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# Sigma Level Shift

Sigma Level Shift is the difference between the within (also known as short term) performance and the overall (also known as long term) performance. Basis the process and data observations Motorola estimated that overall capability of the process is 1.5 less than the within capability (though this number may vary from process to process and industry to industry).

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Himanshu Sharma on 22nd Nov 2022.

Applause for all the respondents - M V Ramana, Anuj Bhatnagar, Gulshan Kumar, Himanshu Sharma, Mohamed Safir, Rahul Arora.

## Question

Q 523. Short term capability is sometimes calculated as Zlt + 1.5. This 1.5 is typically known as Sigma Level shift. What is the rationale behind this 1.5 shift? Is it a valid assumption?

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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Short term capability is sometimes calculated as Zlt +1.5 – Mostly when data is discrete because discrete data is almost every time long term or long term capability is sometime calculated as Zst -1.5. More often we have short term data and we calculated short term capability and to calculated long term capability we subtract 1.5 as Zst-1.5, because long term variation is more than short term variation. If focus on Y=f(X) , ie Output is function of Input than in short term there is very less variation in input factors as Man, machine, material , method, Mother Nature (Environment) and we get better Sigma level or capability but in long term input factor changes and effect output Y as well. So in long term 1.5 sigma level shift considered. Below is the Example of long term variation:

Y=f(X)

Output Y = f( man, machine, Material, Method, Measurement, Mother Nature)

Man – If we see man variation on long term is very significant because a man may be consistent for a month but for a year or six month he takes leaves, or leave the job then new man join or time to time change in his thoughts, mood and efficiency & focus, leads to more variation compare to short term. Although manpower training and competency process is to take care this factor but still variation will more in long term.

Machine – Machine may be consistent in short term means we get less variation in output  of a hour or a shift compare to variation  in output of six month or one year. Machine wear tear increase with the time, inadequate preventive maintenance, machine breakdown etc leads to more variation in long term. Machine Setting is also important aspect if machine setting change on every setup then process shift and more variation in long term. Companies do preventive maintenance and machine accuracy checks but still as machine gets older, consistency in output reduces and results in more long term variation.

Material – Input material consistency also change with the time for example – a sheet metal coil having less variation within coil but more variation between coils. Means one coil is not same as other coil. Another factor is Batch to Batch variation so batch to batch variation is more than within batch variation. Within coil, Coil to Coil, batch to batch variation is more than output may not be consistent in long term.

Method - To control method we have defined SOP and training to operator but if manual process and lack of poka yoke then some inconsistency can take place which can affect output. Ineffective root cause analysis also affects methods, which may lead to wrong countermeasure and unintended change in methods. Adherence of SOP by operator is also important if sop is available, displayed but operator does not follow consistently then variation will be more in long term.

Measurement – Consistency in measurement is also important in long run. In long run instrument or gauge wear tear take place so time to time Gage R&R and gage Stability study require. If our measurement is not consistent in long term then variation will be more in long run.

Mother Nature – Everyone likes nature variation as different weather, day night, morning evening but sometimes it leads to inconsistency in process and effect output. For Example – Investment casting process require stable environment as less temperature and humidity. So Investment Casting industry located in Nashik, INDIA because of more suitable environment compare to other part of INDIA. We all know long term variation in always more than short term variation in Environment which lead to high variation in long term.

So from above examples we understood that no process can be static over the time, even excellent process. By Convention, this long term variation is defined as 1.5σ Correction in short term α level. A 6σ short term process is considered 4.5α long term process. One more way to look at It:

·         Variance Within each subgroup can be pooled to determine an average of the within subgroup standard deviations.

·         Total Standard deviation is calculated from all of the data without regard to subgroup.

·         Pooled standard deviation does not account for between subgroup variation. Total standard deviation does.

·         Pooled Sigma is best estimate of within group variation.

Short term and Long term Data Collection

Short term

·         Gathered over limited number of cycle of interval.

·         Gathered over limited number of machine or operator.

·         Almost Always Continuous variable data

Long Term

·         Gathered over many cycles, intervals, equipment, operator.

·         May be discrete or continuous

·         Discrete data is Almost always long term.

Below is the short term and long term Sigma Level and respective yield:

 Sigma Level Short Term Sigme Level Long Term % Yield 2 0.5 69.15 3 1.5 93.32 4 2.5 99.38 5 3.5 99.98 6 4.5 99.99966

So it is a valid assumption that consider 1.5σ shift in long term.

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The two distinct interrelated dimensions of a process capability are

1.      Short-term capability, or simply Z-st and

2.      Long-term capability, or just Z-lt.

Z-shift = Z-st – Z-lt or

Z-st = Z-lt + Z-shift and

Z-lt = Z-st – Z-shift.

For Z-shift, we consider the underlying mathematics. The Z-st is given as Z-st = |SL – T| / S-st, where SL is the specification limit, T is the nominal specification and S-st is the short-term standard deviation.

The short term standard deviation is computed as S-st = sqrt [SS-w / g (n – 1)], where SS-w is the sums-of-squares due to variation occurring within subgroups, g is the no:of subgroups, and n is the no:of observations within a subgroup.

Z-st assesses the ability of a process to repeat any given performance condition, at any arbitrary moment of time. Instantaneous reproducibility is measured by Z-st.

The sampling strategy must be designed such that Z-st does not capture or otherwise reflect time related sources of error. The metric Z-st must echo only random influences.

1.5 SIGMA SHIFT

1.5 sigma shift is a statistical correction or in simple words a buffer which is created to protect processes and products from variation which is a constant companion of long-term projects.

Example:

Planning to go to a remote town for two days and this place has limited supplies of all major amenities and need to stay there for two days, assuming will keep extra phone batteries or battery banks, extra set of clothes and whatnot, basically preparing to face any kind of deviation or variation and this is what 1.5 sigma shift is used for.To improve process performance by a data-based methodology to bring down the number of defects to 3.4 defects per million opportunities is the concept of Six sigma and this includes 1.5 sigma shift.

3.4 defects ideally means near to zero defects but statistically is equal to 2 defects per billion opportunities.

Six sigma is a measure of variation and if a process is at 6 sigma capacity indicates working at an efficiency that it is 3.4 defects Per million opportunities which has near-zero defects and statistically 6 sigma is equal to 2 defects per billion opportunities, the reason behind this is 1.5 sigma shift.

All processes are designed to meet the specification limits but as a law of thermodynamics, it states entropy enters and variation plays its role and when this happens

Either Process Standard Deviation goes up OR Mean of the process moves away from the center and this standard deviations will fit in between mean and spec limits which decreases sigma level, so to accommodate this variation, the concept of 1.5 sigma shift was introduced.Although control limits are maintained but “Control limits are not enough or a better word is Sufficient”

Reasons are (1) Sampling errors (2) Control charts will not detect each and every movement in process Avg and (3) Variability in data collection

Other reasons are, the biggest error in production is OVERSIMPLIFICATION as it estimates sigma based on short-term variation or data.

All activities of the value chain which substantially add to variation such as shipping and handling effects are not considering

Customer requirements are incompletely understood

Environmental factors to which product is exposed to not counting or how the customer will handle /use or misuse the product.ontrol charts cannot keep track of all variation, we need to consider a few things when we work on any process

Changing environmental conditions which may result in variation so in planning stages itself we consider a compensation factor to accommodate unavoidable variation and this compensation factor is 1.5 sigma shift. This means that

St goal = Lt goal + appropriate compensation factor

Which can be better understood as

1.      Short term goal is 6 Sigma

2.      Long term Goal is 4.5 Sigma

3.      Compensation Factor is 1.5 Sigma

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Short term capability i.e., sigma level based on short term data collected over a narrow inference frame (daily, weekly, 1 shift etc.) is predominantly influenced by common causes of variations (variations by chance, cannot assign causes) and often reflects optimal performance levels. These measurements involve relatively less sample data having typically 30 or 50 data points. This is denoted by Z (st or within).

On the other hand, long term capability i.e., sigma level based on long term data collected over a broad inference space (monthly, quarterly, multiple shifts/ machines/ operators) is influenced by both common as well as special causes (when a cause can be assigned to variation) of variations. This reflects the actual level of performance as experienced by the customers. This involves relatively large data points in the range of 100s and 1000s representing the whole population. This is denoted by Z (lt or overall).

Short term capability is generally higher as the process operates at an optimal level under a controlled or supervised environment. But when a process is observed for a longer duration the performance dips and hence the long-term capability is lower. That is why, short term capability is sometimes referred as ‘potential’ while the long-term capability is referred as actual ‘performance’.

As a thumb rule, short term capability is calculated by adding 1.5 sigma to long term capability –

Z (st or within) = Z (lt or overall) + 1.5

This is known as Sigma Level shift. It was calculated by Motorola as long term dynamic mean variation. The logical rationale behind this is the degradation of process capabilities when the system is left alone over a long period of time. This implies the target short term sigma levels must be 1.5 levels higher than the expected long-term performance.

This is sometimes considered a crude thumb rule when considering pure statistics. It is suggested to measure the sigma levels based on data. But a lot of times it is not possible to monitor and measure a process for a very long term. Additionally, philosophical consideration of sigma level shift gives an opportunity to revisit and apply the DMAIC cycle on a process repeatedly. In such cases, this comes as a valid and handy assumption to continuously improve the performance in long term.

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Background

Mean is the arithmetic average of a data set.
Central tendency is the tendency of data to be centred around this mean.
Standard Deviation (also known as Sigma or σ) determines the spread (deviation) from this mean/central tendency.

The more the number of standard deviations that fits between process average and acceptable process limits, the less likely it is that the process performs beyond the acceptable process limits, and it causes a defect. It is for this very reason that a 6σ (Six Sigma) process performs better than 1σ, 2σ, 3σ, 4σ, 5σ processes.

Specification Limits &Control Limits
LSL and USL refer to “Lower Specification Limit” and “Upper Specification Limit”. Specification Limits are obtained from the customer requirements, and they specify the minimum and maximum acceptable limits of processes. Control limits are the indicators of the variation in the performance of the process. It is the actual values that the process is operating on or a real time value.

Sigma Shift

In 1980s, practitioners of Six Sigma at Motorola analysed samples of their processes and deciphered that process capability tends to drift over time. In order to ensure the long-term process achieved a target defect rate, and realising that they could only measure their processes in the short term, they concluded that the short-term process tended to ‘accommodate’ more standard deviations between the mean and the specification limits, and concluded that an additional 1.5 standard deviations from the short-term process was about right. The ‘additional’ 1.5 standard deviations is known as the Sigma Shift. The difference between the Sigma Levels of a process over the short and long term is called the Sigma Shift. Allowing 1.5 sigma shift results in the generally accepted six sigma value of 3.4 defects per million opportunities (DPMO). If we ignore the 1.5 sigma shift, it results in a six sigma value of 2 defects per million opportunities (DPMO).

Processes tend to behave in a different manner over the short and long terms:

• A high Sigma Shift suggests that process could be improved significantly through better control measures.
• A low Sigma Shift suggests that process is well controlled already and no further control methods are necessitated.

Process Capability & Stability

A capable process is one that gives an output that meets customer specifications. A stable process has controlled variations and operates within the control limits.

There are several methods to measure process capability index and ratio including an estimation of the PPM (defective parts per million) .Capability indices such as Cp, Cpk, Pp, Ppk are the most prime ones. The Cp and Cpk indices are primary capability indices. Cp{Capability Index} shows whether the distribution can potentially fit inside the specification, while Cpk{Capability Ratio} shows whether the overall average is centrally located. If the overall average of the process is in the center of the specification, the Cp and Cpk values will be the same. The higher the value of Cpk, the better it is. Cpk values less than 1.0 is considered pretty poor and the process is considered not capable. Values between 1.0 and 1.33 of Cpk is considered barely capable, and values greater than 1.33 is considered capable. Process Capability Pp measures quantitatively the process spread against the specification spread. In other words, what the requirements are versus the distribution of the process outcome. The difference between the USL & LSL is the specification spread; also sometimes referred to as the Voice of the Customer. The process spread is the distance between the highest value and the lowest value generated, also sometimes referred to as the Voice of the Process. Ppk is a performance index that measures how close the real time value or the current process is to the specification limits.

Think of the Specification Spread as the sides of the garage – those are static, they are not moving, and it is important that the process puts values inside those bounds. The Process Spread is the size of the car we are trying to fit in.

Process Stability refers to the consistency of the process with respect to critical peocess parameters. If the process behaves consistently over time, we might say that the process is stable or in control. A process is said to be stable when all of the response parameters that we use to measure the process show both constant means and constant variances over time, and also have a constant distribution. Statistical Process Control Charts are utilized to determine Process Stability. Some charts are used to assess the stability of the process location .Example, Xbar charts that monitor the process average etc, other charts are used to assess the stability of the process variation. Example- range or standard deviation charts.

Process stability and process capability are different concepts altogether and there is no inherent relationship between them.

Although, there exists no direct relationship between process stability and process capability, there is an important connection: Process capability assessment should only be performed after process stability has been ascertained.

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Sigma Shift

An improved process using DMAIC methodology is expected to be impacted by variations due to Common Causes & Special Causes.

Common Causes

Common causes also known as natural problem and random cause is a stage when the process measure shows random pattern within the control limit. This may be due to changes in machinery, measuring patter, employee, or environment. Although, the process is said to be statistically control or stable, a process capability to be conducted to understand the exact cause of the of the random patter.

Special Causes

Special cause variation is predominantly due to defect, breakdown, shortage, delay in the process. Special cause variation shows a non-random pattern of variation or process measures are plotted outside the control limit.

In the short run an improved process is impacted by common cause variation as we are calculating the Standard deviation and Sigma level for shorter period. During short term, the process will show good capability as compared to long term which is impacted by both Common and Special causes. This shift is the 1.5 Sigma shift.

While calculating Long term Sigma level, it is obtained by subtracting 1.5 Sigma from the short Term Sigma Level.

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A universally accepted fact for a six sigma process is that it attains a process capability of 3.4 defects per million, however statistically a six sigma process translates to 2 defect per billion opportunities. So this 2 defect per billion becoming 3.4 defects per million is attributed to the 1.5 sigma shift.

This 1.5 sigma shift is empirically determined by Motorola through years of data collection for processes & it was observed that the processes tend to vary & drift over a period of time, which they called as Long-Term Mean Variation, this variation typically falls between 1.4 & 1.6.

Statistically speaking, 2 defects per billion opportunities correspond to six sigma & 3.4 defects per million opportunities correspond to 4.5 sigma, also the overall goal in long term is a near-zero defect process or a 4.5 sigma level. The variation due to changes in environmental conditions causes a shift in the process in the long run & this shift corresponds to 1.5 sigma.

No matter how stable a process is, over an extended period of time, the environmental conditions change, which causes variation in the process. Thus at the initial onset, the process capability needs to be balanced by a compensating factor in order to account for the changes in order to ensure that the long term goal is met.

Thus Short-Term Sigma Level (6 sigma) = Long-Term Sigma Level (4.5 sigma) + Compensation Factor (1.5 sigma)

This overall phenomena of 1.5 sigma shift can be visualized as shown below:-
Also after a process has been improved, we calculate the standard deviation & sigma value, however these are considered to be short-term values as the data only contains common cause variation, whereas in the long run a process can have both common cause as well as special cause variation. Since the short-term data does not contain the special cause variation, thus it has higher process capability than long-term & the 1.5 sigma shift attributes to this difference in the short-term & long-term process capabilities.

Although the original work was done by Motorola led to the discovery of the 1.5 sigma shift, however lean six practitioners have concluded that the size of the shift depends on the industry & type of process being studied, although the general concept that processes drift over time & the short-term capability needs to be better than the long-term capability remains valid everywhere.
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Himanshu Sharma has given a very unique explanation of the 1.5 Sigma Shift and hence his answer has been selected as the winner.

My 2 cents

1. Another reason for Sigma Shift is the fact that we typically do not control all factors of a process. We could keep a handful in control however the others might go out of control and the gradual shift

2. Concept of Sigma Level makes perfect intuitive sense, however do not go by the number 1.5. It was what Motorola observed. It is very likely that you might observe a completely different number. So go on, observe your own data and identify your number

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