Z-Score, Z-Value

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Standard Normal Z Score


Purpose

Z-Score, or z-value, is used to determine the probability of defect for a random variable critical characteristic which is measured in continuous data. The Z transform is an important tool because it enables this calculation without the complex mathematical calculations which would otherwise be required.


Anatomy

Z Score

Standard Normal Curve - Normal Distribution

Reference: Juran's Quality Control Handbook

Terminology

A. The standard transform, Z, transforms a set of data such that the z-value mean is always zero (0) and the z-value standard deviation is always one (1.0). By virtue of this transformation, the raw units of measure (e.g. inches, etc.) are eliminated or lost so the Z measurement scale is without units.

B. X is the value that a random variable CT characteristic can take.

C. The mean of the population. When using a sample, an estimate of mu such as Xbar will be used to substitute mu.

D. The standard deviation of the population. When using a sample, an estimate of sigma such as "s", will be used to substitute mu.

E. The area under the z-value Normal Curve Table, for normal distributions where mu=0 and standard deviation =1.0, is the reference used to find the surface that lies beyond the value of X.

F. This area represents the probability of a defect. When X takes the value of a performance limit, for example a specification limit (SL), the area under the normal curve which lies beyond the Z value is the probability of producing a defect.


Major Considerations

The use of the z-value Standard Normal Deviate assumes that the underlying distribution is Normal. When establishing a rate of nonconformance with the Z value, if the actual distribution is markedly skewed (i.e. non-normal), the likelihood of grossly distorted estimates is quite high. To avoid such distortion, it is often possible to mathematically transform the raw data.

Application Cookbook

1. To calculate the Z value from sample data, apply the formula and replace population average and the population standard deviation with Xbar and "s" respectively.

2. Use the following Excel function (NORMSDIST) to obtain the probability related to a Z value. Note that Excel gives the probability to be lower than the Z value. In order to obtain the probability of being greater than a Z value, simply use 1-NORMSDIST.

3. In Minitab use the function "Calc>Probability Distribution>Normal" to obtain the probability to be lower than a Z value.

Cumulative Distribution Function

Normal: mean = 0 and standard deviation = 1.00000

X--------P(X < SL)
2.91----0.9982

1-0.9982= .0018 (.18%) which represents the probability of having a value greater than Z.

Z-Value: Short-Term


Purpose

To be able to evaluate short-term process performance. To rate performance based on benchmarking.

Anatomy

Z Value

Z-Score Short Term


Reference: Mikel J. Harry, The vision of six sigma

Terminology

A. General equation of Z

B. Z short-term for Upper specification limit

C. Z short-term for Lower specification limit

D. Specification limits, which are an expression of the CTs

E. Central tendency to be used in the calculation; Target = T for short-termartificially centered process

F. Short-term standard deviation


Major Considerations

  • Z is a metric
  • Z is always in term of "how many short-term sigma"
  • Z makes a bridge between process and Normal probability
    The Six Sigma objective is to achieve a ZST level of 6 or higher

Application Cookbook

1. Take the reference for the central tendency. Target value is used because Z short-term reflects "process capability" under the assumption of random variation.

2. Estimate the standard deviation short-term

3. Compute both specification limits

4. Apply the proper formula to compute Upper and Lower Z short-term
Note: Z upper = Z lower if the target is the specification mid point.

Z-Value: Long-Term


Purpose

To be able to evaluate long-term process performance. To statistically estimate the PPM, DPMO.

Anatomy

Z Value

Z-Score Long Term


Reference: Mikel J. Harry, The vision of six sigma

Terminology

A. General equation of Z

B. Z long-term for Upper specification limit

C. Z long-term for Lower specification limit

D. Specification limits, which are an expression of the CTs

E. Central tendency to be used in the calculation

F. Long-term standard deviation


Major Considerations

  • Z is a metric
  • Z long-term is always in term of "how many long-term sigma"
  • Z makes a bridge between process and Normal probability
    The Six Sigma objective is to achieve a ZLT level of 4.5 or higher

Application Cookbook

1. Choose the reference for the central tendency

  • for the actual distribution, use m (i.e. the overall mean)
  • for artificially centering on the mid point of the specification, use T(i.e. the target )

2. Estimate the standard deviation long-term

3. Compute both specification limits

4. Apply the proper formula to compute Upper and Lower Z long-term

Z-Test – One Sample


Purpose

To compare the population mean of a continuous CT characteristic with a value such as the target. Since we don't know the population mean, an analysis of a data sample is required. This test is usually used to determine if the mean of a CT characteristic is on target when the sample size is greater or equal to 30.

Anatomy

Z Test

One Sample Z-Test


Reference: Basic Statistics by Kiemele, Schmidt and Berdine

Terminology

A. Null (H0) and alternative (Ha) hypotheses, where the population mean (mu) is compared to a value such as a target.

For the alternative hypothesis, one of the three hypotheses has to be chosen before collecting the data to avoid being biased by the observations of the sample.

B. Minitab Session Window output.

C. Hypotheses tested: H0 is equal to 10 vs. Ha is not equal to10.

D. Descriptive Statistics - Name of the column that contains the data, sample size, sample mean, sample standard deviation (StDev) and standard error of the mean.

E. Computed Z statistic using the formula below to obtain the z-value.

Z-Score Formula

Calculating the Z Score

Calculating The Z-Score

F. P-Value – This value has to be compared with the alpha level and the following decision rule is used: if P < alpha, reject H0 and accept Ha with (1-P) 100% confidence; if P is equal to or greater than alpha, don't reject H0.


Major Considerations

The assumption for using this test is that the data comes from a random sample with a size greater or equal to 30. It can also be used when the standard deviation of the population is known but this case is quite rare in the practice.

Application Cookbook

1. Define problem and state the objective of the study.

2. Establish hypothesis - State the null hypothesis (H0) and the alternate hypothesis (Ha).

3. Establish alpha level. Usually alpha is 0.05.

4. Establish sample size (see tool Sample Size - Continuous Data – Z Test - One Sample).

5. Select a random sample.

6. Measure the CT characteristic.

7. Analyze data with Minitab:

  • Use the function under Stat>Basic Statistics> 1-Sample z.
  • Select the Test mean option, input the target value and the desired alternative hypothesis (>, <, not equal: the default setting).
  • Enter a sigma value. This value can be the sample standard deviation (s) or a known value of the population standard deviation.

8. Make statistical decision from the session window output of Minitab. Either accept or reject H0.

9. Translate statistical conclusion to practical decision about the CT characteristic.

From Z-value to Six Sigma Tools.

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