# Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-Bench

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#### ottomotto

Hi,

can you please tell me which role the Inverse cdf of a standard normal distribution in formula for Z-Bench plays?

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Z.Bench = F-1 (1 - P1 - P2)

where:

P1 = Prob (Observations < LSL)

P2 = Prob (Observations > USL)

P1 and P2 probabilities are based on the nonnormal distribution used in the analysis with parameters you that you specify or estimated from the data.

F (X) = cdf of a standard normal distribution

F-1 (X) = Inverse cdf of a standard normal distribution
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Does it ouccr because of the numbers of standard deviations between the middle and the upper or lower limits is represented by this inverse?

Thank you very much!

#### Miner

##### Forum Moderator
Staff member
Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

The CDF takes a z-score and converts it into a probability. The inverse CDF takes a probability and converts it to a z-score. It is similar to the relationship between SIN and ARCSIN in trigonometry

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#### ottomotto

Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

The CDF takes a z-score and converts it into a probability. The inverse CDF takes a probability and converts it to a z-score. It is similar to the relationship between SIN and ARCSIN in trigonometry
Thank you, but why is actually the Inverse cdf of a standard normal distribution used in this formula for non-normal distributed values?

#### Miner

##### Forum Moderator
Staff member
Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

The entire concept of a Z_bench score is based on the standardized normal distribution.

The idea behind Z_bench is to convert disparate measures (e.g., Cpk, DPMO, etc.) into a common metric. Hence the name Z_bench for Benchmark Z score. This is used in Design for Six Sigma to promote prioritization of improvement efforts, similar to using SEV, CRIT and RPN to prioritize efforts on an FMEA. Given that, the Z_bench score can be an estimate versus a pure mathematical calculation.

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#### ottomotto

Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

The entire concept of a Z_bench score is based on the standardized normal distribution.

The idea behind Z_bench is to convert disparate measures (e.g., Cpk, DPMO, etc.) into a common metric. Hence the name Z_bench for Benchmark Z score. This is used in Design for Six Sigma to promote prioritization of improvement efforts, similar to using SEV, CRIT and RPN to prioritize efforts on an FMEA. Given that, the Z_bench score can be an estimate versus a pure mathematical calculation.
AH ok nice! I understood thanks.

So can I say the standardized normal distribution is in this forumla because of the entire concept in Z_bench an in special to calculate the number of deviations between average point and the curves?

It is very interesting for me which actual task the inverse of the standardized normal distribution does in this case in special.

#### Miner

##### Forum Moderator
Staff member
Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

Would you please rephrase your question? I am unsure what you meant, and I would rather not guess and add confusion.

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#### ottomotto

Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

Would you please rephrase your question? I am unsure what you meant, and I would rather not guess and add confusion.
Thank you very much for your help and patience No problem, i will try to rephrase my question.

On the one hand you can say the standardized normal distribution is in this forumla because of the entire concept behind the Z_bench value.

An the question on the other hand is which special task does the inverse cdf fulfill in the described formula above. Is it in special to calculate the number of deviations between the average point and the curve?

In a whole aspect i am understanding now the cause why the cdf is even in the formula for non-normal data.

But there should be an actual mathematically task of the inverse of the standardized normal distribution in the formula above. Which task does the inverse cdf fulfill mathematically exactly in the given order of the formula?

#### Miner

##### Forum Moderator
Staff member
Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

On the one hand you can say the standardized normal distribution is in this forumla because of the entire concept behind the Z_bench value.

An the question on the other hand is which special task does the inverse cdf fulfill in the described formula above. Is it in special to calculate the number of deviations between the average point and the curve?

In a whole aspect i am understanding now the cause why the cdf is even in the formula for non-normal data.

But there should be an actual mathematically task of the inverse of the standardized normal distribution in the formula above. Which task does the inverse cdf fulfill mathematically exactly in the given order of the formula?
As I said in an earlier post, the inverse cdf converts the probability of observations within the specifications into a standardized z-score. You can then interpret that z-score as the sigma level of your process, which is the number of standard deviations between the average and the specification limit.

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#### ottomotto

Re: Which role plays Inverse cdf of a Standard Normal Distribution in Formula for Z-B

As I said in an earlier post, the inverse cdf converts the probability of observations within the specifications into a standardized z-score. You can then interpret that z-score as the sigma level of your process, which is the number of standard deviations between the average and the specification limit.
Ok thank you very much!
You gave me very helpful support. 