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Sigma Level


Sigma Level (Z) - is a representation of the process capability in terms of opportunities and defects. It refers to number of defects per million opportunities (DPMO). Higher the sigma level, lower the number of defects

1σ = 691,462 DPMO
2σ = 308,538 DPMO
3σ = 66,807 DPMO
4σ = 6,210 DPMO
5σ = 233 DPMO
6σ = 3.4 DPMO


Also, by definition, Sigma Level (Z) is the number of standard deviations that can fit between the mean and the specification limit. E.g. for a process operating at three sigma level, three standard deviations can be fitted between the process mean and the specification limit.



An application oriented question on the topic along with responses can be seen below. The best answer was provided by Arunesh Ramalingam on 9th November 2017. 




Q. 41 Sigma Level is sometimes used as a method of assessing the performance of a process. The reporting of Sigma Levels as Long Term (Overall) and Short Term (Within) has continued for a long time. If both of these can be computed independently, why do we sometimes need to derive one type of sigma level from the another by using 1.5 sigma shift?


This question is based on a suggestion by Aman Kumar Jha.


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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In any organization, processes are generally designed for the long term, but most of the times only short-term process data is available.

Also, short-term data mostly contains only common cause variation, while long -term may contain both common cause variation and special cause variations.

Data collection for long term Sigma level calculation would be very difficult as it would need to be collected from several lots, many shifts, many machines and operators and so on.

A reliable estimation of the long-term performance of the process can be made, by estimating the variability that would be experienced over the long term as a function of the short-term variability.

This approximation of the drift of the process mean in the long-term can be used along with short-term sigma level to calculate the Long-term sigma level.


Why 1.5 Sigma Shift?

As indicated in figure below [1], the normal probability distribution predicts a 3.4 defects per million opportunities (DPMO) for a Sigma Level of 4.5 and 0.002 DPMO for Sigma Level of 6.




Motorola has determined, through various studies on process data collected for years, that the process average (mean) is likely to shift over the long term by +/- 1.5.  

The worst-case performance of a process in the long term can be estimated by shifting the short-term process mean by +/– 1.5 and then estimating the fraction of defects i.e. non-conforming to the specifications. If the business could accept that level of defects then the process can be considered capable. This is indicated in figure below [2]




[1] http://www.six-sigma-material.com/Tables.html

[2] https://www.dmaictools.com/what-is-six-sigma/the-15-sigma-shift

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By shifting sigma value by a std dev of 1.5 on either side, the adjustment takes into account what happens to each process at multiple cycles of product manufacturing. Using 1.5 std deviation gives a strong advantage in improving quality not only in industrial processes but in commercial and services as well.

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I used the internet to understand the answer and I am writing. Thanks so much for asking this question so that It is very  helpful for me to understand and get clarity.

When we do a DMAIC project, we will only consider short term aspects of the data. We also know for the fact that the data will keep varying time to time for the same process and the mean will keep drifting. So when we collect long term data we will experience seasonal impact and specific critical factors impact on the data.

Six sigma actually translates to 2 defects per billion opportunity and 3.4 defects per million opportunity. We define as 6 sigma however ends up in a 4.5 sigma value. This difference of 1.5 is considered and to be able to consider the short term impact as well as the long term impact in the calculations that we do, we use this 1.5 sigma shift especially when we calculate one sigma value.

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Every company has a mission and vision.  Mission and vision can always tell us about the standards of the company in terms of there standards. The hidden factor is profit. 

   Zst and Zlt can be calculated separately,  Z is always based on the standard deviation.  In short term(Zst)  every company wants to prove themselves which attracts the investors. So 1.5 which is added can be divided into two parts (1+0.5), i.e., one being the standards one level above their competitors and second part is the process variance. 

  In the long term the process gets diluted minimizing the hidden factor. Long term(Zlt) when calculated shows the reverse effect i.e., their shift has always been less in terms of the std.dev and variation when compared to other competitors which boosts & pulsates the hidden factor. The planning and implementation of the practices will be done accordingly which keeps the hidden factor always climbing. 

14 hours ago, Vishwadeep Khatri said:

Q. 41 Sigma Level is sometimes used as a method of assessing the performance of a process. The reporting of Sigma Levels as Long Term (Overall) and Short Term (Within) has continued for a long time. If both of these can be computed independently, why do we sometimes need to derive one type of sigma level from the another by using 1.5 sigma shift?


This question is based on a suggestion by Aman Kumar Jha. It is a part of the November Episode and can be answered by approved Excellence Ambassadors till 10 PM on November 9, 2017. There are many rewards amounting to 0.5 million INR or more. Just being regular here earns you a reward. Even a streak of 3 great answers can get you a reward. All rewards are mentioned here - 



All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/ 


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Suppose the tolerance limits on the dimension are 5.000±0.012, i.e. 4.988 to 5.012.

Data collected from the process during second shift indicates that the process mean is 5.000 and its standard deviation sigma=0.004.

±3 sigma fits inside the the tolerance because ±3 sigma= ±3x0.004= ±0.012.

Capability Cp = Cpk = 1.

The process mean doesn't remain constant. The process mean may shift 1.5 sigma to the right or 1.5 sigma to the left.

If we assume a 1.5 sigma shift to the right, the yield is the area under the normal curve to the right of -1.5 sigma or about 0.9332.

Suppose if the process variation is reduced so that sigma= 0.002. Now ±6sigma exist between the tolerance limits and the process can be called 6sigma process. To calculate the yield for the six sigma process, we allow the mean to shift ±1.5sigma. Suppose the mean shifts 1.5 sigma to the right so the yield is the area under normal curve to the right of -4.5 sigma which turns out to be 0.9999966. Defect level =1- 0.9999966= 0.0000034 or 3.4 ppm. The mean may not shift exactly 1.5 sigma on each side and no process is truly normal to the sixth decimal.

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Sigma level Long Term (Zlt)

Sigma Level Short term (Zlt)

A Z value is a data point's position between the mean and another location as measured by the number of standard deviations. For example, a 6 sigma process means that six standard deviations lie between the mean and the nearest specification limit. Six is the Z value.

Zlt is calculated from data set which can have special causes or assignable causes

Zst is calculated from a data free of any special cause or assignable cause

Zlt is calculated for a longer duration of time in which there are some changes in the process.

Zst is calculated in a shorter time frame when all the parameters remain the same


We can measure Zsl and Zlt separately for a particular data set. But generally we measure one and compute the other with the formula Zst= Zlt+1.5. This formula is derived by Motorolla, by conducting many research from their historical data.

For measurement purpose we can measure both separately, but when we want to check the improvements in a process. We should always measure Zst, because it gives correct reflection of our data. And estimate Zlt from Zst.

In a six sigma project, we should always make Zst as our bench mark and make improvements on this.


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Overview of Process Capability and Sigma levels

From good old days, we have been using the term process capability denoted using Cp and Cpk. The Cp is the process potential when the process is subjected to only chance cause variations, with the assumption that the process mean is perfectly centered. When the process is well centered, and if the distribution spread with 3 times standard deviations from mean on both sides falls exactly with in the tolerance, then the Cp is equal to 1. When this spread (+/- 3 std deviations from mean) falls within half of the tolerance the Cp is equal to 2. These scenarios are depicted below.




For the case where the Cp = 2, it is a process whose “sigma value” is equal to 6. It means the spread represented by 6 std devns on both sides of the mean will fall exactly with in the tolerance limits. If we statistically work out the defect rate for such a process, it will be 2 per billion.


However, in reality the mean could shift due to special causes.





That is why the process capability index Cpk was evolved, which also takes into account the mean shift, during the time the Cpk is calculated. It is calculated as the distance between the mean and the closer tolerance divided by 3 std deviations. Thus Cpk becomes a more stringent measure of process capability and considers not only the inherent variation of the process, but also how well the mean is centered.


Concept of Long Term Sigma


Reality does not stop here.


Though the mean shift has been considered while calculating Cpk, the questions arise:

1.     “Will the mean shift remain constant over a period of time?”

2.     “How much change of mean shift will occur over time?”


Motorola has determined, through years of process and data collection, that processes vary and drift over time – what they call the Long-Term Dynamic Mean Variation. This variation typically falls between 1.4sigma and 1.6sigma. Thus, 1.5sigma mean shift is taken as a standard which is also to be considered as a variability that has to be taken into account in the long term.


So, even if a process operates at 6sigma level, when influenced by only chance causes, the mean shifts that are bound to happen in the long term can practically influence the sigma level. Taking 1.5sigma mean shift as standard, the sigma level of a process, would drop from 6sigma in short term to 4.5sigma in long term. The defect rate associated with 4.5sigma is 3.4 parts per million.


Why do we sometimes need to derive one type of sigma level from the another by using 1.5 sigma shift?


At any point of time if we do a sigma level assessment on a process, it will be short term sigma only. However, when we certify a process we have to give an assurance that the process will maintain its capability in the long term. Thus, the derivation of long term capability (sigma level) is extremely important, considering the mean shift that would take place over time. This is done by using the 1.5sigma shift principle as explained. By ensuring the short term capability i.e. process potential as 6sigma, we are assuring that in the long term the process will deliver at 4.5 sigma level.


On the other hand, if we have a long term data from a process, we can work out its sigma level and using the 1.5sigma shift principle, estimate its inherent short term capability. Otherwise, carrying out a specific short term trial to understand the short term capability could cost us additional effort and cost.

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I see the calculation to be of One Way : Calculate the current Sigma Value and anticipate the 1.5 sigma drift over Long Term. Not sure if we can calculate the Long Term Sigma and derive the Short Term Sigma from the same.

From existing literature it is evident that years of study and data has yielded the outcome that the drift would be anywhere between 1.4 to 1.6 sigma. I rest my understanding here. :)


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The objective of a Six Sigma process improvement project is to reduce defects. A defect is a failure to meet customer standards for quality. Measuring process sigma for a business process provides a means of assessing how well the business meets its customer requirements .Process sigma (also referred to as sigma level) is a measure of process capability and the higher the process sigma, the more capable the process is. Sigma is a statistical term that measures how much a process varies from perfection, based on the number of defects per million units. Sigma is a measure of the process variability or spread. Sigma is the symbol for Standard Deviation. A Six Sigma process has a short-term process sigma of 6, and a long-term process sigma of 4.5.


Determining sigma levels of processes (one sigma, six sigma, etc.) allows process performance to be compared throughout an entire organization, because it is independent of the process.

The theoretical defect rate for Sigma process levels 1 to 6 is as follows:

One Sigma = 690,000 DPM
Two Sigma = 308,000 DPM
Three Sigma = 66,800 DPM
Four Sigma = 6,210 DPM
Five Sigma = 230 DPM
Six Sigma = 3.4 DPM,


To put it very simply, the process sigma indicates how many Sigmas can fit inside the gap between the process average and the nearest specification limit. Any value beyond the specification limit indicates a defect or unacceptable result. In cases where both lower and upper specification limits exist and the process is not capable on either side of the distribution, the process sigma can be calculated by adding the theoretical DPM levels on each side of the distribution (using the Sigma Conversion Chart) and then finding the corresponding process sigma for the combined DPM level. It aims at reducing variation and the associated defects, wastes and risks in any process.

Sigma level is an attribute capability measure. The maturity of a process is described by a sigma rating indicating its percentage of defect-free products it creates.

 If the defect data is collected over a short period of time z-value is the sigma level. If defect data is collected over a long period add 1.5 sigma to the z-value before reporting the sigma level.  Thus long term data capability is short term z-value plus 1.5 sigma. This is why a six sigma process has 3.4 DPM or a .0000034 probability defective, which is a 4.5 z-score.  The assumption is that a short-term capability for a single opportunity is equal to a z-score of 6; then at the end of a year, it would be closer to a z-score of 4.5. A more common measure of process capability is Cpk, which is equal to the process sigma divided by 3.  So a Six Sigma process has a Cpk of 2.0.


In a stable process, the mean naturally shifts as much as 1.5 sigma in the long term on either side of its short-term value. A Six Sigma process is actually 4.5 sigma in the long term. A typical process has been proven to have a shift in its average performance of up to +/- 1.5 sigma over the long term. A long term Six Sigma process that is rated at 4.5 sigma is considered to have a short term sigma score of 6 sigma.

The real motivation for implementing a Six Sigma program is to make more money. Six Sigma strives to improve customer satisfaction, increase sales and reduce defects with the ultimate goal of increasing profitability.

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Motorola has determined after years of of study of process and data that process vary and drift over a period of time and they call it as long term dynamic mean variation.  Its value varies between 1.4 to 1.6

When we calculate sigma level and standard deviation after improvment of a process through dmaic methodology, we consider data over a period of months not for years which is short term data.  Short term data contain 's only common cause variation on the other hand long term data contain common as well as special cause variation both. Because of this short term data is of higher process capability than long term.  This drift in between long term and short term is 1.5.

In o e way while calculating short term. Sigma level we are considering only common cause.  Over a period of time probability of increasing of special cause variation increase and we ignore it in calculation in short term data. 


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Process sigma or sigma level is a measure of process capability - the higher the process sigma, the more capable the process is. Simply, the process sigma indicates how many standard deviations (“Sigma”) can fit inside the gap between the process average and the nearest specification limit

Statistically a six sigma process means 2 defects per billion opportunities however popularly accepted definition of a six sigma process is one in which there are about 3.4 defects per million opportunities which  is almost negligible in number and considered a near-zero defect process but it corresponds to 4.5 Sigma level.

It indicates that the Six Sigma process has a short-term process sigma of 6, and a long-term process sigma of 4.5 with mean-shift of 1.5 standard deviations (sigma) over the long run.

However stable any process is, over an extended period of time (Long Term), variation happens due to several sources of variation i.e. different shifts, different suppliers, different lots of material, etc.

That additional variation is generally going to make a process appear less stable than a short run.


Based on years of process and data collection, Motorola determined that processes vary and drift over time – and called it as Long-Term Dynamic Mean Variation and factored correction of 1.5 standard deviations (sigma) in short term sigma level rather than waiting for years of long time observation for the sigma level calculation.

Using 1.5 sigma as a standard deviation gives us a strong advantage in improving quality not only in industrial process and designs, but in commercial processes as well. It allows us to design products and services that are relatively impervious, or ‘robust,’ to natural, unavoidable sources of variation in processes, components, and materials.”

 Conclusion:  Six Sigma process capabilities is always reported in short-term sigma and the Long-term sigma is determined by subtracting 1.5 sigma from our short-term sigma calculation to account for the process shift that is known to occur over time.

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Before getting in to crux of this question, little exploration on sigma long term and short term. Long term sigma (Zlt) is (USL-Xbar)/Standard Deviation or (Xbar-LSL)/Standard Deviation. Standard Deviation considered here is from long term data. Long term data comes from a population of data where special cause of variation is primarily dominant. In case of Zst (Sigma level short term) standard deviation comes from short term data. Short term data comes from a sample data, which primarily will have common cause of variations. Now to distinguish what is short term and long term data, reference point would be, once a special cause gets introduced in to a process that’s the transition point of short term to long term. Research reveals that, over period of time center of process moves by 1.5 times sigma towards USL which is termed as ‘Long term dynamic mean variation’. So Zlt = 1.5 + Zst or Zst = Zlt - 1.5. If the data set is continuous and long term data is available and there are no grouping of data then Zlt becomes reference. In case if the data can be logically grouped, then average weighted standard deviation of all groups can be taken as reference and Zst can be arrived at. Weightage for a group in this instance will depend on the number of samples in the group. Zlt can never be greater than Zst as special cause of variations are introduced in to data over long term.

With above background it can be derived that Zst is the practical sigma level of the process considering sampling of data is the practical approach, to project this in to future Zlt can be derived considering 1.5 sigma shift. Similarly vice versa, if population data sigma level (long term sigma) can be derived then the short term sigma level can be derived using short term sigma level. Hence though there are direct method to arrive at short term and long term sigma level are available this method of conversion of one sigma value to other would help.

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Background of 1.5 Shift:

When a process is improved through a Six Sigma DMAIC methodology, the sigma value and standard deviation are calculated. These values are deemed as short-term values as the data collection done by the project team,  for the process happens over a period of months, rather than years and the data contains only common cause variation, by and large [there could be one or two outliers(Special causes) but in general this is all about common cause ].


Now it would be interesting to observe how the process behaves over a longer period of time (over years), and then the calculation on the sigma value and standard deviation is done. These values are deemed as long-term values as the data collection for the process happens over a longer period. The data can contain both common cause variation and some special causes (assignable cause) variation (as you expect discrepancies during longer period) .  As short term data (Sigma level and Standard deviation) do not have this special cause variation, it will have higher process capability than long-term data.   


 It is good to have a long-term view of how the process works or to put in other words, observe how the process behaves over a longer period of time, as controlling a process in the longer run is difficult. The process owner/project team can identify special causes (if any) and then address them.  But in reality does this happen? The answer is a firm no.  Why is it so?    Calculating Long term is difficult because

       i). It takes time, energy, skill, dedicated human resources to work on
       ii).Analysis of the data measured needs to be done twice
                 a). First for a shorter period of time (in months)
                 b). Second for an extended period of time /day-to-day (over years)
       iii).Management/Leadership/Stakeholders approval needed for doing # 1 and # 2

Six Sigma pioneers in Motorola realized all these aforementioned difficulties and adjusted this with a “1.5 sigma shift” .


How to make use of 1.5 sigma shift concept to derive one type of Six Sigma level from another ?                                         

Any process will have a natural degradation over a period of time, i.e. the centring of the process will tend to shift over a period of time.  Six Sigma pioneers at Motorola found out through years of process and data collection that processes vary and drift over time and deemed these process variations as long term dynamic mean variations. They made the calculation and found out that under the worst conditions, the performance degradation of a process would go by 1.5 sigma.


Hence the pioneers wanted to allow in their calculations for this worst-case scenario and to compensate for process shifts over time, they corrected in advance for a probable shift of 1.5 sigma. Because of this, even though the statistical tables show 3.4 defects per million opportunities(DPMO), when the distribution between the mean and the closest specification limit is 4.5 sigma, the target is raised to 6 sigma and still thereby achieving a maximum of only 3.4 DPMO. 


This 1.5 shift is good enough to ensure that the process will meet the goal of 3.4 DPMO over a long run.  


The short term sigma level is calculated first. Then a shift of 1.5 is used. Short term sigma is represented as Zst.   Long term sigma is represented as Zlt and can be calculated as Zlt = Zst – 1.5



There is no hard and fast rule on 1.5 sigma shift. You can use different value when setting the sigma target for a process, if you have the data on experiences obtained from other similar processes.   But as we saw, 1.5 sigma shift is good enough to meet the objective of 3.4 DPMO.    

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Sigma Level:


The literal meaning of the term itself means a lot. It is the determination of the level of deviation/s in the process measured statistically. This is accompanied with suitability of interpretation with deviations per million. 


Since it is accompanied with the suitability with a million outputs, the requirement for short & long term arises. Because 1 million out puts or more than 1 million needs a considerable amount of time. sometimes months or years or decades or even more. 

Also it is not common that the outputs of a million or more than a million happens in a short period of  time, meaning it takes a considerable amount of time as well,  and hence the deviations or differences in the product/service values are likely to occur & could be crucial for the overall interpretation. 


Hence the Shift of 1.5 Sigma might have agreed after a several detailed & regular analysis & after incorporating a lot of experience.

Even though both can be measured separately , it is more appropriate to practically consider that the shift is common & likely in all the process where a longer time period is required to get a million of outputs.


In case of a manufacturing setup or production setup, a lot of parameters leads to the shift of the value of the MEAN itself. This could be for various reasons.


1. Wear & tear of the tools & equipment

2. Fatigue of workers

3. Seasons or seasonolity

4. Varying temperature

5. Variations in the raw material specifications & other input parameters

6. The absolute gravitational forces & natural forces themselves

7. Time itself - Time is a bigger factor that leads to all the changes

8. Social & emotional factors

9. Many more..


Hence all these factors are very likely, the consideration of the SHIFT is crucial.

It is not necessary that it has to be only 1.5 Sigma. It can be any other value such as 1.0 Sigma, or 0.9 Sigma, or even 2.0 Sigma.


In order to accurately choose, we need to go in detail of what & why the shifts could be there.






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Process Sigma is the measure of variation in a process which is relative to customer requirement and or process output. Defects in a process are measured on a scale of Defects Per Million Opportunities(DPMO). So any chance or scope to fail in meeting the customer’s requirement or desired process output is a defect, so a high DPMO is detrimental to the process.

It is easier to calculate the sigma level in a production related process, as the steps become as simple as:

-          Define the number of units produced

-          Define the number of defect opportunities per unit

-          Count the number of defects in actual

-          Calculate DPMO using the formula.

Sigma is  a statistical term that measures to what extent a process varies from the desired output from the process, based on the number of defects per million units.

Sigma Level allows to understand at what level of accuracy the output of the process is. Thus the performance of processes can be assessed across the organization.

Level of Sigma

Defects per million Units

One Sigma Level


Two Sigma Level


Three Sigma Level


Four Sigma Level


Five Sigma Level


Six Sigma level



In a process Improvement or sigma level improvement effort, there is a general rule that that Six Sigma performance is a long term process and as mentioned above, there are only 3.4 defects per million units.


A short term(within)sample of a process i.e. as it is now – let’s say a counter service at a QSR, will be free from assignable or special causes which means it will represent random causes only. It normally constitutes a group of similar things which are collected across a narrow inference space , maybe many orders from a single cash register/ Till Machine and it’s operater.

A long term(Overall) sample of the same process as mentioned above will have random and assignable causes  and is collected across a broad inference spaces with data across many different day parts and many different cash registers and operators.

Sigma levels can be calculated individually in both the above data sets.


In a classic process as mentioned above(Service time) in which "time"  is the measure for performance, it is established that the process will shift in it’s average performance of up to +/- 1.5 sigma over the long term. A long term six sigma process that is rated at 4.5 sigma is generally considered to have a short term six sigma score of 6 sigma  currently and the combination of all the short term samples that make up the long term performances will give an output of not more than 3.4 DPMO.

Though it is not necessary that the long term performance has to be 1.5 sigma lower that the short term performance, but studies show that commonly that is the case.

Hence, even if both sigma levels can be computed independently, it is acceptable to derive one type of the sigma level from the other using the 1.5 sigma shift as it gives almost a near accurate feel of the overall process performance when the short term performance is measured and gives a near accurate performance baseline  for a short term process performance study / review when the overall process performance is already defined.


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Q. 41 Sigma Level is sometimes used as a method of assessing the performance of a process. The reporting of Sigma Levels as Long Term (Overall) and Short Term (Within) has continued for a long time. If both of these can be computed independently, why do we sometimes need to derive one type of sigma level from the another by using 1.5 sigma shift?


Process sigma is also referred as sigma level. It provides you the high level baseline to understand whether the process is capable of delivering the product or not in order to satisfy the customer. When the sigma level is high, the process is then more capable of producing goods and services. If the process sigma level is low, then the process is incapable of producing such products and services.


There are two levels of sigma, which are short term process level of 6 and long term process level of 4.5

Before understanding clearly the 1.5 shift, it is mandate that we need to know the standard deviations, mean, short term an long term variations.


The process sigma clearly states the gap between the average and the nearest specification limit set by the customer. Statistically, to say, how many standard deviations or sigma levels fit inside the gap between the process mean and the nearest specification limit.




DPMO is defects per million opportunities. It the average number of error rate or defect rate with the opportunities defined. This will let us reduce the process variations on a long term basis to satisfy the customer.


DPMO = Defects * 1000000 / (( # of defect opportunities / unit ) * number of units)


Mean ( μ) = x1 + x2 + x3 + … + xn) / n

Where X1, X2 are data values and n is the number of data points.


Standard deviations =image.png.50728d2bb3d864b4ed3eb52d7a06473d.png

 Standard deviations are denoted by Sigma(σ) . Standard deviation tells us the spread of the data. If larger the standard deviation, the more is the spread and vice versa. It the data point is far away from the process mean, this is denoted by the sigma level or Z score. It is calculated using the below formula.

Z = (x – μ) / σ



A chart is being audited with 52 opportunities like ICDs, CPT, Physician names , etc. 1000 charts are randomly audited for quality checks, with what 975 defects were identified.

DPMO = (975 / (52*1000) ) * 1000000 = 18750

Using Z table, the sigma score is 3.6


Sigma Performance Levels – One to Six Sigma

Sigma Level

Defects Per Million Opportunities (DPMO)














Is this process good?  6000 to 66000 defects are there per given million opportunities. If the process is not improved and continues to work at 3 and 4 sigma, then client may seek some other vendor with greater sigma level for a long run. Any organization running at 3 and 4 sigma will not stay with the client on a longer run. If organization A works at 4.5 and Org B works at 6 sigma levels, client would prefer choosing 6 for its stability and capability on a longer run.


Short term and long term Z


The process sigma level is calculated by

Z = (x – μ) / σ = (SL – μ) / σ


Short term samples collected consists of random causes only and not assignable causes and it is usaually collected from one lot, one shift, one machine, one part / operator. This is also called as within variation.


Long term samples are usaually random and assignable causes and data is collected from multiple machines, part, many opertaors, and many shifts. This variation is called overall variation which includes short term variation also.


Short term is denoted by Zst and long term is denoted by Zlt.

Zlt = Zst – 1.5


In sigma level calculations, use Zst. A Six Sigma process is 6 sigma in the short term and 4.5 sigma in the long term or:

Zst = 6

Zlt Zst – 1.5 = 4.5


With Zlt at 4.5 sigma level the DPMO would be 1350. And Zst at 6 sigma level the DPMO is 3.4 Now which would we prefer?

Any typical process would have natural process variations over aa period of time and exhibits 1.5 shift in the future. Hence the process at 4.5 sigma level would shift 1.5 from the process average to the nearest specification limit with 3.4 ppm defects.


This 1.5 shift will make the 0 defects approach with 1.5 lower sigma level.


Why 1.5 shift?

The normal distribution of the process or data will always predict he 3.4 DPMO when the sigma level operates at 4.5 and away fro mthe process mean. 1.5 shift along with 4.5 sigma level is required. Any normal quality six sigma process requires 6 standard deviations  between the process mean and the nearest specification limits. It is because the process is likely to shift over the long run from 1.1 to 1.5 shift. It is the process natural behavior assuming the maximum tolerable limit of the process to meet the customer requirements, the shift happens.


It is still acceptable as a long run measure for defect free approach. If the underlying causes are identified and rectified then the process with defect free approach runs longer with Zlt at 4.5 sigma level.


Process sigma Vs sigma level

Process sigma is process variation. It is measured in terms of data units as standard deviations. Whereas, the process sigma level Z is count with no unit of measure.



For reporting purpose, the process capability is reported in terms of short term sigma with no special cause variations occurring. Long term is calculated by subtracting 1.5 from the short term considering the natural process shift of 1.5 between the process mean nad the nearest specification limit.




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Short term and long term sigma levels can certainly be calculated using short term process capability index (Cp) and long term process capability index (Pp) respectively. But generally, calculation of long term standard deviation is bit tricky as it requires collection of data over a long period of time. Most of the time, what we calculate is short term standard deviation and hence we calculate short term sigma level, whereas we are infact interested in long term performance of the process and thus we are more interested in long term sigma level of the process. Easiest way of calculating long term sigma level is by deducting 1.5 from the short term sigma level, which is done to accommodate shift of process mean by +/- 1.5 sigma in the long run. 

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Long Term and Short-Term Sigma levels can be calculated from Ppk and Cpk, which in turn can be calculated using Long term and Short-Term Standard Deviations.


Short term Sigma considers primarily common causes, while Long Term Sigma considers special causes.


Very often, it is difficult to assess Long Term Standard Deviations as gathering data over a sufficiently longer duration of time can be challenging. The time may extend to many months, during which time, many things can change including, the market demand, business scenario, departmental and organisational leadership, observers, sponsorship of the study and so on. Therefore, to cut the lead time for the Long Term Sigma Assessment, the relationship between Short term and Long-Term Sigma can be used.


Moreover, when setting a target for any process, the following need to be considered. One would be the target under standard environmental conditions. The other would be changing environmental conditions which may result in variation.


Even highly stable processes, over an extended period of time may feel the impact of changing environmental conditions, which causes variation. These environmental changes need to be balanced by a compensation factor in order to account for these changes to ensure that the long term target is met.


Therefore, the Short-term target would be the Long-term target plus a compensation factor. This compensation factor has been empirically arrived at by Motorola as approximately 1.5, originally referred to as “Long Term Dynamic Mean Variation”. This was arrived at under some assumptions.


Thus, a process operating at 3.4 DPMO would be at a short-term Sigma level of 6, but in the long term would be only at a Sigma level of 4.5. For a process to be at a Long-Term Sigma level of 6, it needs to operate at 2 DPBO (Defects per Billion Opportunities).

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The sigma level is always an interesting area of discussion and Long-term Vs Short-term sigma is one of the first things you come across while calculating process capability using Minitab. Some relate this to the time period but most of you got this right linking the difference to common cause Vs special cause variation.


In fact, it was tough to choose the best answer with Mohan, Arunesh and R Rajesh providing an appropriate answer. My vote goes to Arunesh for his crisp narration, highlighting the keywords and providing references from where he did his research. Congratulations Arunesh and all the best everyone in pursuing Excellence!

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