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# Control Limits

Go to solution Solved by Venugopal R,

Control Limits

Control Limits - are calculated as three standard deviations away from the mean on either side of the mean. Also known as process limits, these are derived basis the process data and represents the area within which natural variation can be expected (common cause variation). Any observation/data point outside the control limits is a reason for investigation and will always have an assignable cause (special cause)

Upper Control Limit - three standard deviations above mean
Lower Control Limit - three standard deviations below mean

An application oriented question on the topic along with responses can be seen below. The best answer was provided by Venugopal R on 10th November 2017.

## Question

Q 42. This question relates to control charts for defects data. While the upper control limit has obvious importance for a control chart drawn for defects, what is the importance of lower control limit in c chart or u chart? Are there certain situations where the LCL is of relevance and other situations where it has no meaning?

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The control limits for Control charts are derived based on its own data, applying the statistical principles applicable for the distribution under which the data falls into.   ‘c’ charts and ‘u’ charts are used for ‘count’ data, such as number of defects in as part / sample. The choice of ‘c’ or ‘u’ are made based on fixed or varying sample sizes.

It goes without saying that, when these charts are used for monitoring count of defects, anyone will only want the defect count to be as low as possible. Hence the UCL for defect makes sense, but the question is “why do we require a lower control limit for defect count?”

LCL - little significance:

Some times when the limits are worked out, the lower control limit might assume a negative value; in such cases, the calculated LCL, being negative has no meaning and the LCL is taken as zero. Obviously, no point is going to fall below zero, and hence the LCL is of little significance here, except when the count is zero.

However, if we are using the ‘run’ patterns for our study of stability as per its rules, then the 1sigma and 2sigma limits are also used, apart from the LCL.

LCL - Could unearth important finding:

Where we do have a positive LCL, and if some data points fall outside, it indicates a situation that may be “too good to be true”. It will be worthwhile to investigate the special cause(s) that could have resulted in this occurrence.

1. It could be measurement a error. For eg. a wrong gauge could have been used and it was failing to detect defects.

2. It could be a change of an inspector that added subjectivity in the defect identification, especially if the defect was to be visually identified.

3. Or it could be some genuinely favorable condition that brought down the defect count. These could be opportunities of unearthing some favorable factor that we have been missing or ignoring.

• One example from my experience is when we were using ‘u’ chart for plotting the count of character errors in captured data, processed from multiple sites. Few consecutive days we observed the count falling below the LCL. Upon investigation, we realized that one particular processing site was down during those days. Further probe revealed that this particular site was performing with an operating application, whose version was obsolete. Once the correct version was installed, we were able to sustain a reduced mean error count and the control limits could be narrowed.

LCL - More important (than UCL?)

4.    It is not necessary that c and u charts should always represent defects, which are always “lower the better”. For eg. a consumer goods company selling a popular brand of shaving cream, wants to do a study to see the number of individuals out of sample who use their product. They pick a sample of individuals in a city every day and find out how many of them are using their brand. In this case, since the sample varies every day and it is a count data, ‘u’ chart applies. However, this is a case where "higher the count, the better". Hence the LCL and the count falling below LCL is of utmost importance.

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The role of a Lower Control Limit in the case of defects or defectives control chart is a very relevant question as who would not like having a process with a defect or defective rate as low as possible. Anyone would probably be happy if some data points fall below the Lower Control Limit. It would be a “Out of Control” problem, which would be good to have. It would create an opportunity for the process owner to investigate the reasons why the process’ defect or defective rate went below the control limit, identify if any best practices had been effective and then replicate these practices elsewhere.

Yet, there could be situations where the Lower Control Limit becomes relevant for other reasons.

Outsourced Process – You have to only meet the requirement, cannot exceed it

There could be an outsourced process, where the customer requires the vendor to inspect and remove defects or errors to the levels the customer has agreed to with the end user. By good process control practices and by using the right methodology and equipment, the vendor may be able to bring down the error rate even below the agreed limits. The customer could react in two different ways. He may accept the output quietly and leave it. Or he may have some other points to worry about.

He may feel that if the vendor’s good work in bringing down the error rate far below the agreed limits is accepted and acknowledged let alone appreciated, the vendor may use such events to try to negotiate a higher rate and increase costs, which as far as the customer is concerned, may not add any value as his contract with the end user is at a higher defect rate. Therefore, the customer if he control charts the vendor’s performance, he would need the Lower Control Limit to tell the vendor that his process is “not in control” and that his “performance has to improve”. The customer may also be worried if the vendor, buoyed by this acknowledged performance, may quote this performance with other potential customers, who would be his own competitors, get more business, become less dependent on him and so on.

Furthermore, the customer may be worried that if he “spoils” his end-user with such “Super-Quality” deliveries, the end-user may get used to this and then start cribbing if the deliveries are within the agreed defect rates, but not significantly better. To avoid this, the Customer may prefer to stick to the agreed norms.

“Negative” Lower Control Limit

In another scenario, the Lower Control Limit when calculated could be negative. Obviously it is not possible to have negative defect or defective rates as this would mean that when the process is run on the input material or information, defects in the input are removed. Negative Lower Control Limits could mean that the process is occasionally capable of operating at zero error or zero defect levels. When a process has a positive Lower Control Limit, this may mean that the process in its present form is not capable of zero defect production and will need to be improved upon considerably.

While in the above two situations, the Lower Control Limit may become more relevant than usual, the following Quality Story would make interesting reading in the context of “Lower Control Limit”.

Quality Story

An American firm scouting globally for buying an automotive spare signed a deal with a Japanese firm @ 25 cents per piece including all packing, transport, taxes, duty etc. and placed an initial order for three million parts. Wanting to impress their vendor on how strict their quality standards were, they added “We accept just three defects for every thousand parts”.

The order was delivered as per the agreed schedule, but accompanied by a bill for \$750,900. It did not take the American accountants a very long time to figure out that the bill was \$900 more than what was agreed to.

A bit perturbed about this (especially since it was believed that sticking to deals was part of both Japanese tradition as well as Japanese management practice), the firm rushed a cable to the Japanese supplier, requesting for an explanation.

Back came a letter from the supplier, “You had ‘asked’ for 3 defectives for every 1000 parts. At this rate, you will require 9000 defectives for three million parts. We have made extra efforts to produce the 9000 defectives, which works out to an additional 10 cents per piece. The extra \$900 is on account of this!”

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Defect as such is an Undesired Outcome.

Importance of lower control limit in c chart or u chart : I believe the LCL for a C & U Chart is more of a lower base as the universal wish is not to have a defect and the "0" line captures the count of instances over the timeline where the defect count is 0.

Are there certain situations where the LCL is of relevance and other situations where it has no meaning?

LCL is of relevance where all achieving a defect count as near to or a zero is Important. In a Hotel Operation, in a Multicourse Meal, the next course being served ahead of its turn/ sooner than the stipulated time  is as much a Defect as a course getting delayed !!

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Control Charts are basically for

1.      To study how a process changes over time.

2.      Whether the variation in process are because of common causes

3.      How the process has improved because of the improvement done in process.

4.      Predicting the expected range of the outcome

In a C chart & a U chart center line is for average, there is an upper control limit and a lower control limit.

There are three rules for control chart which determines that process is stable

1.      None of the reading is above USL/ below LSL

2.      There are no more than 6 points consecutively above/ below center line

3.      There are no more than 6 points consecutively trending upward/downward.

Now if we don’t have a LSL in control chart than we might miss if the process is breaking one of the rule for a stable process. i.e “None of the reading is above USL/ below LSL”

In case of U Chart we can neglect LCL is it is coming negative.

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Process control cannot be possible without some sort of a reliable system in place for identifying and understanding the variations in the business processes. A control chart can help identify these variations and see how to use them for future improvements. It is one of the seven basic tools of quality control.

Control chart are graphs used to study how a process changes over time. Data are plotted in time order. A control chart always has a central line for the average, an upper line for the upper control limit and a lower line for the lower control limit. These lines are determined from historical data. Control charts are graphs for attribute data and are plotted basis the progress in quality of the product.

When a point falls outside the limits established for a given control chart, those responsible for the process are expected to determine whether a special cause has occurred.  Determine whether the results with the special cause are better than or worse than results from common causes alone. If worse, then that cause should be eliminated if possible. If better, then special cause should be retained within the system.

Statistically based control chart is a device intended to be used

- at the point of operation

- by the operator of that process

- to assess the current situation

- by taking sample and plotting sample result

So as   to enable the operator to decide about the process. A control chart is like a traffic signal, the operation of which is based on evidence from samples taken at random intervals.

A green signal - Process be allowed to continue without adjustment as only common causes are present.

A yellow signal - Wait and watch trouble is possible. Be careful and seek more information.

A red signal      - Process has wandered. Investigate and adjust. and take corrective actions else defective items will be produced. There is practically no doubt a special cause has crept in the system.

The u and the c control charts use the Poisson distribution to model the results.

A U-chart is an attributes control chart used by collecting data in subgroups of varying sizes. U-charts shows how the process, measured by the number of nonconformities per item or group of items, changes over time. Nonconformities are defects are occurrences found in the sampled subgroup. A U-chart is particularly useful when the item is too complex to be ruled as simply conforming or nonconforming. For example, a car could have hundreds of possible defects, yet still not be considered defective. U-charts are used to assess the systems stability, analyse the results of process improvements and for standardization.

C-charts are used to look at variation in counting type attributes data. They are used to determine the variation in the number of defects in a constant subgroup size. The opportunities for defects to occur must be large but the number of defects that occur must be small.

Control chart views the process in real time, at different time intervals as the process progresses. It helps in keeping the cost of production minimum. By enabling corrective actions to be taken at the earliest possible moment and avoiding unnecessary corrections, the charts help to ensure the manufacture of uniform product or providing consistent services which complies with the specification.

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A control chart illustrates the dynamic performance of the process and helps us to

1. Know the historical trend or behavior of a process;

2. Monitor a process for stability;

3. Detect changes from a previously stable pattern of variation;

4. Signal the need for the adjustment of a process;

5. Detect special causes of variation.

Control chart consists

A central line - The center line is the horizontal reference line on a control chart that is the average value of the charted quality characteristic. Use the center line to observe how the process performs compared to the average. If a process is in control, the points will vary randomly around the center line.

An upper line for Upper Control Limit (UCL)– Upper Control limit is the upper horizontal lines. The UCL is based on the random variation in the process with 3 standard deviations above the center line.

A lower line for Lower Control Limit  (LCL)– Lower Control limit is the lower horizontal lines. The LCL is based on the random variation in the process with 3 standard deviations below the center line.

For  defect data both c-chart and u-charts are used.

The c chart is used to monitor the number of occurrences of an event. It requires that the opportunity for events remains the same from observation to observation while the u chart is used in the same situations as the c chart when the opportunity for events varies from observation to observation

Both UCL & LCL are used to judge whether a process is in control or out of control.

Control limits are calculated from the data and it is the voice of the process –how the process is capable of producing.

If the LCL comes out negative in calculation, then there is no lower control limit and LCL is considered to be Zero.

We know that defects cannot be less than zero (that is negative). Therefore, even though mathematically (in case) we get negative value (lower control limits) - logically we have to take zero.

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In cases where LCL is below zero and in negative range it has to be taken as zero for practical purpose, but it indicates the asymmetrical and non-normal data, which indicates that tests run may have some problem or the pattern of data is difficult to analyze. This implies need of analyzing the outliers and study the process stability, and that’s the relevance of lower control limits. If focus is about reduction of defects to lowest level then LCL may not be relevant, focus would only be to operate below UCL and optimize it further down. If goal achievement is the only focus then LCL will not relevant, but if reduction of process variation in the focus then LCL comes to play.

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The c charts are statistical tools used to evaluate the number of occurrences per unit produced by a process.  The c chart can be applied to any variable where the appropriate performance measure is a count of how often a particular event occurs and samples of constant size are used.  The c chart answer the questions has a special cause of variation caused the central tendency of this process to produce an abnormally large or small number of occurrences over the Tim emerged observed.  Not that unlike p or np charts,  c charts do not involve counting physical items.  Rather they involve counting of events.  Like all control charts c charts also consist of three guideline centreline,  a lower control limit,  a upper control limit.  The centreline is the average number of occurrence per unit and two control limits are set at plus and minus three standard deviation. If the process is in statistical control then virtually all subgroup occurrences per unit should be between the control limits and they should fluctuate about the centre line randomly.  If lower control limit is not there then central tendency predictions will not be possible because it derived from ucl and lcl  both.  Absence of lol is of no means g of central tendency and stability of process by c chart.  Eg in a plant chance of injuries are there,  it can go maximum any level in number but it can't go beyond zero so it's lcl will be zero any how but it will be there while calculation because injuries can't be negative.

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Control limits are different from specification limits. They typically lie between the specification limits and represents 99.7% data in case of a normal distribution. To me LCL will be much more important when we don't want the process to go below the lowest control limit. LCL can also be 0 in some cases where what we are looking at is 0 defects. For example in case of airline industry we would want the no of accidents data to nearing the LCL or ideally should be 0. The UCL and other ranges would be important but not as important as the LCL.

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Control Limits:

For defects data obviously the UCL makes a lot of sense.

I feel the importance of LCL in C or U chart is mainly to see the anticipated defects / non confirmaties that could occur going forward.

This will help us to see or identify the assignable cause for defects.

In C chart the LCL is set for one defect.

LCL could be set Zero also.But there are some instances where LCL is relavent to be kept & in some situations not.

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c-Chart & u-Chart:

The c-chart is used for "Poisson" processes. These are used with random arrival models, or when "counting" attributes. This type of chart, for example can monitor the number of "defects" in each of many equal samples (constant sample size). Occurrence Reporting data (number of reports per month) empirically appear to fit the "Poisson" model, and the c-chart is recommended when charting occurrence report counts.

The u-chart is used when counting "defects" per sample when the sample size varies for each "inspection." A good example at DOE facilities is the number of lost or restricted workday cases per 200,000 man-hours. The number of cases is counted for fixed time intervals, such as monthly or yearly, but the sample size (number of man-hours worked during each time interval) changes.

Calculate and Plot the Upper Control Limit:

Add three times the standard deviation to the average. This is the Upper Control Limit (UCL). Plot the UCL on the graph.

The UCL will be a horizontal line on c-charts.

The UCL will be variable on u-charts. Do not plot the UCL on a p-chart if it exceeds 100%.

Calculate and Plot the Lower Control Limit:

Subtract three times the standard deviation from the average. This is the Lower Control Limit (LCL). Plot the LCL on the graph.

The LCL will be a horizontal line on x-charts and c-charts.

The LCL will be variable on p-charts and u-charts. Do not plot the LCL on a p-chart if it is below 0%.

SPECIAL NOTE: If the LCL is negative (less than zero), and the data could not possibly be less than zero, e.g., a negative number of reports written, or a negative time period, then the LCL is assumed to equal zero.

EXAMPLE C-CHART

The C-chart

The chart below is an example of a C-chart. It is counting the number of occurrence reports per month. Since there are multiple dates on Department of Energy occurrence reports (discovery date, categorization date, final report date), the date used for the basis of the graph is identified on the x-axis label.

The c-chart is used when counting discrete events, as in a random arrival process. Counting events is a good example of a random arrival process, as long as each of those events are independent from each other, and overall occur at a constant rate. This is also known as a Poisson process.

In this graph, an initial 24 month baseline was established at the end of 1994 for January 1993 through December 1994. This baseline remained valid until the end of 1995. Then a shift occurred, with 10 of the next 11 points below average (starting with November 1995). A new baseline was established for November 1995 through September 1996. Note the gap in the two baselines from January 1995 through October 1995. This is acceptable. One does want to avoid overlaps between adjoining baseline averages, however.

The current baseline (Nov 95 - Sep 96) was based on much less than 25 points. It does appear that the new data is coming in lower than the new baseline (but so far, no statistically significant difference has been detected). If a statistically significant difference does generate, perhaps the choice of November 1995 through September 1996 was insufficient. A better alternative may turn out to be to continue to 21.8 average through March 1996, and calculate a new baseline starting in April 1996. This serves to illustrate how control charts can evolve over time, and baselines with less than 25 points may prove to need to be readjusted.

EXAMPLE U-CHART

The U-chart

The chart below is a typical chart used by a safety department - cases per 200,000 hours. Rather than just plotting the number of accidents, this graph plots accidents per 200,000 hours. This allows the rate on the graph to be consistent even with different size work forces.

In this example, the number of cases each month is determined, then divided by the number of hours worked in the month, and finally multiplied by 200,000. The control limits vary from month to month as hours change. When hours worked are low, the control limits are far from the average line. One expects a large amount of variability in the data when there is a small work force. When hours worked are high, the control limits move inward. One expects a small amount of variability.

Note that the initial average was calculated for a time period starting prior to the beginning of the graph. This is typical for an existing graph which has accumulated many years of data, and a decision is made to remove some of the "old" data. In this case, it was decided to remove the data prior to fiscal year 1995. However, the baseline average was left as is, rather than recalculating it for Oct 94 to Jan 95.

There are several significant trends in this graph. First was a decrease from 2.28 to 1.64 in early 1995. The five points from Feb 95 to Jun 95 were greater than one standard deviation below the previous average. However, there was then a significant spike in August 1995. August 1995 reflects special cause variation, and we will assume the cause in question was identified. When developing the proper average to use, it was decided to remove August 1995 from the average. Note the average line cuts through the center of the remaining data for the time interval.

A more recent shift of seven points in a row below average occurred from Sep 96 to Mar 97. As April 1997 remained below the 1.64 previous average rate, an initial baseline was calculated for Sep 96 through Apr 97. During this interval, the Lower Control Limit was mathematically less than zero. Since you cannot have a negative case rate, the Lower Control Limit is not plotted.

This new baseline contains less than 25 points, so it may not be stable. However, this eight point average does provide some basis to determine if a decrease in accidents is continuing, or if the accident rate has steadied out.

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The better three answers are provided by Mohan PB, Anirudh Kund and Venigopal R. The best answer is by Venugopal R as it covers the various scenarios for the LCL being seen as important limit for a c chart or u chart.

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