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  2. Hi All, Could you please explain how to calculate the Cp, Cpk for the below mentioned data. its derived from the team on a daily basis quality of day. We would like to know about the capability of the process Date % of Quality 21-Jun-19 100 22-Jun-19 99.24 23-Jun-19 99.82 24-Jun-19 99.76 25-Jun-19 99.82 26-Jun-19 100 27-Jun-19 100
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  5. The chosen best answers are that of Bhavesh Wakde and Mohamed Asif. Both answers use simple language to explain the concept and graphics to support the answer. Benchmark Expert View has been provided by Venugopal R.
  6. Drum, Buffer, Rope is a supply chain technique to define the flow of processes based in pull system of production. Drum is synonymous with the first link of the supply chain which defines the pace of flow or pace of manufacturing which in order is followed by rest of the processes. This is in line with the concept of Theory of constraints which states the business performance or supply chain effectiveness is controlled by one or few critical factors which are actually bottleneck in the entire pipeline.By identifying that area of constraint, effort is made to improve that so that Throughput and effectiveness of the line can be improved. In Drum, Buffer, Rope system, Just like Theory of constraint, pace is set by the bottleneck operation i.e., the process or operation with least cycle time or efficiency is placed in the front and other processes / operations follows the front line. This helps to make sure that the preceding operations work in tandem with the first operation and hence maintains the optimum inventory in result. The speed of preceding operations is increased based on the improvement in the effectiveness of the front bottleneck process. Thus Drum refers to the process operation which sets the pace and stage for assembly line or supply chain pipeline. Buffer refers to the number of amount if inventories in between the operations. These intermediate inventories are nothing but the business cash entrapped in between the processes. Rope is an imaginary line which connects the entire set of operation in the pipeline. Toyota introduced the functioning of Kanban which resembles the Drum, Rope and Buffer. Kanban set the pace of the pull system. It acts as the tool to develop the pace for manufacturing and regulates.
  7. Drum Buffer Rope is a concept popularized by Eliyahu Goldratt in his book Goal. It was used in analogy to a Boy Scout team, to ensure that the team doesn’t get dispersed wide due to the variations in speed of the fast movers vs the slower ones. A synchronized approach ensured that the team is in step. Once the rope is stretched to the max, the fast movers have to slow down or stop. The same concept can be applied to manufacturing, where speed is controlled by the drum, buffer adjustments by a rope with a goal of increasing throughput. The drum relays the information of the bottleneck to the group. The rope controls the speed of the set of processes ahead of the bottleneck from running away and creating excess WIP. Rope is like a signal to the front of the line to start producing at the front as consumption is happening in the bottleneck. The limitation of DBR approach is due to shifting bottlenecks. Kanban is a Lean Management technique to achieve Just In Time. Kanban controls the upper limit of inventory, thus avoiding over capacity. It is achieved by going from the consumer of the process, taking demand from the end consumer up thorough the chain, with the rate of demand controlling the rate of production. Conceptually, for DBR, this is like keeping the drum at the last person and making it to a demand driven process.
  8. Drum-Buffer-Rope Approach Goldratt introduces the concept of Drum-Buffer-Rope (DBR) approach in this book “The Goal”. This is one of the TOC concepts which are used to manage production systems. Widely used in supply-chain teams and manufacturing systems. TOC concepts are very useful with combination of Agile, Lean in production floor and provides huge developments. How it works: For a general example: We have got a line of kids who are up for a cycle ride in the woods. In the middle we have got the kid D who is the slowest or the bottleneck. Everyone else wants to go faster than D, now if that happens all the kids will be lost. Now how do we keep the line from spreading? Suggestions: Tie the rope for the kids so the speed is constant for all the kids Keep kid D at the front of the line. Industry processes: The drum is the slowest process in a system. It can be the limited resource, the bottleneck or any constraint. The first step in a process is to identify the drum in the process. Once identified, the drum tells us the upstream system to identify when to more items to the system and stopping maximum progress at once or unnecessary inventory at the start of the system. The rhythm of the drum also gives the maximum delivery rate because any system can deliver maximum value as its slowest part. Once the drum is identified, the next thing is to make sure the drum is beating faster as fast as possible. This is where the “rope” comes into picture and ensures the drum doesn’t exhaust out or disappear with the velocity of the factory production. Many practitioners also believe in Drum-Rope only. Ideally, the drum will beat at a steady rate, a consistent amount of materials will be delivered to the drum and downstream steps will be receiving the drum’s output and delivering to customer. But in reality input to drum might be disrupted so buffer is needed. Adding a buffer before the drum gives the flexibility to the drum to pace up with the slowness and adding buffer later will add value to the downstream process. Business Approach- In an agile environment of multiple sprint teams been into production system of delivering business models to the Business annually, the bottleneck or the Scrum Sprint Team which is delivering less process models is the drum. Driving the velocity of the sprint teams equally can be identified as the rope. The Buffer is the additional time given at the start of the sprint when the project is in scope and timelines are being set.Limiting work-in-progress is an effective rope as it stops parts of the Sprint Team shooting ahead. 1) Identifying the Scrum Masters and shuffling the Scrum masters and teams for a better pace in delivering models can be a possible solution. 2) Also No Scrum Team should maximize delivering models in any of the sprints at the start of the cycle. Disadvantages: DBR follows a fixed drum approach. If the bottleneck differs in a process which usually happens, then the DBR becomes rigid in the process. The Kanban methodology becomes more effective in this scenario and acts as a better solution.
  9. Q. 186 Provide some of the latest examples where breakthroughs with improved accuracy or reduced turnaround times have been achieved due to use of effective algorithms for predictive analytics. Do provide your reference source of information at the end of your response. Please remember, your answer will not be visible immediately on responding. It will be made visible at about 5:00 PM IST/ 11:30 AM GMT/ 4:30 AM PST on 20th August 2019, Tuesday to all 54000+ members. It is okay to research various online sources to learn and formulate your answer but when you submit your answer, make sure that it does not have content that is copied from elsewhere. Plagiarized answers will not be approved (and therefore will not be displayed).  All Questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/  All rewards are mentioned here - https://www.benchmarksixsigma.com/forum/excellence-rewards/
  10. Benchmark Six Sigma Expert View by Venugopal R Drum, Buffer and Rope is a phase used in ‘Theory of Constraints’. It refers one of the methods for ‘synchronous’ flow in a production line. The Drum refers to a workstation that could be a constraint or bottleneck. For instance if we have a production line, where we have 3 workstations, A, B and C of which 2 of them (say A and C) are capable of producing at the rate of 120 units per hour and one workstation (say B), can produce only 100 units per hour, then the overall production rate for the line is 100 units per hour. Here, the production rate of the entire line is determined by workstation B, which is the constraint, or known as ‘DRUM’. We can afford some slackness for workstations A and C without impacting the overall production rate, but no slackness is affordable for station B as it will directly impact the production rate of the production line. Since the workstation B is a constraint, it is important to place some ‘Buffer’ stock of the material being fed into it, so that there is no chance of this workstation to run out of stock. This precautionary inventory before the DRUM is know as the BUFFER. The buffer does not mean that more material will be produced than required, but it is an inventory in terms of time. This means that the buffer stock is an advance quantity of input material that will be made available to workstation B, even as it starts producing the first unit. If the buffer inventory level decreases or increases, the communication that is provided to the previous process(es) is know as the ‘ROPE’. The ROPE ensures that the system is always maintained with a required level of inventory, neither excess not short. The DBR approach helps to plan the production schedules based on the output requirements. The quantity to be produced is determined by the market / client requirement. The maximum production rate that the production line can produce is determined by the DRUM workstation. The starting time for each workstation can be worked backward from the output requirement, taking into account the production rate of each workstation and the buffer requirements. The representation shown above is a simple one for purpose of easy understanding. We can also apply the DBR on multiple lines leading to a final assembly line. Once we address a DRUM workstation with buffer, it is quite possible that we may have the possibility of another potential constraint to emerge. This would call for buffer requirements for other workstations as well. The entire network of the process flow can be mapped with this concept and the timings and buffer requirements be worked out. The DBR approach helps as a planning tool and provides inputs to Kanban dashboard for regulating the production flow, scheduling the timing for workstations and to prevent inventory build-up.
  11. Drum Buffer Rope commonly known as DBR is an application from Theory of Constraints (ToC) used during planning and scheduling. Drum as constraint, Buffer as Inventory, and Rope for scheduling. DBR was developed by Eliyahu Goldratt. He also spearheaded Optimized production techniques, ToC and Critical Chain Project Management (CCPM). His famous writings include “The Goal” and “The Race”. In “The Race” he describes about Logistical system based on metaphor developed in “The Goal” DBR is similar to Kanban or CONWIP (Constant Work in Progress) DBR is similar to Kanban or CONWIP (Constant Work in Progress) Primary Objective of DBR is to ensure Throughput Expectations are met at the same time managing the operating Expense and the Inventory. In an assembly line, typically manufacturing setup, the constraint is referred as Drum (Slowest step/activity). This constraint acts as a Drumbeat to set the speed of the system. The process of releasing new work into the system is referred as Rope. Drum and Rope work together, as drum completes load, rope allows new work orders to be released. To ensure Drum is always engaged with load (Work @ full capacity), buffer is created in front of Drum. System Throughput is dependent on Drum’s working. Considering the below production line, Here A3 is the constraint, as it is used as Drumbeat, pace is set based on A3. A3 can process only 35 pieces per hour but A1 and A2 produces more than 35 pieces per hour. Rope monitors work load and pulls work from supplier and keeps buffer ready. Referring to 5 focusing steps in Theory of Constraint (ToC) 1) Identify the Constraint 2) Exploit 3) Subordinate all decisions to exploit 4) Elevate the constraint 5) Avoid inertia Drum: (High throughput as the constraint churns out work @ max capacity) Identifies constraints in the system Set’s pace Schedule Capacity Buffer: (High throughput as constant buffer is maintained before constraint) Exploits the constraint Shields the constraint (Ensuring buffer is created) Rope: Rope subordinates work to the constraint It Alert’s/Signal’s to release work to constraint Maximum output and by effectively using DBR we get less idle time In general, Objective of DBR in manufacturing setup to have minimum inventory, maximum output and to have delivery dates obvious or predictable. Processes which are predictable can be well planned and continuously improved. Both DBR and Kanban are systems to control production, pull based and is very similar to each other. However there are few difference. DBR assumes on single constraint in the production system whereas it need not be the case for Kanban. Kanban controls inventory level at each production stage, however DBR does it only at constraint. Kanban can be simply improved by applying DBR (ToC), combining lean & JIT manufacturing along with planning tool to have improved throughput. Thus preventing over production and having an highly efficient system. Kanban with Super Market before constraint is similar to DBR. Kanban can have more than 1 rope for effective processing steps and hence has an upperhand over DBR.
  12. Drum-Buffer-Rope (DBR) is a method derived from the concept Theory of constrain(TOC) from the book written by Eliyahu Goldratt . As book indicated that slowest scout of mountaineering team decided the pace of climbing on mountain of the troupe. Troupe had identified that scout was loaded heavily (constrain) which decides the pace of team. After load of the scout was distributed to the team & same scout had been put up to lead the team. Overall pace of the troupe got increased and scout boy (Drum) become the frontier to lead the troupe & decide the pace which others follows easily . Similar pattern is being co-related with the methodology of DBR. Terminology are : Drum : The pace setting resource/Production pace(Constrain) Buffer : The amount of protection in front of bottleneck resource Rope : The scheduled staggered release of material to be in the line with the Drum’s schedule. As shown in the picture(Source: https://www.marris-consulting.com/en/training-news/training/training-theory-of-constraints-in-production) considering the industrial scenario set of processes are involved to make the output from the system. In which one of the process will be considered as bottleneck which is having higher cycle time or lesser throughout , compare to other processes is being considered as constrain(Drum).This bottleneck decides the throughput or rate of production of entire cycle/system. To avoid shortages at bottleneck process buffer is getting planned ,as an hour lost for the sake of inputs bottleneck process will affect the entire output of the system .Once the buffer depleted in front of bottleneck process signal (though rope) is getting pass on to non- bottleneck process to initiate to feed up to bottleneck process. This systematic approach protects weakest link in production system against process variation and dependency, which maximize system’s overall production effectiveness. The Drum is a schedule for the bottleneck process in entire system and the Rope is a schedule for releasing raw materials to the shop floor and derived according to the Drum and Buffers. The Rope ensure the proper subordination of non-bottlenecks. Any operational plan needs to be productive, reliable, realistic and robust. - If it aligns with market requirment, we consider it as productive. - If it is aligned with our resource (capacity and add-on capability), we consider it reliable as well as robust. in case of inevitable disturbances or disruptions that will hit it; “realistic” means that it is capable of being done with the available resources, including material supply, maintenance etc. In Toyota Production System. Rope (signal) concept is similar to KANBAN (signal-card) which provides inventory details at different process & refurbishing the stock at operational levels . Both the concepts provide the system for the proper planning of inventory management except the buffer description in DBR is taken up w.r.t. time whereas in KANBAN relevant to parts quantities. Since KANBAN provides better trace-ability which consider physical count of parts it gives faster and better result than Drum-Buffer-Rope method.
  13. This post is really helpful for all Six Sigma aspirants. Many thanks.
  14. Hi All, I completely agree with the views shared by predecessors. I would like to add, it is true for initial stage as it is difficult to convince all employees either in top-down or bottom- up approach.But, once we start the Six Sigma initiative without mentioning technical jargon or statistics concepts, we can take buy-in from Senior Management. I would like to mention some typical roadblocks and solutions (the same I mentioned in different blogs). 1. Lean Six Sigma is a fad .By who – •Expressed by top leadership. When and why – •After hearing preliminary things about Lean and Six Sigma. .How to deal with it – •Show them the benefits of Lean Six Sigma by implementing a small scale pilot project or suggesting an implementation. 2.Lean Six Sigma is too statistical .By who – •Expressed by top leadership and also employees. .When and why – •After learning Six Sigma is a statistical approach. .How to deal with it – • Black belts can help them overcome this resistance by explaining one or two statistical tools. 3.Why should one change? .By who – •Expressed by top leadership. .When and why – •After hearing Lean Six Sigma will enforce a culture change. .How to deal with it – •By explaining change is imperative. It helps in keeping an organization competitive in the market. .By who – •Expressed by employees. .When and why – •Employees enjoy being in a comfort zone. .How to deal with it – •By changing, competitiveness to the organization can be ensured, along with upgrading the skills. 4.Non – cooperation from employees .By who – •Expressed by employees. .When and why – •Typically happens while implementing new process, due to resistance to adapt to new processes. •Also, due to fear of admitting mistakes of the past. .How to deal with it – •Build trust in employees that they wouldn’t be penalized for past mistakes. •Encourage the “Move Ahead” philosophy. •Ensure employees are involved in designing and setting up the new processes. The starting point is: To communicate and ensure employees are convinced not to resist change. Important: No use of ‘force tactics’ to convince employees. 5.Wrong team members .By who – •Observed within the team members. .When and why – •Happens when a team has several members possessing the same skill-sets. Duplication of thoughts leads to change efforts falling flat. No creativity or lack of creativity in ideas is perceived. .How to deal with it – •Choose team members on different skill-sets. •Choose team members based on four factors: .Capability; .Creativity; .Willingness; and .Ability. Wish you all a good luck for Lean Sigma journey. Many thanks, Srijit Chatterjee (Process Improvement Manager, PWC)
  15. Drum is the Constant which we in the process that is always the whole process is nearly dependent (main factor ) ,buffer is the back up (inventory )Which we create to manage the drum ,rope ensures us that the buffer is always maintained ( ensure that inventory is never out of limit ). Kanban (pronounced as KAAN BAAN) is a process to maintain inventory If I m working in a night shift and some inventory is out of stock I will write it down on the card and in morning when other person take over he will first read the card and than fill the inventory.
  16. I need to obtain a finished product that product had to go through 3 process Process 1 Process 2 Process 3 Process 1 produce 50 product per hour Process 2 produce 50 product per hour But process 3 produce 40 product per hour... So the pace of our final product is 40 product per hour which is set by process 3... So that makes process 3 a constraint.. Process 3 is our drum Now what does a making process 3 our drum means? It means that it is producing 40 product per hour so we cant let it go down.. if there is some problem in process 1-2 and their output reduce ultimately the output of process 3 will reduce... How to avoid that? We place some already processed product from process 2 as an inventory to process 3.. this is called as Buffer What does buffer do..? When there is a reduce in input to process 3 the buffer provide an extra input so as to keep the process on its pace What does rope do? Rope is a signal to maintain the buffer level... Suppose i have 5 product in my buffer and when the buffer product reduce to 3 a mechanism will send the signal which will fill my buffer inventory back to 5 The same way we use Cards Kanban to ensure the inventory is upto the mark and the process didnt lack because of it
  17. The chosen best answer is of Natwar Lal for clearly explaining and highlighting the need for Bessel's Correction using a data set. Also go through the answer provided by Mr. Venugopal - Benchmark's in-house expert.
  18. Q. 185 Explain Drum, Buffer and Rope approach and describe how it relates to Kanban. Please remember, your answer will not be visible immediately on responding. It will be made visible at about 5:00 PM IST/ 11:30 AM GMT/ 4:30 AM PST on 16th August 2019, Friday to all 54000+ members. It is okay to research various online sources to learn and formulate your answer but when you submit your answer, make sure that it does not have content that is copied from elsewhere. Plagiarized answers will not be approved (and therefore will not be displayed).  All Questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/  All rewards are mentioned here - https://www.benchmarksixsigma.com/forum/excellence-rewards/
  19. Variance calculation is used to find out dispersion of population(N) which contents entire set of data. This can have multiple sub set of data which can be represented by sample size(n). Formula for the population of variance : σ2 = Σ ( Xi - μ )2 / N N = Population σ2 =population variance X =Set of data which can range from i=1 to the population(N) μ = mean of population Formula for the sample of variance is similar but it can be categorized as biased and unbiased : Biased Unbiased s2n = Σ ( xi - x_bar ) 2 / n n = Sample size of the Population s2 = Sample variance x =Set of data which can range from i=1 to the sample (n) x_bar = mean of sample s2n-1 = Σ ( xi - x_bar ) 2 / n-1 n = Sample size of the Population s2 = Sample variance x =Set of data which can range from i=1 to the sample (n) x_bar = mean of sample If sample drawn data are evenly getting picked up from population then its mean will be very much close to true population mean, whereas sample data drawn are picked up from particular zone there are high chance of sample mean will be far away from population mean such cases leads to underestimation to the true variance of the population. This leads to sample mean will seats between the sample data but far away from population mean which also indicates that gap between sample variance to the population variance. This phenomenon is known as biased. To overcome this issue unbiased variance formula is getting used to find out the sample variance . This also provides the high liberty of degree of freedom during sample data collection which is getting restricted during biased variance calculation.
  20. What is Bessel's Correction? - It corrects the bias in the sample variance and standard deviation where the denominator is changed from N to N-1. Why is it required? - It is required because it is difficult to determine the population variance and standard deviation. We all know, that due to the constraints of time and money, we prefer to work with samples and not populations. Once we have the sample, we apply Descriptive Statistics to get the sample statistics. These sample statistics are then used to make inferences about the Population Parameters (which happens to be our interest area). Assume, that we need to determine the average and variance in weight of the Indian male. Here the average weight and variance in weight are the Population Parameters. And it is also obvious that we will not be able to get these numbers by considering the entire male population of India. Hence we revert to doing sampling. For example sake, let us assume that the average weight of the entire male population in India is 70 Kg. BUT we do not know it. Instead we need to determine it. So we picked a sample with a sample size of 10 and measured the weights The sample average is 77 Kg. This is the sample statistic that is then assumed to be the population parameter. Therefore, one would assume that the population average weight is 77 Kg (instead of the actual 70 Kg). Now, we need to estimate the population variance and standard deviation as well (remember weight is a continuous metric and hence we also need to determine the spread in the data). Now in column 4 (in my example), I am working with the sample average of 77 kg and then computing the variance and standard deviations (variance = 201.8 and standard deviation = 14.2). While in column 6, I worked with the population standard deviation of 70 Kg (variance = 250.8 and standard deviation = 15.8). Ideally if the population mean was known, one should be working with 70 Kg (or column 6) but since it is unknown, one could only estimate it using the sample data and sample mean. You will notice that the variance in column 4 (201.8) is less than variance as computed in column 6 (250.8). This highlights two important facts 1. The difference is the bias 2. This bias will always make the variance or standard deviation less than what it should be if population mean is considered I have also calculated the variance and standard deviation with Bessel correction where (N-1) or 9 is used in the denominator (variance = 224.2 and standard deviation = 15.0). You will notice that the bias is corrected to some extent. This happens as the denominator is decreased, the overall value increases. This is Bessel's correction which is applied when population variance and/or standard deviation is to be estimated from sample mean. Bessel correction is to applied only when population mean is unknown. Another way of understanding the Bessel Correction is by the concept of 'Degrees of Freedom'. In my example, I had a sample size of 10. Now if I pick another sample and want to keep the sample mean same, then I have the freedom to change only 9 values. I will need to keep one value fixed. This fixed value is the pivot around which the other observations can change. The same concept is applied to a population. In order to keep the population mean same while picking multiple samples, one would need to keep at least 1 value fixed. Therefore if the population size is N, the degree of freedom becomes N-1. This same N-1 is used in the Bessel Correction.
  21. Bessel’s work includes: correction to seconds pendulum corrected observation of personal equation correcting effects of instrumental errors Bessel functions (Cylinder function) Measuring stellar parallax and many more...…... Application of Bessel’s correction in Statistics, includes corrections in sample variance and sample standard deviation calculation. Correction is on the formula to use (n-1) instead of n; where n is number of observations (in sample) Why n-1 instead of n? Sample Variance/Standard Deviation gives biased estimate of Population Variance/Standard Deviation. Minor Bias! Bessel's correction is to eliminate this minor bias in sample standard deviation and sample variance calculations. So, as we use n-1, accuracy in these calculations increase significantly. Let’s refresh standard deviation and variance calculation and deep dive with an example, Both for Standard Deviation and Variance for sample data, N is replaced with (n-1). For Population we considered N and for sample it is (n-1) In order to get more clarity, lets consider data set of 100 observations (Population) and sample data randomly picked @ 5% from population data. For the same sample data, excel throws 2 different values, "STDEV.P" Formula -> Considering N Samples "STDEV.S" Formula -> Considering (n-1) during calculation R Code: Considering same sample Vector using R function: sample_data <- c(82, 31, 95, 33, 92) > mean(sample_data) [1] 67 > var(sample_data) [1] 1021.30 Verification: > sum(sample_data)/length(sample_data) [1] 67 > sum((sample_Data - 67)**2)/length(sample_data) [1] 817.04 Manual data is different from direct calculation. This is because, R directly applies Bessel's correction during calculation Applying correction to Manual calculation: > sum((sample_Data - 67)**2)/(length(sample_data)-1) [1] 1021.30 As the data sets are not same, As the Samples are representations of Population Variations Exists.... This is referred as "Sampling Variation" SAMPLES can Vary POPULATION Is Fixed Usually for Normal Distributed data, when we do sampling, we most likely select samples around mean and miss out lower and upper extreme values. Why this Bias? This downward Bias is corrected further, by dividing SAMPLE numerator by (n-1) and divide POPULATION numerator by N This is simple because of considering DoF (Degrees of Freedom) which means Sample data loses an (1) observation [for 6 samples, DoF is 5; for 5 samples DoF is 4; for n samples DoF is (n-1)] Dividing by (n-1) results in deriving unbiased variation. That is, diving by DoF (Degrees of Freedom) against the Sample Size. Less Biased Estimator and thus moving towards more accurate calculations.
  22. Bessel correction in statistics is used to correct the bias in the estimation of the population standard deviation. Variance calculated from sample is usually smaller, when used to estimate for population. Adjustment needs to be done to account for this. To give a context, we can calculate the mean and variance of all students in say class 10. Now if we randomly pick a small sample of 3-4 students and calculate their standard deviation, it may not capture extremes. It is more likely to get more of the same type of students. This standard deviation will likely be smaller than the population. Adjustment needs to be done for the bias Normally Variance is calculated as mean of the sum of squares of deviation of sample values from sample mean. Standard deviation is calculated as square root of the variance. When the population mean is unknown, to estimate the population variation, (n-1) is used instead of (n) in the denominator. This method is used when estimating population mean and variance from sample. The logic is that 1 degree of freedom is used in the sample mean and hence only (n-1) for the sample variance. When (n-1) is used, the variance will increase. References for more mathematical reasoning and formulas: https://www.statisticshowto.datasciencecentral.com/bessels-correction/ http://mathcenter.oxford.emory.edu/site/math117/besselCorrection/ https://proofwiki.org/wiki/Bessel%27s_Correction
  23. Benchmark Six Sigma Expert View by Venugopal R As most of you would have gathered, Bessel’s correction refers to using n-1 while computing standard deviation from a sample. The answer to this question can lead to an in-depth statistical discussion. However, let me try to make it simpler by elucidating the below two points of view. It is assumed that the readers are familiar with the equation used to compute standard deviation and the term ‘degrees of freedom’. Readers are also advised to do more thinking and work out some examples to gain more clarity. Whenever we take samples from a large population, the sample means can vary, whereas there can be only one population mean. Many times, the population mean will not be known, and hence we have to use the sample mean from each sample value (xi – xbar) in the calculation for sample standard deviation. You can observe that if you substitute the sample mean (xbar) with any other value, you will get a larger standard deviation. So, assuming we know the population standard deviation and we substitute it for the sample mean, we would be getting a larger value for the standard deviation, (unless of course the population mean happens to be exactly equal to the sample mean!) This implies that the sample standard deviation calculated using the sample mean will always be lower than the true standard deviation based on population mean. We may correct this bias by reducing the numerator in the equation for standard deviation; Bessel’s correction reduces the denominator by using n-1 instead of n. Another related aspect is about the ‘degree of freedom’. When we have ‘n’ samples, for a given mean value, except for one sample, all others have the freedom to assume different values and still achieve the same mean. In other words, n-1 samples have the freedom to vary and hence the degrees of freedom is n-1. When we calculate the ‘expected’ value of the standard deviation, we need to divide the summation of the square of differences from mean by n-1, instead of n. When the sample size becomes larger, the difference in the standard deviations computed based on n and n-1 will narrow down. The Bessel’s correction is applied to reduce the bias on the standard deviation calculation for a sample.
  24. Usually, when you calculate any parameters on an entire population you tend to get more accurate results. However, when you working on a sample from the population you are working just with a portion of the population and, your std deviation or variance cannot be accurate. Hence, Bessel’s correction i.e., n-1 helps in getting more accurate answers when you involve a sample.
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