Q 607. What is the difference between Zoverall and Zwithin? Which one is more appropriate for determining the baseline performance in a DMAIC project?

 

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Z-score : It basically is the most commonly used measure of process capability in six sigma. It usually indicates the number of standard deviation fitting between the mean and the specification limit

 

Z within : Z within measures the ability of a process in a specific subgroup or sample. It evaluates how well a process performs against its  sample data.

Formulae used : z= (X-Xbar)/s

Where :   X represents the value which needs to be evaluated

                 X bar represents the sample mean

                 S represents the sample one standard deviation

 

Z overall : Z overall measures the overall process capability, its basically gives us an idea that how a particular process is performing irrespective of subgroups.

 

Formulae used :  z = (x – μ) / σ

 

Where :   X represents the value which needs to be evaluated

                 μ represents the overall process mean

                 σ represents the overall process one standard deviation

 

The option of selecting the right Z score in a DMAIC project depends on the  focus, data structure, and goals of the project. Both metrics are valuable for process improvement, but  provide different perspectives on process capability and efficiency. It is important to choose the one that meets your project goals and provides the most relevant insights for your improvement efforts.

However we give better preference to Z overall

As it provides a complete picture of the ability of the process to meet customer requirements in all subgroups or categories. This can be valuable if your primary goal is to evaluate the overall performance of the entire process and determine whether it meets your organization's standards or customer expectations.

The measurement known as the z score is used to determine how a particular value varies, from the average of a group of values. It helps us understand the number of times the variation within the group can fit between that value and the average. The standard deviation represents this variation while the average is referred to as the mean. To calculate a z score for a value we use this formula; z score = (value - mean) / standard deviation.

 

When we may not have access to all values in a group and we only have a sample of them, we can utilize the standard deviation of that sample instead of considering the entire group. The sample average is called the sample mean, while its standard deviation is referred to as the sample standard deviation. The formula for determining a z score using these values would be; z-score = (value - sample mean) / sample standard deviation.

 

We can assess how a process performs using Zoverall and Zwithin measurements, which indicate how closely an average result aligns with our desired outcome.

 

Zoverall examines variations, across all results over time, regardless of when or how they were obtained.

When analyzing the results of conditions, such, as those obtained on the day or using the same machine Zwithin provides insights into the variation within the sample (usually much shorter than the overall time period).

 

It's important to keep in mind that process changes can be normal or abnormal. 

 

When evaluating process performance for baselining anticipating before and after making improvements, the choice between Zoverall and Zwithin depends on two factors; 

 

1. The type of data available i.e. whether it is continuous or discrete.

2. The stability of the process i.e. whether it remains consistent over time or undergoes changes.

 

If we have continuous data like measuring weight or length and the process is stable with less variations over time using Zwithin might be more suitable. This metric reveals variations within the process itself.

On the other hand if we have data, such as counting good or bad outcomes or if the process is unstable and experiences significant changes over time then using Zoverall could be more appropriate. This metric captures variations, in the process that may impact customers.

 

Using both data and comparing them can provide an understanding of the effectiveness of the process and identify areas for improvement. It's beneficial to maintain these numbers as records for our DMAIC project.

Before we go into an explanation of Z.overall, Z.within and Z.bench, We need to first understand what is meant by the letter Z.  “Z” is the number of standard deviations between the mean and the specification limit. In process capability calculation it is the mean distance from USL and LSL measured in Standard deviation.

The formula for calculation Z. score is

The graph below indicates the probabilities associated with the normal distribution and Z values.

For example: our customer lead time is 240 minutes. Historically, lead time average is 150 minutes and a standard deviation of 55 minutes.
Assuming a normal distribution, Z score is,

 

      = (240 – 150) / 55

       = 1.63

i.e., 1.63 Standard deviations above the mean.

The Z score is very useful because it can convert any continuous measurement into a generic value, allowing comparisons between different process is easier.

Z.bench can be used as measure of process of capability. Various Z.bench calculation are Zwithin or Z_ST and Zoverall or Z_LT

Minitab has a selection to calculate Benchmark Z’s or Sigma levels along with the Cp and Pp statistics. By selecting the option in minitab, these graph as shown below will be displayed.


Difference between ZOverall and Zwithin:

Sl. No	Zoverall or ZLT	Zwithin or ZST
1	It is an indices used as an alternate to Pp	It is an indices used as an alternate to Cp
2	Zoverall indicate the actual long-term capability of the process	Zwithin indicate the potential short-term capability of the process
3	Z is calculated using the overall standard deviation	Z is calculated using the within standard deviation
4	Represents long term variation of the process	Represents the potential variation of the process
4	Also known as Short term Z.Bench	Also known as Long term Z.Bench

 

Benefits of Z bench

Ø  Z bench converts continuous data into a number of standard deviations, which can be used as Generic value of measurement between the process.

Ø  It serves as a tool for comparison between different process. If the Process “X” has a higher Z bench than process “Y”, Process “X” is better in meeting specs than process “Y”. It doesn’t matter what the process type, the comparison is relative measure of capability. 

Ø  Process “X”  is Better when compared to Process “Y”

Zwithin is important in DMAIC project for determining the baseline performance because it is essentially a prediction of the ability of a process to meet a specification. However, Zoverall also important because it indicates how the actual process performed over a period of time.

 

This definitely was a tricky one.

The respondents have given the correct explanation of Zoverall and Zwithin. However not all have been able to correctly tell if Zoverall or Zwithin should be used for baseline in a project.

D Nandakumar has mentioned it correctly and hence is the winner for this question.

 

P.S. Small correction in D. Nandakumar's answer - 

last line - Zoverall is not over a period of time. Both Zoverall and Zwithin are process performances at the given moment of time.