 Session 11 of our course on Quality Control and Improvement using Minitab, I am Professor Indrajit Mukherjee from Shailesh J. Mehta School of Management, IIT Bombay. So, previous session what we have seen is basically control charts, we are discussing about control charts. So, we have talked about X bar R and X bar A type of control chart to monitor CTQs and figure out where whether there is some assignable cause because of which the mean is shifting or variation is shifting to abnormal scenarios like that. So, in this section what we will do is that we will talk about one more control chart and then we will move towards attribute type of charts which is also very relevant and frequently used in controlling the quality. So, one of the charts that come to our mind which is very prominent is you see I cannot sometimes in scenario is that I cannot collect multiple observations of subgroups at a given time point like that and sometimes it is irrelevant also to take samples when the readings will not differ much like that in chemical process viscosity. So, if I go at a certain instance and there is a chemical liquid that is being manufactured and at the time point if I take 5 samples like that, all samples will be more or less very very close like that readings will be very close, there will be not much difference over here. So, here brain oil hardness like that. So, whenever a manufacturing process happens, so in that case hardness treatment was done and in that case products are coming out of that. So, everywhere we expect the hardness to be more or less same. So, one sample is sufficient because we want to save the cost and in that case we do not need multiple samples like that. So, single sample observation is there and in that case what is to be done what type of control chart I should implement and try to figure out whether the process is going out of control or is in control like that. So, this kind of scenarios, but in this kind of scenarios also you have to remember that the data may be first observation with the second time point T2 observation there can be some correlation which can exist between the data or auto correlation we talk about. So, there can be scenarios where these two data may be interrelated like that. So, readings of initial observation T1 and second observation may be somewhat related with each other. So, such kind of scenarios so when we have, so in that case we may use different types of charts. So, one of the options that is available is individual moving range chart. So, here you can see 20 observations are taken in a given process at different time points T1 to T20 let us say and only one observation was selected over here and we want to monitor the mean and variance of this. So, we can calculate the overall mean if you see this one I can always calculate so what is the overall average over here. So, this can be calculated like that and these are the individual observations that you see over here. So, x bar observations can be collected which is shown over here and then upper control limit line and these are calculated based on some measures which is r bar over here and this r bar is basically calculated based on moving range like that. So, how moving range is calculated over here 36.3 over here minus 28.6 that is the second observation moving. So, this is the moving range what you are seeing so observations from moving range 2 like that. So, second observation it will start and let us say there are k observations k moving average we are having like that. So, formula says from 2 to k like that all moving average if you take the average of that and we will get and this will be k minus 1 because one observation will be not there and the other observation we can we can calculate moving range over here and average of moving range will be treated as r bar over here and moving range means range means maximum minus minimum. So, next observation with the previous observation absolute difference between these two is the range of this. So, otherwise maximum minus minimum also you can think of as the range over here moving range basically. So, average of moving range is calculated over here and then with this average which is treated as r bar over here I can calculate the limit lines for individual chart and which is x bar minus 3 r bar by d 2 where d 2 will be defined like previous one x bar r chart the d 2 value will be will be influenced by the value of n equals to 2 over here. So, corresponding to n equals to 2 I will figure out because every 2 observation we are calculating the moving range. So, this subgroup size will be treated as 2 over here. So, in this case what will be happening is that x plus 3 r bar by d 2 and x minus of this as the lower control limit line and this will be upper control limit line will be x bar plus 3 r bar by d 2 where d 2 will be defined by n equals to 2. Similarly, here also you will find d 3 r bar and d 4 r bar like x bar r chart similar to that. So, because d 3 is 0 over here for n equals to 2. So, in this case this becomes 0 and this is d 4 m r bar which is the average of moving range basically which we are calculating over here. So, overall average x bar of the individual observation we can get and then we can define the upper control limit line lower control limit line and similarly moving range average will be done we we can calculate over here. So, moving range can be calculated over here and and this is the expression for moving range x i minus x i minus 1. So, previous observation absolute value of that. So, in this case this will give me moving range and average of this moving range we are just monitoring over here. So, their upper control limit line will be there lower control limit line will be there and the central line will be m r average basically m r average. So, average of this or r will be the average of this, this is the central line what you see over here. So, Minidive does it automatically for you and in that case we will try to demonstrate how the Minidive gives the output when we are just defining that one. So, I am opening a file where this information is already there like that. So, I am just opening that one. So, in this case what we can do is that we just this is the data C2 column is having the data over here and we can go to stat and then control chart and then variable chart for individuals over here I am a chart option is going to come over here. So, I am a chart when you click and then identify the variable that we are in the hardness we want to monitor over here and I am a option will be also test that one point goes outside or not remains same everything remains same over here and then I click ok like that and immediately what will happen is that you will get an output in Minidive and in that case you will try to figure out that these observations are shown in this on the top it is individual values and the moving range is shown over here. So, moving range is happening in limit over here and also individual chart is having upper control limit line and the calculation even same formula that I have shown it will be same like this. So, this formula is used over here to calculate the upper control limit upper control limit lower control limit line like that ok. So, all points are within the control limit line. So, there is no problem and the process is under statistical control and if this is a stable process we can we can just define like that ok. And although this charting techniques is quite robust, but we have to remember that charts also are built based on certain assumption like plus plus or minus 3 standard deviation what we are considering over here and that can be violated also. So, in this case we need to cross check that whether the data of hardness over here follows normal distribution or not. So, how do we check that there are ways different ways of checking that one probability plot papers are there. So, otherwise statistical tests are also there which confirms that this data set is non-normal like that or normal data set like that. So, for that knowledge of hypothesis testing is required which we will discuss because experimentation design of experimentation requires that knowledge like that and so we will give brief about that afterwards. But if the data is non-normal although it is it can handle moderate deviation from normality over here IMR chart like that. But if scenario comes that it is skewed in that case there are options that we we can change this data set that you are seeing complete data set into a different variable. So, if we are saying this is the CTQ which we can define as y, we can change this data set or convert this data set by some transformation over here. So, which can be y dash over here. So, we will use some transformation like let us say y dash equals to square root of y or y dash equals to square of y like that. So, when you convert this data over here and in that case what happens is that the data the converted data will become normally distributed like that. Then what we do is that this can be implemented in x bar R also because if the data is non-normal in that case we have to we need to do something on the data. So, there are many ways of doing this and we will discuss it is afterwards like that. But at present you you can I can assure you that there are checks which ensures that whether the data coming from normal distribution or not it is possible to check. So, in this case what happened is that I have checked the data initially and I saw that there is some it is deviating from normality over here. So, some transformation was used over here. So, I am writing that as trans hardness over here. Some transformation was done over here and the data seems to follow normal after I have done some transformation. So, 36.3 let us say some transformation was done over here with some expression and that is the standard way we do transformation of the dataset. Then what we do is that on the transform data when it is normal then we plot the transform data and try to figure out whether that is following and that is under control or not. So, what we do is that first we check the data normality like that if it is normal we plot it in IMR chart. But if it is non-normal in certain scenarios what we will do is that we will try out some tricks and we do the transformation of the data it can be linear or non-linear transformation of the data. And when we transform the data into some other variables and then we try to monitor the and that follows normal distribution then what we do is that on the transform data we will apply our control chart techniques. So, IMR chart will be used over here now I will use transform data hardness. So, then I will again click ok and try to see whether that is following within the control limit line or not. So, first so, this is only done when there is a high amount of skewness and we want to but moderate deviations can be handled by IMR chart. So, that you have to remember and in this case also after transformation also what we see is that data is within the control limit line in individual and also in range over here. So, we can assure that the process is in statistical control. So, this is a stable process like that. So, whenever you find that there is some deviation you can always ensure that whether I can do transformation on the data set which will convert the which will assure the normality assumptions like that and then I can apply the charting techniques what we are using over here IMR like that. So, everywhere it is true. So, certain scenarios we need to transform it is non-normal scenarios because everything is based on assumption of plus minus 3 standard deviation that assumptions of building the upper limit and lower limit which differentiates between control and out of control. So, in that case what happens is that we try to ensure that one. So, second case also is like that. So, I have taken another example over here. So, this was taken by Amitabh Umitra's book this was taken from that book and next example also we have which talks about IMR implementation like that. So, here hours between failure was monitored over here. So, monitoring the occurrence of failure over here. So, number of hours between failures. So, how many hours are passed before it fails like that. So, this observation was collected over here and there are 20 observations sample 20 samples observation and this is also a single observation that we are having at any given time point like that. So, this is T1, this is T2 like this and here also with the original data set we can try IMR chart and because the data was seen to be not normal in that case we have also transformed the data. Here one transformation was used which is known as which is square root of the original values like that. So, if you 286 if you square root that it will be 16.91 like that it was also not following normal distribution. So, we will do that afterwards and try to figure out that how to check normalities because we need to understand hypothesis and then only we can understand the statistical test which confirms whether it is normal or non-normal like that scenario. So, here what we are assuming that this moderately it is deviating from normalities over here and then we have applied control chart. So, IMR chart we will apply first and let us try to see that whether everything is in control. So, when we do that we see some amount of observation which is going outside the control limit line for time between failure over here and after transformation let us try to see then what happens. So, here it is reflecting that without transformation some variables are going outside and but after transformation let us try to see this data set when I have used a square root type of transformation on Y what happens to the data set. So, here you can see that after I have done the transformation something extraordinary happened over here. So, in this case what is happening is that all the data points are falling within the control limit line you see. Initially without transformation we are seeing that abnormal scenarios and there is a there are observations which is going outside, but when I have converted the data into and it follows normal distribution what happened is that I saw and I plotted the IMR chart then everything is in control. So, if you have taken the action based on the assuming that is normal and in that case we may have over adjusted the process because process has not gone outside, but because the data is non-normality showing outside of the control limit line, but when I when I convert the data and converted data is plotted in IMR chart what happens is that everything is fine. So, I do not need to adjust the process like that ok. So, that is one observation quick observation what we can make out of this is that first we have to see the assumptions of normality is everything is fine no problem and if it is not there let us convert the data into some so that it is normal normal some transformation we will apply over here linear or non-linear transformation or log transformation whatever transformation is required and that we will understand what are the different types of transformation that can help in this regard and whenever I have done that and then plot the IMR or any any types of control chart and then we will see that whether actually we need to take any action or we do not need to take any actions like that ok. So, we have applied such kind of method. So, this is an observation and you will find in any books that they will refer that you can convert the data into normal so that it follows normality and then you apply the control limits or apply the control chart techniques like that on the converted data like that ok. So, these are the main charts which are generally used in manufacturing any other processes. So, in that case and the magnitude of shift is quite large and in that case we prefer to use this one, but if the magnitude of shift is less we have other types of control chart to monitor that one. So, those things those I will not discuss over here like to some chart exponentially weighted moving average chart. So, some charts are available even they are robust against failure of normality assumptions like that. So, their researchers have proved that they are quite efficient like that, but this that will not be our topic of discussion over here. So, we will stop over for the variable type of control chart we have only discussed x bar R, x bar S what scenario we will apply, then individual moving range when I have single observations and then how to deal with that and then we will discuss about another types of control chart which is attribute control chart which is attribute control chart and that will be our discussion topic now onwards. So, IMR chart and now we will discuss about attribute control chart. So, what is attribute control chart? Here generally what we do is that this is the initial phase of analysis when we implement preventive measures like that what we do is that in a process we try to figure out or in assembly lines how many defects are coming in the process like that. So, out of 100 assemblies that is inspected and within that how many defects or defective items is coming in the process like that. So, defects arrival of the defects. So, if you have done statistical course you know that arrival of defects follows Poisson distribution and this chart is approximation of Poisson distribution and based on that plus or minus 3 standard deviation was implemented and the formula is given over here to calculate the upper limit line and lower limit line. So, this data set what you are seeing is that 100 pinterest circuit board was at a given time point we are taking 100 pinterest circuit board and within the circuit board how many defects was observed like that. So, you understand this term you have to know this is for defects only. These are the defects and not defectives over here. So, when it is defects it follows a Poisson distribution. So, it will follow Poisson distribution and arrival of the defects is basically Poisson that is the underlying assumption and then what we do is that we try to build a control charts for that and and this is the rate of defect arrivals like that. So, this is C bar or average of defects that you count over here. So, summation of this and summation of this and divided by total number of observation that is 26 that will give me C bar over here and plus 3 standard deviation of this process Poisson process. So, that is also mean and you understand that mean and variance is same say mean and variance of the Poisson distribution is same. So, in that case those are same. So, this is C bar plus 3 standard deviation and this C bar can be calculated easily. So, here there is no need of putting any n n over here. So, because n is constant over here. So, we do not need to take care of that. So, over here every time I have taken 26 successive samples over here of 100 pinterest circuit board. So, this is the sample size n that we are taking at a given time point like that. So, at t 1 time point 100 was inspected out of that 21 was defective. Then again 100 was inspected at time point t 2 and what has come out is 24 defects over here. Then average of the defects will give me C bar observation over here. So, summation of this plus this will give me C bar observation. So, we will get C bar observation over here and then I can calculate the upper control limit line and this will be the central line over here C bar will be the central line. So, C bar is the central line upper control limit line and lower control limit line that will give me some idea that whether the defects had shoot out in certain scenarios like that. Here you can see that when we plotted the data in control charts, one has gone outside the upper control limit line and one has gone below the lower control limit line over here. Now, if it goes below the lower control limit line, you see number of defects has gone down drastically over here. So, in this case what happens is that I can take this as an opportunity. So, this is an opportunity. So, I need to understand why defects has gone down over here. So, sometimes what happens operator may not be able to differentiate between defects. So, if that is the scenario then the way we are measuring defects is something is going wrong. So, we need to correct that one, but otherwise if the process everything is fine, same inspector, same same personnel is doing the inspection over here and it has gone down. So, what was the scenario in the process at the time point? So, what is the process condition at the time point? That can be taken as an opportunity and that may be the optimal scenario what we will try to implement in every time we start the process like that. So, this is not an abnormal situation for me until unless the there is evidence that the reading that was taken was quite incorrect like that. So, then we need to correct that one otherwise this gives me an opportunity to develop the standard operating practice like that which may be the optimal one and to arrive at the optimal scenarios like that. But if this is shooting out over here and defects are very high over here and in that case this is a reasonable cause this is a reasonable cause and we need to correct this one what has happened over here we need to cross check and try to see that one. So, this data I have in Minitab and that we will try to plot and see charting of that. So, here you try to understand that here n is not used to calculate upper limit line lower limit line n is not required because this is constant and because we are using Poisson assumption in that case n is not required it will be required at certain other instances like that ok. So, I will go to the Minitab option. So, this we have shown so this is not required now and then we go to another file where we have c charts observations that is already given there and we will try to illustrate c over here. So, in this case 21 defects. So, let me go back to the is it the same example here 26 observations are there and we have over here 26 observations. So, this is the data set that we have taken over here and this is taken from Montgomery's book and then we try to see whether it is under control or there is some abnormalities in the process. So, I go to stat and we have seen that one. So, then what we do is that control charts I go to control chart attribute chart and then there are options of c at the bottom over here you will find. So, when I click c over here it will ask which is the variable I will say defect is the what I want to monitor and c chart option again test will be one point going outside. So, other things you can just explore. So, I am just showing that only I am interested in defects. So, and in this case defects follows Poisson and that is the based on that control limits will be developed I click ok and what will happen is that I will get a chart like this. So, same chart what we have generated and shown in the power point the same. So, one point has gone outside over here and over here over here. So, two points have gone outside over here ok. So, if you have to implement that you want to build the natural limit lines like that and give it to the process for next. So, what we will do is that we will eliminate this point we will eliminate this point and re-evaluate whether all the points are within the control limit line like that. So, if that is so, then those are the limit lines we will use. So, because of this point this limit line is changing over here. So, if I eliminate this one limit line will automatically change again. So, then again I will see whether all the points are I will remove this point and this point over here I want to capture the natural variation to build up the control limit line and that will be used in actual manufacturing and that I will hand over to the process personnel who is monitoring or owner of the process basically. So, in that case what we will do is that we will eliminate and recalculate. So, that was the trial control limit line concept that we have used. Over here two points are going outside that we have noted down over here and n is fixed over here. So, n is not changing over here. So, in this case similarly defects and sample in textile this was another example. So, maybe so, this is the second observation over here and also here also you see 25 samples and there are every time I have number of occurrence of foreign particles are recorded 25 samples each is 100 meter square selected. So, this is the units that we are selecting over here 100 meter square like that. So, in this case also we are counting the defects only and units remain same n remain same. So, in this case textile we can again draw this. So, what I have done is the control chart attribute control chart and I go to C type of control chart because then defect textile over here and options remain same. So, I am not changing the options over here and I click ok and I click ok like that and what I get is that one point is only falling outside the limit line. So, that is an abnormal scenario. So, we need to take care of that and try to figure out what has gone wrong. So, in this case and always you see these lines are joined over here that does not mean that this is a continuous over here just for your we can visualization of this all are discrete points over here because in control chart what we do is that we do not continuously monitor the process like that. We take some observation and then what we do is that we try to go to the process again after 30 minutes 25 minutes or 15 minutes based on the frequency what is decided and that depends on the number of samples to be taken and the total production lot that is happening and based on that these are all discrete time points we are making an observation and just noting that one. But for visual impacts what we are doing we are joining the points to see any trend is appearing or not like that. There are different trends like cyclic behavior is there or not or continuous increasing trend of defects is happening or not and all these things also indicates some abnormalities. There are different trends that can be trend analysis also can be done on this control charts and that can be seen also whether there any any other different types of abnormalities there or not even if it is within the control limit line we do those analysis like that ok. So, this is C type of control chart when we are talking about defects when we are talking about defects we use C type of control chart like that and the N is fixed also N is fixed over here. The scenario can be the defects number of defects may not be fixed like that what we are seeing over here. Every time I go to the process and in that case what will happen is that sample size can also change like that. What you see over here is fixed that is sample size, but this can always change see this sample size can always change like that. Here it is 50 that you can see it is non-confirmities over here is given like that. So, sometimes somebody may be interested that I want to plot that per unit with respect to this is the sample number over here I am checking. So, how many per unit? So, in this case 2 by 50 like this this is 3 by 50 like this. So, I will get some UI measures over here. So, then what I can do is that I will I will monitor this because that makes more sense per unit sometimes what happens that gives more impression. So, sometimes somebody also prefers to use per unit like that. So, in that case it is not much difficult formula remains same. So, in this case only n comes in picture. So, in this case number of observations as an unit what I am taking over here that will that will be counted over here. So, average per unit will be counted over here. So, this is the formulation that you see for u chart which is known as u chart, but mostly used when the sample size changes whenever sample size changes like this example what you see over here this example what I am just demonstrating there 20 samples are taken and every time sample size is changing over here you see every time sample size is changing over here and in this type of scenarios the formulation that you have seen. So, here it will be UI. So, every time the upper control limit line and the lower control limit line will keep on changing over here. So, what will happen is that we will have u bar as the central line over here, but upper control limit line because n i is changing over here. So, n i is this one. So, n i is changing every time. So, every time point when you go this is the first point and the observation is this is the upper limit line this is the lower limit line like that. So, then this is the point over here this is the upper control limit line and this is the lower control limit line. So, limit position is changing basically earlier it was this like this for the first point second point it is from here to this point like that ok. So, this is the this is the area where we are we need to concentrate and see whether the point is going outside or not. Here you see the limit line is somewhere below over here a point has gone outside. So, this is a variable control limit line that can be generated and we can see this generation when we are seeing using Minitab same data points what we want to use over here and maybe u chart that data set may be over here with variable sample size like that. And this is the sample that is C 5 to C 7 it is it is that well sample size changes and I am monitoring non-confirmities and that is defects let us say. So, in this case stat if I go and quality tools and I go to this control chart and I go to attribute chart and you go to u charts like that one option is there and then you mention that variable is non-confirmities and subgroup size is given in sample size carpets like that. So, if you give this one and options over here I am doing the testing of one point going outside. So, that is the only condition I am taking and when I click ok over here what you see that this is the same graph what I have just plotted like that. So, every time the control limit lines are changing and because of the NI is changing over here and we have different control limit lines but operator does not prefer to do like that way because every time they have to change the control limits and check that one. So, sometimes what happens is that we take average of this control limits that you are seeing. So, some average line will be plotted over here. So, one straight line may be plotted. So, if I go to this one sometimes what operator does for their benefit or something like that even the personal quality personal can what they can do is that they can take a average line over here. Similarly, they can take a average line over here based on the data and information so that it is easy for the operators to see also. So, there will be mistakes for this, but sometimes they have or maybe the maximum of this points maybe that will be the upper limit over here and this is the lower limit I will follow like that. So, there are we can implement those types of otherwise the best one is to have variable control limit lines like that variable control limit lines ok. So, limit lines and calculation of this is given by this one. So, this will be ui over here and ui over here. So, this will be central line. So, this is taken as overall average over here and this will be this will change based on ni conditions like that sorry ni conditions will be there. So, in this case u average. So, we will have different upper limit line and lower limit line for this. So, this type of u charts is generally used like in software industries lines of codes and number of defects that comes like that at different phases like that. So, I have different phases and each phases we have this many lines of codes and out of that how many defects like that. So, it will have variable number of lines of codes and accordingly how many errors that is appearing like that. So, we want to represent that one in control chart. So, they use generally prefer to use u type of control chart in their quality control divisions like that. So, they prefer to use u because sample size changes or the unit changes basically unit of inspection changes like that ok. So, that is why they uses this type of control charts like that. So, we will continue from here attribute type of chart. So, we have done with defects kinds of chart when there is defects it follows poison and and we have if it is constant defect number of samples is constant or in an unit unit if fixed like that in that case we use c type of control chart and in case the sample size is varying mostly we prefer to use u type of control charts like that. So, we will continue from here and we will start with other types of attribute control chart like p chart, n p charts like that which are also very prominent and used in many scenarios like that. So, we will discuss that in our next session. Thank you for listening.