# Is it possible to get Natural Tolerance (Tn) with Non Normal Distribution?

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#### Mark_Navigator

Hi all,

this is a question so long in doubt for me and that I need......
If I get a non normal distribution, Am I able to calculate and communicate to the designer the actual process natural tolerance, if the distribution found is non normal?
Specially with new processes and no Tp (project tolerance) given, the natural tolerance is the crucial information..........
Thank you whoever explain me and to all this "heavy" question............

#### Bev D

##### Heretical Statistician
Super Moderator
if this is for design and you aren't doing statistical tolerancing (which you shouldn't do if the distibution isn't known) AND you have representative data (several lots, set-ups, different operators, raw materials etc.) you can probably safely use the min and max values of the observed data

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#### Mark_Navigator

The question is to get the natural tolerance of the process with non normal distribution, specially in the case I tried, with a not previous designed dimension in a new product zero series production........

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#### Barbara B

The so called natural tolerance is the area which covers hte middle 99.73% of a distribution. (0.27% of the distribution values will be outside of this area). The limits of this range are defined by the 0.27%/2 = 0.135% quantile (lower limit) and the 100% - 0.27%/2 = 99.865% quantile (upper limit), independent of the distribution itself.

In case you have normally distributed data, the lower limit is equal to
mean - 3*stddev
and the upper limit is equal to
mean +n 3*stddev

For other distributions the quantiles could be calculated using a statistical software package (e.g. Minitab, JMP, R) if thes provide quantile calculations for the specific distribution you're interested in.

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#### Mark_Navigator

Thank you so much Barbara, I know, but how can MINITAB (that we use in our company for statistical analisys) calculate the 99,73% area for non normal distributions?
I'm a Clements' method sympathizer, but it's not mentioned in MINITAB and I tried by myself to use it but I got some doubts about the usage....(like the "teta" table selection for specified skewness and kurtosis data).......

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#### Barbara B

Thank you so much Barbara, I know, but how can MINITAB (that we use in our company for statistical analisys) calculate the 99,73% area for non normal distributions?
1. Choose a distribution for your data (it should suit the measurement situation, e.g. Weibull for reliability data)
2. Estimate the parameters of the distribution, e.g. using
Graph > Probability Plot > Single
Graph variables: [name of the data column]
> Distribution: [select the distribution from the drop down list on tab "Distributions"]
> OK > OK
3. Check if there are any relevant/significant deviations visible on the probability plot (points should lie within the 95% confidence limits, points should follow the blue line in a linear manner) and if the distribution test reveals any significance (p-value should be above 5%). If the distribution could be used to model the data, proceed with the next step. If not, ask yourself what systematic effects are present in your process and/or if your process is stable at all. Afterwards eliminate all systematic effects and/or stabilize your process and go back to step 1.
4. Calculate the 0.135% and 99.865% quantile:
1. Fill in the percentage values (e.g. 0.00135 for the lower quantile and 0,99865 for the upper quantile) in an empty colum of your worksheet (e.g. c2)
2. Calc > Probability Distribution > [select appropriate distribution, see step 3]
select "inverse cumulative probability" (=quantile function)
fill in the estimated distribution parameters (e.g. shape and scale for Weibull)
select the input column (e.g. c2) and an empty column for the calculated quantiles (e.g. c3)
> OK

I'm a Clements' method sympathizer, but it's not mentioned in MINITAB and I tried by myself to use it but I got some doubts about the usage....(like the "teta" table selection for specified skewness and kurtosis data).......
I appreciate this apparent "lack" of feature in Minitab, because the Clements/Pearson method is not useful to obtain reliable capability indices.

In ISO/TR 22514-4:2007: "Statistical methods in process management -- Capability and performance -- Part 4: Process capability estimates and performance measures" it is stated that a Johnson transformation or the usage of the Pearson/Clement's method shouldn't be used as a standard calculation (see p.20):
5.5.3 Pearson curves method
[...]
This method is not preferred but is presented here for completenesss due to its occasional use.

This approach, and a similar one based on Johnson curve, should be regarded with considerable caution, especially when it is a procedure within a "black box" computer program used to analyse large sets of data.[...]

But you can use this approach and say that your method is in compliance with a standard (or technical report TR). In the end you'll get non-informative numbers which can be called "process capability indices". (Sorry for being that blunt. I know it is hard to find a useful and reliable statitistical method for capabilities.) Just take a look at the attached paper to get an impression for yourself how "reliable" the Pearson/Clement's-method and the Johnson as well as the Box-Cox transformation are.

Your're welcome

#### Attachments

• How reliable are different methods for capability indices.pdf
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#### bobdoering

Trusted Information Resource
Just take a look at the attached paper to get an impression for yourself how "reliable" the Pearson/Clement's-method and the Johnson as well as the Box-Cox transformation are.

Awesome paper - it is the approach I have been supporting for a long time! If you don't use the right model the results realy have a hard time being correct!

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#### Mark_Navigator

Hi Barbara,

I'm trying your instructions but I'm not able to go on with point 4:

It's not clear for me at subpoint No.1 -> "Fill in the percentage values (e.g. 0.00135 for the lower quantile and 0,99865 for the upper quantile) in an empty colum of your worksheet (e.g. c2)".
Which percentage values do I have to fill in there? How can I find that values?

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#### Barbara B

Which percentage values do I have to fill in there? How can I find that values?
The limits of the natural process spread (or natural tolerance) are given by the 0.135%- and 99.865%-quantiles of a distribution. You don't have to choose other values, but simply use those two numbers for the calculation.

Attached you'll find 3 screenshots with an example for simulated Weibull data (first column / c1 / "data").
• Fig 1: Fill in the percentage values for process spread quantiles (0.00135 and 0.99865)
• Fig 2: Use Calc > Probability Distribution > [select distribution, here: Weibull],
1. Choose "Inverse cumulative probability"
2. Fill in the distribution parameters provided in the probability plot (e.g. shape 1.294 and scale 183.1) and
3. select "Input column:" c2 (second column, the one which contains the percentages for quantiles).
4. To store the quantile values within the worksheet, fill in a number of an empty column in the worksheet (e.g. c3 for column 3).
• Fig 3: Result of the settings: quantile values are stored in the worksheet column 3 (c3): 1.110 for the 0.135% quantile, 787.805 for the 99.865% quantile
• Finally you can calculate the 99.73% process spread or natural tolerance (nt) by subtracting the lower 0.135%-quantile from the upper 99.865%-quantile
nt = Q(99.865%) - Q(0.135%) = 787.805 - 1.110 = 786.695

Hope this helps

Barbara

#### Attachments

• Quantiles in Minitab 1.png
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• Quantiles in Minitab 2.png
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• Quantiles in Minitab 3 - result.png
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#### Mark_Navigator

Thank you so much, Barbara!!!

Now I finished to test the procedure and it's so.....wonderful!!!!
I got the "nt" of one my non normal distribution found!!!!!
You open me and probably in my factory a very, very important way!!! No I'm able to get all indicators for also non normal distributions!!! GREAT!!!!
Another time, thank you so much, you've been so useful!!!!! You're my Guru!!!!!