123782Vinny,

You’re on the mark with your response. The E-rule applies to symmetric bell-shaped distributions providing approximate probabilities between the mean and xSDs from the mean. Good call!

Ken0July 28, 2005 at 2:09 am #123786Try Graph/Probability Plot……try a few different distributions and see if one fits the data better than another. The p value should be of some help.

0July 28, 2005 at 4:03 am #123794“Distribution ID plot” will run the tests for about 11 different distributions all in one whack. Also look at https://www.isixsigma.com/library/content/c050725a.asp this week.BTDT

0July 28, 2005 at 2:02 pm #123841Darth and I have argued about this in the past, but I’m still a liberal when it comes to the empirical rule. For example, I fully agree that a perfect normal distribution will have 99.73% of the points between +/- 3 sigma. However, distributions such as a perfect uniform and right triangle distribution will have 100% within +/- 3 sigma, so they don’t viloate the rule; very “heavy tailed” distributions like Chi-square w/ 2 degrees of freedom and exponential will NEVER have points below -3 sigma and only about 2-3% over the +3 sigma level. To answer your question: no, they don’t fit the rigid statistical rule, but they sure do come close…

0July 28, 2005 at 2:20 pm #123844Thanks to everyone. Very helpful.

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