0:14

We're gonna look at a particular type of control chart here,

Â and this is based on attribute data or more specifically proportion kind of data.

Â So proportion defective.

Â So let's take a look at it from the point of view of an example and

Â see what we can do with it.

Â So we have a car parts supplier who is testing weekly samples of

Â 15 sub assemblies so the sample size is 15.

Â These subassemblies are taken from the assembly line and

Â these are taken over 10 weeks.

Â The inspector records the number of defects of subassemblies.

Â So what is the inspector doing?

Â Looks at these 15 subassemblies every week and puts them in two different categories.

Â Something that is defective and something that is not defective.

Â So you have a truck door that's defective, a truck door that's not defective.

Â It can be one or the other.

Â The inspector does not care about the extent of defects.

Â So a little scratch on a door versus a door that is has

Â a much higher degree of defect, it's really the same.

Â They're both called defective, regardless of the extent of defect.

Â So if that's all that you care about is looking at defective versus non-defective,

Â this is the kind of chart that would be appropriate.

Â 1:32

Coming back to the data, so there are over 10 weeks of getting samples of 15 each.

Â The data that we have is described over here.

Â So week one through week ten.

Â Each week 15 doors are taken, 15 subassemblies of doors are taken and

Â the number of defective ones are being recorded.

Â You can total these up and you'll see that there are total of 12 defective

Â subassemblies and so these are out of a total of 10 times 15, 150.

Â So that's the kind of data that we have to start off with.

Â So we're moving towards calibrating a control chart.

Â So let's do the numbers here and see what we can find.

Â 2:16

So first thing, we want to get an average proportion

Â of all of the proportions that we saw over here.

Â We had 10 samples of each sized 15.

Â Each of the number of defects in those samples can be converted into

Â a proportion.

Â Each one divided by 15 is going to give us proportion, and then we're also going

Â to get for purposes of constructing the control chart, the standard deviation.

Â 2:42

The reason we need to get that is because the upper and

Â lower control limits are based on plus or minus three standard deviation.

Â The formula that you see over here is basically

Â telling us how do you get the upper and lower control limits.

Â So it's the mean plus or minus three times the standard deviation.

Â Standard deviation is computed as whatever proportion you got as the average

Â proportion, multiply that by one minus the average proportion and

Â divide that by the sample size all under square root.

Â The sample size here, this is something that you need to pay

Â close attention to, the sample size is 15.

Â 3:22

It's 10 samples of size 15.

Â So the sample size here is a constant 15,

Â that's what you'll be using to compute a standard deviation here.

Â So let's go through some of these computations here.

Â So in order to get the average proportion over those ten samples

Â you can do it two ways.

Â You can either take each of the proportions, so

Â the first sample had 3 defective subassemblies out of 15, giving us 0.2.

Â The second one had one out of 15, giving us 0.067 or

Â you can simply take the total 12 defective

Â subassemblies over the 10 samples of size 15.

Â So your going to get 12 divided by 150 and that will give you the average.

Â 4:09

So either way you're going to get an average that you'll use for

Â the center line.

Â And if you were to take the other method, which is taking each of the proportions

Â and averaging them out, you would get the same thing.

Â So here you have each of the proportions shown to you for each of the samples.

Â If you calculate the center line for this, it's based on 12 divided by 150, which

Â 4:37

is 0.08 and the standard deviation is based on 0.08 times 1

Â minus 0.08 divided by 15 which is a sample size all taken under square root.

Â So the standard deviation works out to 0.07.

Â The upper and lower control limits are simply commuted based on plus or

Â minus 3 standard deviations.

Â 4:57

Now, the point that you might wanna note here is that the lower

Â control limit actually turned out based on the computations,

Â strictly based on the computation.

Â It turned out to be negative 0.13.

Â Now as you know, negative proportions are not going to make sense.

Â There are no negative proportions, so we're going to bump that up to a 0.

Â So the lower control limit is going to get bumped up to a 0.

Â What this should also tell you and something that you might wanna note

Â is that this chart is not gonna be symmetrical.

Â You're keeping the mean at 0.08 the upper control limit is at

Â 0.29 based on these calculations.

Â The lower control limit is now going to be 0.

Â So it's gonna be an a symmetrical kind of control chart.

Â And that's what we can see in terms of the chart and

Â here I have the chart generated based on Minitab Software generated here.

Â 5:59

What do we see here?

Â We see that there is a point that's marked in red that's

Â outside of the control limit.

Â This is the 8 defective subassemblies out of

Â 15 that was found in that one particular sample.

Â So if you go back to the data over here, you can see that sample number 8

Â had 5 defective subassemblies which gave us 0.333.

Â So staying on this slide, you can see that there is going to be a problem

Â because 0.333 is above the upper control limit,

Â which is also what is shown to us in this picture.

Â That there's a point outside of the upper control limit.

Â The question that you should be asking is so what?

Â What do we do next?

Â We found a point that's outside of the control limit.

Â Well it's going to depend on whether your trying to calibrate the chart at

Â this stage or

Â not, and here it's clear that were trying to calibrate the control chart.

Â Were using data from the process to come up with the upper and

Â lower control limits.

Â So what do you need to do?

Â You need to figure out what happened at that week eight,

Â there was a proportion defective that was higher than the upper control limit.

Â Can we figure out what the reason was for that?

Â If it is reasonably clear.

Â If it is clearly a special cause variation,

Â something that should not be impacting the process on a day to day basis,

Â then we can simply delete that sample, and we can re-compute the control limits.

Â So we throw out that sample and you re-compute the control limits based on

Â not having that sample in our calculations.

Â If that's not the case,

Â if you can not really eliminate that particular observation.

Â That particular sample based on a cause that's clear, then

Â you you have to go back to the drawing board and re-compute the control limits.

Â 7:48

So let's take the easy route here and let's say that we could figure

Â out the reason for weaker proportion being outside the control limits.

Â So eliminate that point and then you go in and you figure out the new control limits.

Â So here you see that there is no week eight being represented here,

Â you're going from seven to nine and ten and because

Â you've thrown off week eight you've got the same data that you had earlier.

Â You come up with a new mean proportion,

Â which is going to be based on now a what was it, we had 12 defects earlier.

Â We took out the one that had 5, so we have 7 defects out of a total of now 135.

Â So 7 out of 135 gives us a mean proportion of 0.052,

Â and then you get the standard deviation based on that.

Â And then you get the upper and lower control limits based on that.

Â 8:44

Now, what you can see from here based on the proportions that you have for

Â each of the samples as well as the upper and

Â lower control limits that we've already computed, you can see that there's gonna

Â be no problem in terms of points outside the control limit.

Â All the points are going to be within the control limits based on the fact that

Â 9:10

Putting it, plotting it in terms of the chart, you can see the same result,

Â you can see that.

Â All the points are within the control limits, so what you can say from this,

Â what you can infer from this is that the current process

Â 9:25

is expected to have between a 0 to 22% defect rate.

Â We got the 22% defect rate based on the upper control limit.

Â So given the current technology, given the way the process is designed,

Â given the kinds of training that you have for your people,

Â you are expecting a 0 to 22% defect rate from this particular process.

Â 9:48

Based on what your context is,

Â it may not be appropriate for you to use just 10 samples

Â that we used over here in order to compute the upper and lower controlled elements.

Â So even moving beyond the particular problem, you might wanna reflect on

Â the fact that, was 10 samples enough for you to come up with upper and lower limit?

Â And that's going to be contact dependent in the sense that

Â what are the different type of things you want to cover in the number of samples.

Â If it's many samples during many ships

Â in the day many days of the week then obviously you have to have coverage for

Â those sorts of things in your data collection.

Â So then I would have samples that are collected over a couple

Â of weeks to make sure that every day of the week is represented at least twice.

Â And then if there are multiple shifts there being represented in the sampling,

Â that is being used to come up with the inherent capability of the process.

Â You also want to think about whether you want to separate different days,

Â separate different shifts.

Â And those are the kinds of managerial decisions that you will need to make

Â in addition to the mechanics of computing these control charts.

Â 11:08

So finally, when and how do you use the p chart?

Â The proportion control chart, you use it to compute control limits for

Â attribute kind of data.

Â It's a discrete distribution.

Â It's data that's dealing with conforming and non conforming items.

Â All its telling you is whether there was a product that was defective or

Â non defective or defective or as expective.

Â You may use different subgroup sizes in order to come up with a process with

Â an attribute kind of control chart with a p chart.

Â 11:47

What do I mean by that?

Â Well, we used sample sizes of 15.

Â It was a constant sample size across.

Â It's possible for you to construct a p chart based on

Â having different sample sizes for each of those samples.

Â We simply haven't gone through those kinds of calculations, but

Â you can find those in software, you can find those in different

Â sources to be able to compute control charts for different sizes of samples.

Â 12:17

We call that you're only dealing with two possible outcomes so this is based on,

Â if you're familiar with the binary distribution, the binomial distribution,

Â rather, you're talking about a binary decision.

Â A binomial distribution is saying something is either good or bad,

Â it's a zero one kind of decision.

Â 12:35

You want the subgroup size to be large enough to be able to capture defects.

Â If you're finding a lot of zeros then your subgroup size is not large enough.

Â So you want to think about that.

Â And then in terms of the frequency of how you collect data,

Â whether it should be every hour or once every day,

Â that's something that you also have to think about in terms of designing

Â the system of assessing your process over a longer period of time.

Â