 Welcome to session 13 of our course on Quality Control and Improvement Using Minitab. I am Professor Indrajit Mukherjee from Shailesh J. Makar School of Management at Bombay. So, earlier session we have just started about process capabilities. So, we are talking about CP index. So, from there onwards we will continue in this lecture. So, let me just recap what we have told. So, capability index what we mentioned is that it is a index which shows that ratio between voice of the customer and voice of the process. So, how do we express voice of the process? This is voice of the process 6 multiplied by standard deviation of the process and which is to be estimated this sigma has to be estimated over here. And this sigma estimation is also known as sigma within variability within. So, in this case Minitab expresses this variability as sigma equals to R bar by D2. So, this formulation what you see over here is basically within variability, within sample variability over here. So, this is calculated based on statistical process control chart. So, this is estimation of the sigma over here can be done by if we are drawing the X bar R chart in that case. And we have also told that mentioned that this process capability we estimate whenever it is under natural variation that means common cost variability is only and the process is stable. So, in that case we apply this one. So, this will be the delta the difference between upper specification this will be the delta over here and 6 multiplied by standard deviation estimate over here. And this estimation what you see is given over here in this estimation because and this is the subgroup size based on which I can I can define D2 values. So, this subgroup size is also known to us when we are collecting the samples. So, whenever the process is in statistical control immediately we can calculate the CP index over here. So, this is delta by 6 standard deviation because 6 is the common spread that we have assumed that 99.73 of the observation will be falling in case it is a normal distribution. So, the underlying assumption over here is the CDQs follows normal distribution and for that I am assuming this making a simple assumption over here. And we are also assuming over here that the centering of the process is perfectly on the target. So, there is no variation from that. And so, we can calculate a capability index which is known as CP index over here ok. So, and MINITAB does it automatically for you. So, you have to only mention that whether how to calculate the sigma you mention that one and MINITAB will calculate for you. So, then also we mentioned that just capable process if voice of the customer and voice of the process. So, this is the voice of the customer and this is the voice of the process and the ratio is equals to 1 means I am consuming full tolerance basically. My variability of the process is consuming the full tolerance like that. So, in other words that means that if this is the specification and upper specification limit and lower specification limit. So, completely I am consuming the total variability spread across from USL to MSN like that ok. So, this is not a industry standard that equals to one condition just acceptable like that. So, if you have to improve that one you have to go to 1.63 and beyond for not so critical items, but if it is very critical maybe we go about to 1.67 or 2 as the process capability for a given CTQ for a specific CTQs like that. We want to reach that level of capability like that ok. So, if it is a 6 sigma process we can which is well known like 6 sigma methodology what we talk about and it is linked with CP value of 2 like that and then we have other index over here CP 1.63 about 4, CP 1.67 about 5 like that so ok and so that way we can we can link with CP values and also with sigma level of a process like that, but that is not the right way we will we will see another way of defining the and linking with the sigma levels like that ok and MINITAB gives you some options to do that ok. So, in this case what will happen is that so I can calculate this and also we have to remember that this is for both side specifications. So, CP can be calculated whenever USL and LSL is given like that one sided specification in certain scenarios when we have one sided tolerance or like that. So, in that case CP cannot be calculated like that. MINITAB also does not reveal any such calculation of CP whenever I give one sided specification let us say LSL or USL. So, with some example we will try to see. So, here you can see like that what I explained last time also say if it is not capable in that case rejection some some will be outside the USL condition and this is the LSL condition what you see. So, many many data points will fall over here outside this one. So, there will be rejections like that and let us assume this is the target value. So, what we are getting over here ok. So, this is merely just capable what we told and this is ok 1.334 sigma level what is written over here. So, some companies may accept may be considering this as acceptance level of CP values, but excellent performance we always say that CP value should be equals to 2 which is equivalent to 6 sigma level like that ok. So, towards excellence if you are moving for quality we should we should look for CP close to 2 like that. So, and at least more than 1.33 like that many of the industries follow 1.33 as the basic standards of accepting a CTQ performance like that ok. So, based on this we are just taking an example last time we showed that in MINITAB how we are going to do that. So, I am taking a piston ring example which is having 25 observations and each of the observations has 5 subgroup size over here and at a given time point T 1. So, this was collected at T 1. So, this will be collected at T 2. So, T 1, T 2 observations like that and we have T 25 observations which are at certain gaps the observations are collected at certain intervals like that. Intervals are pre-decided and can be calculated also how much should be the interval like that we are not going to that details, but that is possible and how many subgroup size to be taken that is also rational subgrouping is another concept which is used to define that one ok. So, if that is correct in that case we can analyze the data and we can analyze it using control chart techniques like that ok. So, I am taking the same example the specification is given as 74 point plus or minus 0.035. So, tolerance is about 0.07 total tolerance if you see from USL and USL like that and this is the total delta values that we have or voice of the customer that is given and if it is within this we will and the distribution over here what you see is that this is Minitab output. So, this is LSL this is USL this target I have defined let us say 74 is the target value over here and CP index is calculated 1.17. So, USL this is LSL is 73.965 and USL is 74.035 this you have to enter Minitab and this is 74 we have mentioned. Sample mean is this total observation what you are seeing over here 125 and 25 observations. So, 125 observation. So, overall mean X double bar is 74.035 which we have mentioned are 74.0012 like that ok. 125 observation and you see standard deviation within is calculated over here this is based on Minitab we will ask you what basis I will calculate sigma within. So, we have mentioned over here that you calculate from control chart and the formula to be used is R bar by D2 or from sample range chart you calculate this one. So, Minitab you can assume and also you will find that performance index will be provided over here there is a term which is known as PPM less than LSL and greater than USL like that and PPM total will be this is PPM means parts per million basically less than LSL and PPM more than USL. So, number of items that will fall outside USL and fall below LSL that will be given as performance over here. So, what is observed over here means actually the data set will be plotted over here and Minitab will see if something is going beyond the tolerance and that will be counted there those counts and that will be converted into PPM 10 to the power 6. So, this will be converted into that number how many fall out in this much. So, in million how much it is. So, that will be calculated as observed and expected performance what you will find is that Minitab will place a normal distribution curve over here what you can see dotted line. And so, if this is the and based on certain mean that we have X double bar and we can always calculate that corresponding to USL. So, USL will be converted into Z value. So, Z USL over here and based on that it will it will find out that what is the probability of Z greater than Z values for USL over here and that will give me fraction non-confirming over here and that will give me the PPM that is falling outside. So, a normal distribution assumption will be used over here to see what is the expected performance and also the standardization calculation that will be used for Z conversion will be within that R bar by D to that formulation will be used for calculating this one. So, corresponding to value of X bar and standard deviation that sigma hat over here what I can do is that I can calculate Z USL over here and from there I can I can reach to this PPM level over here on this side and I also I can reach on this side. So, expected within performance is shown over here how much will be less than this. So, probability and then accordingly convert to PPM and here also we can do that. So, expected within and expected PPM more than USL also. So, this is the expectation in case you are just super imposing and thinking that this is a perfect normal distribution with the mean and standard deviation that is calculated from the data set how much will be the fallout from LSL and USL like that and total summation of this two will give you the total values that you see over here ok. So, what are the options that we that we use over here. So, I will just show you the options. So, I will go to the data set and in this case what we will do is that we will use process capability data set and we will try to illustrate the same thing. So, that we can we can see how we are. So, this data set is already I have created. So, I mean it have file I am opening this is the observations of one data set this is the other observation which we will analyze now sample pistons ring 1 to ring 5, 5 subgroups and these are the observations over here. So, I go to stat on top and what I do is that I go for quality tools and over here capability analysis and what I will do is that normal capability because I am assuming normality over here. So, let us assume the data follows normal. So, I am using normal capability then subgroups across row I will use and because all observations are in different columns like that. So, I will use from here to here and then I will say select this one. Then lower specification I will write this is 73.645 maybe and 73.645 and upper specification is 74.035 that is given plus or minus 0.035. So, 45 like that 645. So, I think this is correct. So, we will have 73. Sorry 965. So, this will be 73.965. So, you got the mistake 965. So, this is if you subtract from 74.035. So, that will come and then here you will find that transformation or no trouble in case it is non-normal in that case what is to be done we will see. So, this is we are assuming normalities. So, I will not click anything over here. Estimation over here. So, within capability estimation which method I will follow for subgroup size greater than 1. So, here it is written that you see within subgroup standard deviation how do we calculate. So, you have to mention pulled standard deviation S method R method. So, I will use R method because R control chart we have seen X bar R. So, used unbiased constant we will keep this one as default over here. So, and also in case subgroup is 1. So, in my case it is not true. So, this area you can ignore like that. So, I will click ok like that. So, over here and options over here target is let us say assume 74 is the target like that and within group analysis we are doing. So, overall analysis we are not considering over here. So, then capability CP values I want to see and parts per million also I want to see other than that we do not understand now at this current time point. So, we will just click ok and k value is taken as 6 because 6 standard deviation is the in formulation how much time standard deviation this we do not check. So, 6 we are keeping over here and I click ok and I click ok what will happen is that I will get this analysis what you see over here and you see that LSL I have given target I have given. So, USL is given sample mean was minute I was calculated this is the sample mean and number of observation based on which sample mean is calculated is 1 to 5. Standard deviation within is calculated as from R bar by D2 formulation ok. So, this is calculated as range average and from that D2 is taken from number of subgroup size that is 5. So, based on that this is calculated and PPE observed and expected performance is shown over here same thing what I explained like that Z it is converted into Z and from that we calculate what is PPM less than LSL what is PPM above LSL. So, total PPM parts per million that will fall outside the specification is basically 502 like that and then CP index is 1.17. So, at this time point we are not concerned about CPL, CPU and CPK. So, let us assume the CP is the only measure we know. So, 1.17 and that is the value I am looking for. So, CP 1.17 is not enough. So, 1.33 I told maybe the standard many industries follows that means improvement is needed over here ok. So, that gives you some basis when we have existing scenarios and whether to take improvement initiative or not for a CTQ that we can define from here. So, CP value is less than 1.33 let us say and in that case improvement is needed. So, in that case I have to reduce the variability you see the measures of CP index says that I cannot do anything on the tolerance, I cannot do anything on the tolerance. So, this measure what I have to do is that I have to reduce the sigma level. So, if you can reduce the sigma over here what you see over here. So, if I if I can reduce the sigma values that is coming over here then my CP will go it is inversely proportional to this. So, this will go up like that ok variation reduces CP index goes up. So, in that case. So, my variability deduction is the target over here from the formulation that we see ok. So, what scenario is maybe one sided specification we are having like this if I if I place this one let us say second data set ok one one one more second. So, let me take the next example over here and we have an example over here that is container over here. So, here in this example what you see is that 5 container subgroup size is 5, but LSL is only given 200 over here. So, this example we will take and we have 20 samples with 5 observation. So, total 100 observations we are having 5 subgroup size. So, total 100 observation and lower specification limit is given as 200 PSI. So, if you see on the right hand side it has not calculated potential within capability you see CP is not calculated and star is given over here. So, when I when I do this one. So, let us let us go to the MINITAB file and let us close this one and I will this is the sample observation C1 to C6 I go to STAT and what I do is that control chart sorry quality tools and in this case capability analysis normal capability analysis over here and I give that these are not the data I want container data to be placed over here shift this data set is given over here and LSL is given as 200 over here and upper specifications limit is not there and options over here within that is fine sorry target we have to change options target we will remove because this is a another example. So, here we were not doing this and estimation R bar we have mentioned. So, it will do like that, but what do you see is that when I when I click this one results what do you see CP is star information is given over here. So, star means it cannot calculate that one because this is one sided specification. So, if it is one sided we cannot calculate CP values for that something else has to be done ok. So, this cannot be done. So, because I cannot see delta over here. So, in that case in case you have delta then target values and that case can be calculated, but we are not looking to that. So, one sided specification we will assume that CP index cannot be calculated for that. So, some other index has to be done for those scenarios like that ok. So, that I wanted to mention. So, then let us go back and see this diagram over here. So, this is the concept that is will be used by Minitab. So, Z conversion of this. So, Z conversion I mentioned that X double bar will come from control chart and then we will sigma will be known which is S over here. So, when I get this values what I can do is that what is the B probability of this area over here, B can be calculated, A can be calculated and based on that we can also calculate a Z benchmark concept which is used in calculating the sigma level of the process like that ok. And Minitab gives you this option we will discuss more about this ok, but what I am trying to say is that we can calculate capabilities and we can and we can also calculate that what is the performance expected performance for that some Z conversion is required. So, Z and corresponding to this Z what is the probability that can be converted into PPM on both the sides like that. So, this area and this area can be calculated and then that can be converted into PPM and that is what you see in Minitab's expression what you see over here say expected performance over here this is the portion what you what you can see. So, in case this is normal what is the expected performance in median. So, in long run what what do you expect. So, if this is normal so in population what do you expect this is sample information sample information is giving you a performance observation is around 0 no fallout over here, but if you see if you if you consider the normality assumptions over here then in that case some fallouts are expected that is around 502 ok. So, that is the that is the idea of placing this expected within performance like that in PPM. So, some idea you will get how much PPM fallout will happen parts per million. So, this is the way they calculates and but what you see is that sigma is 2 over here which is freezed and this is the specification what you see upper control limit line and lower control limit line over here and in that case what you see is that sigma is kept same over here, but the distribution is shifting. So, location was here which is the target value let us assume t, but it has moved from here to somewhere over here. So, this may be 50 53 let us say. So, mean is shifting basically. So, x bar was somewhere over here which is on the target. So, specifications was like this. So, sigma remains same and in this case, but the mean is shifting over here towards upper specification line. So, it will move 2 over here, but CP index you see remains same because CP is equals to USL minus LSL divided by 6 into standard deviation. So, standard deviation remains 2. So, it is not changing and USL this is also freezed, but the whole distribution is moving from this end towards this on the higher space upper specification side. So, if you are then the accuracy part is not considered in this formulation what you see of CP. So, accuracy part is missing over here. So, what we have to do is that we have to revise this formulation so that we will penalize if it moves to other upper specification or lower specification limits like that. So, even if CP remains 2, we can we can expect that this is deteriorate performance is deteriorating, but that is not captured is CP. So, what they does is that they use as another index which is known as CPK index what you see over here which is the minimum value of CPL and CPU this minute we will calculate automatically. So, this is for USL how much is X bar from the USL and how much is X bar from this one. So, if the difference between these two comes down and then this CP index also comes down like that. So, whichever way you move it will basically penalize the CP values like that and minimum of this value is taken. So, as to ensure that we want to improve the minimum one. So, in this case so, CP index takes care of this centering concept of where is X double bar. So, not only sigma is considered X double bar is considered over here the formulation is USL minus X double bar we can think of over here and this is X double bar assuming that is the population average what we are getting. So, minimum of these two values what you see will give you the CPK index over here. So, the I want to penalize if it is moving away from the target. So, that is the objective over here and upper capability index is calculated CPU this MINITAB automatically calculates for you because average is known to us and in that case we can calculate sigma is R bar by D2. So, in this case this is also not a problem to calculate and then the MINITAB will calculate CPU and CPL and based on that minimum value it will report CPK index over here. But the assumptions over here what is considered is one of the assumptions is normal distribution assumptions that is taken process is under statistical control that we told that only assignable cause is not there. So, under control so, everything is stable and the mean is centered over here then only we can calculate CP index like that. But if mean shifts in that case we need to consider this formulation which is a revised one and it is measure of potential capability what is mentioned not actual capability. So, in this case just an short term capability you can think of. So, in this case what we can do is that and this can be calculated. So, here when you give the command over here you will find a CPK index that is shown over a 1.33 like that. CP index what we have seen is 1.17 and it will calculate CPL which is based on the previous formulation what we have shown CPL what you see over here this is the formulation X double power minus LSL by 6 multiplied by sigma and the other one is CPU is given over here as USL minus X double power by 66 standard deviation. So, that can be calculated and this is shown over here what you see CPU and minimum of these two is taken as 1.13. So, 1.13, 1.21 the minimum value is 1.13 over here. So, histogram is drawn over here with overlapping normal distribution this is the target which was placed at 74 and this is the upper specification limit with 74.035 and this is 73.695 like that. So, that is the 965 sorry this is the values that is plotted over here and you will see performance observed not a single observation falling outside this one as per the real data and but if you superimpose this and convert into Z and Z USL and Z LSL like that and that some probability we are getting and that can be converted into PPM and this is the number that you get. So, how do you calculate CPK index over here? So, I am just taking you to the same example that is given over here and we will delete eliminate other things then you go to I am showing you again STAT quality tools and capability analysis normal capability analysis and what you do is that you ring 1 to ring 5 and select this one then I then I change the specification as 73.965 and this is 74.035 and then estimation over here R bar is the estimation that we will use other things remain same and options over here target value let us say is 74 we can just mention for observation like that and parts per million will be reported and capability index will be reported over here we will concentrate on that and click ok. What will happen is that you will get this information and this is the graph that will be reported over here and you see standard deviation is calculated this is the standard deviation that you observe over here and this is 0.0099 this is the within standard deviation that means R bar by D 2 and this is the CP index 1.17 and this is the CPK index. CPK is minimum of CPL and CPU that is the formulation we have and minimum is 1.13. So, MINITAB is reporting CPK value 1.13 what is as per the theories that is expected that is the theory that is expected ok. So, within performance is also given over here the actual process spread is represented by 6 standard deviation that MINITAB is expressing because MINITAB has used 6 because I have given 6 as the option to calculate this one. So, that is reported over here and we can take the second example. So, if one sided specification what will happen? So, this example is has a one sided specification. So, I will go to quality tools and then I will go to capability analysis normal over here and then I will give container this example up to this point and I will select those and in this case I will only mention 200 I will not mention the upper capability upper specification and then in estimation we are mentioning R bar formulation and then options over here the target value I am not giving within subgroup analysis we want to do and capability I want to report. So, if you click one sided specification what happens is that you do not get values of CP over here and what you get is CP lower CPL, CPU is not there. So, CPK will be just CPL over here ok. Similarly, if upper specification is given in that case here LSL was given as 200. So, that is why you are seeing this one, but if only upper specification is given it will calculate CPU and that will be the value of CPK basically. So, one sided specification CPK will be reported, but CP will not be reported like that. CPK is the index that we will consider over here. So, CPK is the index that we will consider over here ok. So, that is that is the way we are doing capability analysis. Another option you have in quality tools is that capability 6 pack. Here also there is a normal here we will get one more information that is control chart information also you can see control chart let us say rig 1, this is the 1 and I give specification over here 73.965 and this is 74.035 and then I do estimation same method over here I keep it as it is. So, subgroup size it is not 1. So, within capability analysis so I am doing that one and then what I do in options what I do is that I do not add targets over here because we do not need to calculate CPL. So, that we are not interested into. So, in this case K equals to 6 CP statistics I click ok and then I click ok over here test over here what I see one point going outside 3 sigma that is a condition in control chart that it is asking. So, we are doing that one either all 8 tests that is defined western electric we are not doing that. So, we are taking only one condition over here or other patterns we are not considering any point going beyond plus or minus 3 standard deviation that we are considering as abnormal scenario. So, I click ok and what will happen is that you will get this type of 6 pack many things you will get over here. So, in this case what you will get control charts x bar r chart is plotted over here everything is in control and based on which upper specification lower specification is given. So, within specification CP value 1.17 and CP K is 1.13 that was the value we have calculated earlier and this shows any abnormal day in the 25 subgroup size any observation that is very peculiar ok from the central line. So, nothing peculiar is observed over here. So, and in this case random samples we can say. So, these are there is no problem over here you will also find a normal probability plot that I told that we use for seeing whether the data set is normal or not. So, in this case what is observed from this data set we have we have not gone into that details over here, but we will discuss that in our next session in that and we need to see how to check normality assumptions and over here you will find a value which is mentioned as P and that P value we see to check whether the normality assumption is violated or not because CP and CP K index depends on whether the value whether the data set is normal or not and what we are seeing is that data follows normal over here and all the information in one go 6 pack means all the information in one go you are getting over here. So, you can use that one as a option to see whether x bar r is satisfactory then only we will go for CP CP K index and then we will see whether to improve the process or not to improve the process like that. So, what we will do is that we will stop here, we will start our next session from here and we will try to see some more information on sigma level and how to calculate that one. Thank you for listening to this lecture, we will return back in the next lecture with these topics to continue here.