7QC Tools: Why do we Require to Plot X-bar and R-charts Simultaneously

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The main purpose of the control charts is to monitor the health of the process and this is done by monitoring both, accuracy and the precision of the process. The control charts is a tool that helps us in doing so by plotting following two control charts simultaneously for accuracy and precision.
Control chart for mean (for accuracy of the process)
Control chart of variability (for Precision of the process)
E.g. X-bar  and R chart (also called averages and range chart) and X-bar  and s chart

The Accuracy and the precision

We all must be aware of the following diagram that explains the concept of precision and accuracy in that analytical development.


If you are hitting the target all the time at the bull’s eye is called as  accuracy and if all your shots are concentrated at the same point then it is called as Precision.


Figure-1: Accuracy and precision


You are off the target (inaccurate) all the time but your shots are concentrated at the same point i.e. there is not much variation (Precision)


It is an interesting case. Your shots are scattered around the bull’s eye but, on an average your shots are on the target (Accuracy), this is because of the average effect. But your shots are wide spread around the center (Imprecision).


In this case all your shots are off target and precision is also lost.

Before we could correlate the above concept with the manufacturing process, we must have a look at the following diagram that explains the characteristics of a given manufacturing process.

Figure-2: Precision and Accuracy of a manufacturing process

Figure-2: Precision and Accuracy of a manufacturing process

The distance between the average of the process control limits and the target value (average of the specification limits) represents the accuracy of the process or how much the process mean is deviating from the target value.

Whereas the spread of the process i.e. the difference between LCL and UCL of the process represents the precision of the process or how much variation is there in the process.

Having understood the above two diagrams, it would be interesting to visualize the control chart patterns in all of the four cases discussed above. But, before that let’s have a look at the effect of time on a given process i.e. what happens to the process with respect to the time?

As the process continue to run, there will be wear and tear of machines, change of operators etc. and because of that there will be shift and drift in the process as represented by four scenarios described in the following diagram.


Figure-3: Process behavior in a long run

A shift in the process mean from the target value is the loss of accuracy and change in the process control limits is the loss of precision. A process shift of ±1.5σ is acceptable in the long run.

If we combine figure-1 and figure-3, we get the figure-4, which enable us to comprehend the control charts in a much better way. This gives picture of the manufacturing process in the form of control charts in four scenarios discussed above.


Figure-4: Control chart pattern in case of precision and accuracy issue

Above discussion is useful in understanding the reasons behind the importance of the control charts.

  1. Most processes don’t run under statistical control for long time. There are drifts and shift in the process with respect to the time, hence process needs adjustment at regular interval.
  2. Process deviation is caused by assignable and common factors/causes. Hence a monitoring tool is required to identify the assignable causes. This tool is called as control charts
  3. These control charts helps in determining whether the abnormality in the process is due to assignable causes or due to common causes
  4. It enables timely detection of abnormality prompt us to take timely corrective action
  5. It provides an online test of hypothesis that the process is under control
    1. Helps in taking decision whether to interfere with process or not.
      1. H0: Process is under control (common causes)
      2. Ha: Process is out of control (assignable causes)


6.  Helps in continuous improvement:


Figure-5: Control Charts provide an opportunity for continuous improvement



7QC Tools: Basis of Western Electric Rules of Control Charts

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We all are aware of these famous rule, for beginners let’s understand the basis of these rule. All rules are applied to one half of the control chart. The probability of getting a reaction to the test is ~0.01.



  1. A single point outside 3σ control limits or beyond zone A.
    • Probability of finding a point in this region = 0.00135 if caused by the normal process. Anything in this region is a case of assignable cause.
  2. Two out of three consecutive points in zone A (beyond 2σ).
    • Probability of getting 2 consecutive points in zone A = 0.0227*0.0227 = 0.00052
    • Probability of 2 out of 3 points in zone A = 0.0227*0.0227*0.9773*3 = 0.0015
  3. Four out of 5 consecutive points in zone B (beyond 1σ)
    • Probability of getting one points in zone B = 0.1587
    • Probability of 4 points in zone B and 1 point in other part of the control chart = 0.1587*0.1587*0.1587*0.1587*0.8413*5 = 0.0027
  4. Eight consecutive points on one side of the central line.
    • Probability of getting one points in beyond central line = 0.5
    • Probability of 8 points in in succession on one side of the central line = 8*0.5 = 0.0039

7QC Tools: Case Study on Interpreting the Control Charts

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A process was running in a chemical plant. The final stage of the process was the crystallization, which gave the pure product. There were two crystallizer used for the purpose, each operated by a different individual. The SOP says that crystallizer has to be maintained between 30-40°C and for 110 to 140 minutes. The data for a month is captured below

picture109 In order to understand the process, I-MR control chart was plotted (for simplicity, R-chart is not captured).


As we have learned from the earlier blog, the alternate points above and below the central line represents some short of stratification (see the short connecting arms and the concentration of data points in zone B and C).

We plotted the histogram of the above data set and kept on increasing the number of classes. What we saw was the emergence of a bimodal distribution as we kept on increasing the number of classes.


So, one thing was sure, there were two processes running in the plant. Now question that was to be answered was “What is causing this stratification?”

We started with crystallizer, as soon as we plotted the simple run chart of the process with groups using Minitab®, we could see the difference. Crystallizer-2 was always giving better yield. This should not happen because both the crystallizer were identical and were connected to same utilities. Then we thought about the different operators might be the reason for this behavior, as this was the only factor that was different for both the crystallizer.

picture114When we plotted the same run chart with grouping, but this time operator was used for the purpose of grouping. We got the same result as was found with the crystallizers, the operator-2 working on the crystallizer-2 was producing more quantity of the product. This run chart is not shown here.

We further grilled down to the operating procedure adopted by the two operators. We studied temperature and the maintenance time using scatter plot. The results are shown below


Finally, it was found that operator-2 was maintaining the crystallizer-2 at the lower end of the prescribed temperature and for longer duration. Hence, specification for temperature and the maintenance time was revised.

7QC Tools: Interpretation of Control Charts Made Easy

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Visual Inspection of the Control Charts for Unnatural Patterns

Besides above famous rules, there are patterns on the control charts that needs to be understood by every quality professionals. Let’s understand these patterns using following examples. It would be easier to understand them if we can imagine the type of distribution of the data displayed on the control chart.

Case-1: Non-overlapping distribution

As a production-in-charge, I am using two different grades of raw material with different quality attributes (non-overlapping but at the edge of the specification limits) and I am assuming that the quality attributes of the final product will be normally distributed i.e. I am assuming that most of final product will hit the center of the process control limits.

If the quality of the raw material is detrimental to the quality of the final product then my assumption about the output is wrong. Because the distribution of the final product quality would take a bimodal shape with only few data at the junction of the distribution. Same information would be reflected onto the control chart with high concentration of data points near the control limits and fewer or no points near the center. Here is the control chart of the final product


In this completely non-overlapping distribution, there will be unusual long connecting arms in the control charts. There will be absence of points near the central line.

If we plot the histogram of this data set and go on increasing the number of classes, the two distribution would get separated.


So, whenever we see a control charts with the data points concentrated towards the control limits and no points at the center of the control charts, immediately we should assume that it is a mixture of two non-overlapping distribution. Remember long connecting arms and few data points at the center of the control chart.

Case-2: Partially overlapping distribution

Assume this scenario: A product is being produced in my facility in two shifts by two different operators. Each day I have two batches, one in each shift. There is a well written batch manufacturing record indicating that the temperature of the reactor should be between 50 to 60 °C. The control chart of a quality attribute of the product is represented by following control chart.



We can see that the data points on the control chart are arranged in an alternate fashion around the central line. The first batch (from the 1st shift) is below the central line and next batch (from the 2nd shift) is above the central line. This control chart shows that even we are following the same manufacturing process, there is a slight difference in the process. It was found that the 1st shift in-charge was operating towards 50 °C and the 2nd shift in-charge was operating towards 60 °C. This type of alternate arrangement is indication of stratification (due to operators, machines etc.) and is characterized by short connecting arms.

There are the cases of partially overlapping distribution resulting in a bimodal distribution, which means that there will be few points in the central region of the control charts but, majority of the data points would be distributed in zone C or B. In such cases, it would be appropriate to plot the histogram with groups (like operator, shift etc).

Case-3: Significant Overlapping distribution

If there is significant overlap between the two input distributions then it would be difficult to differentiate them in the final product and the combined distribution would give a picture of a single normal distribution. Suppose the operators in the above case-2 were performing the activity at 55 °C and 60 °C respectively. This would result in an overlapping distribution as shown below


Case-4: Mixture of unequal proportion

As a shift-in-charge, I am running short of the production target. What I did to meet the production target was to mix the current batch with some of the material produced earlier for some other customer with slightly different specification. I hoped that it wouldn’t be caught by the QA!. The final control chart of the process looked like


We can see from the control chart that if two distributions are mixed in an unequal proportions then the combined distribution would be an unsymmetrical distribution. In this case one-half of the control chart (in present case the lower half) would have maximum data points and other half would have less data points.

Case-5: Cyclic trends

If one observe a repetition of the trend on the control chart, then there is a cyclic effect like sales per month of the year. Sales in some of the specific months are higher than the sales in some other months.


Case-6: Gradual shift in the trend

A gradual change in the process is indicated by the change in the location of the data points on the control charts. This chart is most commonly encountered during the continuous improvement programs when we compare the process performance before and after the improvement program.


If it is observed that this shift is gradual on the control charts, then there must be a reason for the same, like wear and tear of machine, problem with the calibration of the gauges etc.

Case-7: Trend

If one observe that the data points on the control charts are gradually moving up or down, then it is a case of trend. This is usually cause by gradual shift in the operating procedure due to wear and tear of machines, gauges going out of calibration etc.


Summary of unnatural pattern on the control charts
Unnatural pattern Pattern Description Symptom in control chart
Large shift (strays, freaks) Sudden and high change Points near and or beyond control limits
Smaller sustained shift Sustained smaller change Series of points on the same side of the central line
Trends A continuous changes in one direction Steadily increasing or decreasing run of points
Stratification Small differences between values in a long run, absence of points near the control limits A long run of points near the central line on the both sides
Mixture Saw-tooth effect, absence of points near the central line A run of consecutive points on both sides of central line, all far from the central line
Systematic Variation or stratification Regular alternation of high and low values A long run of consecutive points alternating up and down
Cycle Recurring periodic


Cyclic recurring patterns of points

For the case study see next blog

7QC Tools: My bitter experience with statistical Process Control (SPC)!

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I just want to share my experience in SPC.

In general, I have seen that people are plotting the control chart of the final critical quality attribute of a product (or simply a CQA). But the information displayed by these control charts is historical in nature i.e. the entire process has already taken place. Hence, even if the control chart is showing a out of control point, I can’t do anything about it except for the reprocessing and rework. We often forget that these CQAs are affected by some critical process parameters (CPPs) and I can’t go back in time to correct that CPPs. The only thing we can do is to start a investigation.



Instead, if we can plot the control chart of CPPs and if these control charts shows any out of control points, IMMEDIATLY WE CAN FORECAST THAT THIS BATCH IS GOING TO FAIL or WE CAN TAKE A CORRECTIVE ACTION THEN AND THERE ITSELF. This is because CPPs and CQA are highly correlated and if CPPs shows an out of control point on its control chart, then we are sure that that batch is going to fail.


Hence, the control charts of CPPs would help us in forecasting about the output quality (CQA) of the batch because, the CPP would fail first before a batch fails. This will also help us in saving the time that goes into the investigation. This is very important for the pharmaceutical industry as everyone in the pharmaceutical industry knows, how much time and resource goes into the investigation!


I feel that we need to plot the control chart of CPPs along with the control chart of CQA, with more focus on the control chart of CPPs. This will help us in taking timely corrective actions (if available) or we can scrap the batch, saving downstream time and resource (in case no corrective action available).

Another advantage of plotting the CPP is for looking for the evidence that a CPP is showing a trend and in near future it will cross the control limits as shown below, this will warrant a timely corrective action of process or machine.


CQA: Critical Quality attribute

CPP: Critical Process Parameter

OOS: out of specification

7QC Tools — The Control Charts


The Control Charts

This is the most important topic to be covered in the 7QC tools. But in order to understand it, just remember following point for the moment as right now we can’t go into the details

  1. Two things that we must understand beyond doubt are
    1. There is a customer’s specifications, LSL & USL (upper and lower specification limits)
    2. Similarly there is a process capability, LCL & UCL (upper and lower control limits)
    3. The Process capability and customer’s specifications are two independent things however, it is desired that UCL-LCL < USL-LSL. The only way we can achieve this relationship is by decreasing the variation in the process as we can’t do anything about the customer’s specifications (they are sacrosanct).
    4. Picture13
  2. If a process is stable, will follow the bell shaped curve called as normal curve. It means that, if we plot all historical data obtained from a stable process – it will give a symmetrical curve as shown below. The σ represents the standard deviation (a measurement of variation)
    • picture88
  3. The main characteristic of the above curve is shown below. Example, the area under ±2σ would contain 95% of the total data
    • picture19
  4. Any process is affected by two types of input variables or factors. Input variables which can be controlled are called as assignable or special causes (e.g., person, material, unit operation, and machine), and factors which are uncontrollable are called noise factors or common causes (e.g., fluctuation in environmental factors such as temperature and humidity during the year).
  5. From the point number 2, we can conclude that, as long as the data is within ±3σ, the process is considered stable and whatever variation is there it is because of the common causes of variation. Any data point beyond ±3σ would represent an outlier indicating that the given process has deviated or there is an assignable or a special cause of variation which, needs immediate attention.
    • picture89
  6. Measurement of mean (μ) and σ used for calculating control limits, depends on the type and the distribution of the data used for preparing control chart.

Having gone through the above points, let’s go back to the point number 2. In this graph, the entire data is plotted after all the data has been collected. But, these data were collected over a time! Now if we add a time-axis in this graph and try to plot all data with respect to time, then it would give a run-chart as shown below.


The run-chart thus obtained is known as the control chart. It represents the data with respect to the time and ±3σ represents the upper and lower control limits of the process. We can also plot the customer’s specification limits (USL & LSL) if desired onto this graph. Now we can apply point number 3 and 4 in order to interpret the control chart or we can use Western Electric Rules if we want to interpret it in more detail.

The Control Charts and the Continuous Improvement

A given process can only be improved, if there are some tools available for timely detection of an abnormality due to any assignable causes. This timely and online signal of an abnormality (or an outlier) in the process could be achieved by plotting the process data points on an appropriate statistical control chart. But, these control charts can only tell that there is a problem in the process but cannot tell anything about its cause. Investigation and identification of the assignable causes associated with the abnormal signal allows timely corrective and preventive actions which, ultimately reduces the variability in the process and gradually takes the process to the next level of the improvement. This is an iterative process resulting in continuous improvement till abnormalities are no longer observed in the process and whatever variation is there, is because of the common causes only.

It is not necessarily true that all the deviations on control charts are bad (e.g. the trend of an impurity drifting towards LCL, reduced waiting time of patients, which is good for the process). Regardless of the fact that the deviation is goodor badfor the process, the outlier points must be investigated. Reasons for good deviation then must be incorporated into the process, and reasons for bad deviation needs to be eliminated from the process. This is an iterative process till the process comes under statistical control. Gradually, it would be observed that the natural control limits become much tighter than the customer’s specification, which is the ultimate aim of any process improvement program like 6sigma.

The significance of these control charts is evident by the fact that it was discovered in the 1920s by Walter A. Shewhart, since then it has been used extensively across the manufacturing industry and became an intrinsic part of the 6σ process.


To conclude, the statistical control charts not only help in estimating these process control limits but also raises an alert when the process goes out of control. These alerts trigger the investigation through root cause analysis leading to the process improvements which in turn leads to the decreased variability in the process leading to a statistical controlled process.

7QC Tools: Flow chart – Have You Ever Took it Seriously?

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We have seen that it is essential to know one’s process before taking any improvement projects. But, the main question remains unanswered

“How process flow can help us?”

Case-1: Department driven company Vs. Process driven company

We have experienced that any company is comprised of many departments.


However, the irony is that the company is made of departments whereas the processes make the business (product and services provided by any company is the result of series of processes that encompasses various departments) as shown below.


As company is divided into departments hence, everyone is responsible for their departmental work-flow but no one is accountable for the entire business flow leading to product & services. Finally because of this myopic vision, people fail to see the entire process that make up the business and couldn’t contribute beyond their department for the betterment of the business. Hence, the process flow described above should not be taken as manufacturing project but as a process starting from order and ends with revenue recognition or simply as “order to payment.”

In the departmental set-up, production can argue that they have completed the task in time and the project has been goofed-up by some other department hence, they are not responsible for the project failure. This is a common scenario where all department works in silos. But they fail to figure out the bigger picture

“If any sub-process of the work-flow fails, it hampers the product/services delivery and as a result, company fails to get the revenue. As the company’s growth is hampered, which in return could hamper everyone’s performance and increment irrespective of the good work by your department.” Simply because if company fails to generate the revenue, from where it would give you the increment? Even if you have performed well! 

I do remember that in my last organization, we were in the customer centric business and after initial struggle we came up with following CFT matrix headed by a project manager. The CFT comprises of the members from each department. Once the project is awarded to the CFT, this CFT is responsible for the end to end delivery of the project. Even though each member was reporting to their departmental head but for the project they were one unit with a common goal. As a result no one has excuse in case project fails, irrespective of any reason.


Advantage of this system is that there is a dedicated team for the delivery of the project and the work-flow is automatically followed. Another advantage is that the respective head interferes with the process in case of criticality otherwise they can so some other value added work for the company (justifying their salary!).

Case-2: Identifying non-value adding steps in the process

Take the case of raw material release by QA for the production (earlier blog). Now if we modify the process flow diagram by adding the time taken for the activity by a department and the person responsible, then flow charts can add value by

1. Increasing the process understanding

2. Identify the problem areas and improvement opportunities by identifying the non-value adding activities.

3. It helps in laying the foundation for value stream mapping (VSM)

4. It helps in establishing the service level agreement (SLA) between the departments.




In the above work flow we can question the QC

why it takes 1 hour for SOP retrieval? Can’t we have soft copies instead?

Why QC head takes 10 hours for approval? Can’t it be delegated to some experienced QC person? if not, can we train some QC persons?

Why QA head takes 5 hours for approval, as he has say only yes or no by looking at the QC data? Can’t it be delegated to some experienced QA person? if not, can we train some QA persons?

To summarize, if we make the process flow in minute details, we can have solutions and we can also eliminate non-value adding steps.


Related Blog

7QC Tools: Flow Chart, Know Your Process Thoroughly

7QC Tools: Fish Bone or Ishikawa Diagram

7QC Tools: How to Extract More Information from the Scatter Plot?

7QC Tools: How to Draw a Scatter Plot?

7QC Tools: Scatter Plot — Caution! Misuse of Statistics!

7QC Tools: Scatter Plot

7QC Tools — How to Prioritize Your Work Using Pareto Chart?

7QC Tools — How to Interpret a Histogram?

7QC Tools — How to Draw a Histogram?

7QC Tools — Histogram of Continuous Data

7QC Tools — Histogram of Discrete Data

7QC tools — Check List

Excellent Templates for 7QC tools from ASQ

What are Seven QC Tools & How to Remember them?

Hammer, M. and Champy, J. (1993). Reengineering the Corporation: A Manifesto for Business Revolution, New York: HarperCollins Publishers.

7QC Tools: Flow Chart, Know Your Process Thoroughly

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Picture52Rightly said by Deming, if we really want to improve our processes then there are two thing to be taken care of. Firstly, we must know our process to an extent that even if someone ask us in the middle of the night, it should come instantaneously from out mouth as if we have witnessed the process every day. Second most important thing is that we must have some measurement system to evaluate our process

Picture53We will deal with measurement in some other day, let’s talk about the process for the time being.

We all must have seen the organizational chart or an organogram that describes how the various departments are arranged within the organization. This chart is valuable in the sense that it enables us to visualize a complete organization, by means of a picture.


Similarly, every organization have business processes that give a pictorial view of the work flow within the organization for delivering a product/services. This helps employees to visualize the movement of men/material/services from department to another. This is also called as process map as it shows the sequence of events/task that are there in performing a given process.

For example: on a macro level, department ‘B’ in an organization has to do some work (adding value)  as shown below by the block diagram


But, for adding some value to the work, department ‘B’ must get some input from some other department say ‘A’ and once ‘B’ has added the value to the input received from ‘A’, it has produced some goods or services which becomes output from department ‘B’. This output in turn becomes input for some department ‘C’ or for an external customer. Same thing is depicted by the block diagram shown above where the flow of goods/services between the departments within the organization is shown.

For example, purchase of raw material by production can be represented by following flow diagram


This is called as flow diagram. The example given above is the flow diagram at macro level. Now let’s see the complete work flow or the flow chart at the sub-macro level for the process “order to dispatch” of some product.


In the above flow chart, we have tried to link the activities of various departments to fulfill the common goal of producing some goods/services for a customer. Point to be noted here is that, each block above represents a separate department hence, it represents a sub-process within that department which is required to be executed by that department in order to achieve the organizational goal of delivering the goods/services to the customer on time.

The micro level flow chart shows the entire sequence of events of a process by using some standard symbols with some specific meaning as shown below


Let’s look at the process of getting the raw material from the ware house for the production at micro level using flow chart


We can see that, it take almost 6 days to release a new batch of raw material for the production. In order to understand it further, the process of “analysis by QC” is investigated as shown below

We can see that the raw material reaches QC on second day, however the raw material is approved/rejected on on the sixth day! Why it is so? let’s get an answer from investigating the QC process


Now we can ask ourselves following questions

Why analysis starts at 3rd day when raw material is submitted on 2nd day itself? whether it is a manpower or machine constrain?

Why review of the QC would require a whole day?

If we can resolve the above issues, we can reduce the approval time by two days.

What we have done above is called as value stream mapping (VSM) of the process, thereby eliminating the non-value adding steps to increase the efficiency of the process.

Above example shows the power of flow chart/process mapping. But irony is that we seldom map the process and in absence of it, it is difficult to start any improvement program. This is because we need to have a baseline for the existing process in order to propose a improvement.

Other outcome of the flow chart is that we can make some decision about the root causes of a problem. Hence, flow chart in combination with fish-bone diagram is a very powerful tool to screen out most probable causes.

Related Blogs

7QC Tools: Fish Bone or Ishikawa Diagram

7QC Tools: How to Extract More Information from the Scatter Plot?

7QC Tools: How to Draw a Scatter Plot?

7QC Tools: Scatter Plot — Caution! Misuse of Statistics!

7QC Tools: Scatter Plot

7QC Tools — How to Prioritize Your Work Using Pareto Chart?

7QC Tools — How to Interpret a Histogram?

7QC Tools — How to Draw a Histogram?

7QC Tools — Histogram of Continuous Data

7QC Tools — Histogram of Discrete Data

7QC tools — Check List

Excellent Templates for 7QC tools from ASQ

What are Seven QC Tools & How to Remember them?