0:14

We're going to look at a particular type of control chart here, and

this is based on attribute data, more specifically proportion kind of data.

So proportion defective.

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

see how what we can do with it.

0:31

So we have a car part supplier

who is testing weekly samples of 15 subassemblies.

So the sample size is 15.

These subassemblies are taken from the assembly line and

these are taken over ten weeks.

The inspector records the number of defective subassemblies.

So what is the inspector doing?

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

Something that is defective and something that is not defective.

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

It can be one or the other.

1:06

The inspector does not care about the extent of defects.

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

a much higher degree of defect is treated the same.

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

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

this is the kind of chart that would be appropriate.

1:32

Coming back to the data, so that over ten weeks getting samples of 15 each,

the data that we have is described over here.

So week 1 through week 10, each week 15 doors are taken, 15 subassemblies

of doors are taken and the number of defective ones are being recorded.

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

subassemblies and so these are auto-factoidal off 10 times 15, 150.

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

2:16

So first thing we want to get an average proportion

of all the proportions we saw over here.

We had 10 samples of each of size 15.

Each of the number of defects in those samples

can be converted into a proportion.

Each one divide by 15 is going to give us a proportion.

And then we're also going to get for

purposes of constructing the control chart, the standard deviation.

2:42

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

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

The formula that you see over here is basically

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

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

Standard deviation is computed as whatever proportion you've got

as the average proportion, multiply that by 1 minus the average proportion and

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

The sample size here, this is something you need to

pay close attention to, the sample size is 15.

It's 10 samples of size 15.

So the sample size here is a constant 15.

That's what you'll be using to compute the standard deviation here.

3:32

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

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

you can do it two ways.

You can either take each of the proportions, so

the first sample had three defective subassemblies out of 15 giving us 0.2.

Second one had one out of 15, giving us 0.067 or

you can simply take the total 12 defective

subassemblies over the 10 samples of size 15.

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

4:09

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

the center line.

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

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

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

If you calculate the center line for this,

it's based on 12 divided by 150 which is 0.08 and

the standard deviation is based on 0.08 times 1 minus 0.08 divide by 15,

which is a sample size all taken under square root.

So the standard deviation works out to 0.07.

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

minus three standard deviations.

4:57

Now, the point that you might wanna note here,

is that the lower control limit actually turned out based on the computation,

strictly based on the computation.

It turned out to be negative 0.13.

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

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

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

What they should also tell you, something that you might wanna note

is that this chart is not gonna be symmetrical.

You're keeping the mean at 0.8, the upper control limit is at 0.29

based on these calculations and lower control limit is now going to be 0.

So it's gonna be an asymmetrical kind of control chart, and

that's what we can see in terms of the chart.

And here I have the chart generated based from Minitab Software shown to you here.

5:59

What do we see here?

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

outside of the control limit.

This is the 8 defective subassemblies

out of 15 that was found in that one particular sample.

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

8 had 5 defective subassemblies which gave us 0.333.

So staying on this slide,

you can see that there's going to be a problem, because 0.333 is

above the upper control limit which is also what is shown to us in this picture.

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

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

What do we do next?

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

Well, it's going to depend on whether you're trying to calibrate the chart at

this stage or not.

And here it's clear that we're trying to calibrate the control chart.

We're using data from the process to come up with the upper and

lower control limits.

So what do you need to do?

You need to figure out what happened at that week eight.

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

Can we figure out what the reason was for that?

If it is reasonably clear, if it is clearly a special cause variation,

something that should be impacting the process on a day to day basis

then we can simply delete that sample and we can recompute the control limits.

So we throw out that sample, and we recompute the control limits based on

not having that sample in our calculations.

If that's not the case, if you cannot really eliminate that particular

observation, that particular sample based on a cause that's clear

then you have to go back to the drawing board and recompute the control limits.

7:48

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

out the reason for week eight proportion being outside of the control limit.

So eliminate that point and then you go in, and

you figure out the new control limits.

8:04

So here you see that there's no week eight being represented here.

You're going from seven to nine and ten, and because you've thrown out week eight,

you've got the same data that you had earlier.

You come up with a new mean proportion which is going to be based on now

what was it, we had 12 defects earlier we took out the one that had 5,

so we had 7 defects out of a total of now 135.

So 7 out of 135 gives us a mean proportion of 0.052 and

then you get the standard deviation based on that and then you get the upper and

the lower control then that's based on that.

8:44

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

each of the samples as well as the upper and

lower control limits that we've already computed.

You can see that there's going to be no problem in terms of

points outside the control limit.

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

9:10

Plotting it in terms of the chart, you can see the same result.

You can see that all the point are within the control limits.

So what you can say from this, what you can infer from this is that the current

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

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

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

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

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

Based on what your context is, it may not be appropriate for

you to use just ten samples that we used over here,

in order to compute the upper and lower control limits.

So even moving beyond this particular problem,

you might want to reflect on the fact that was ten samples enough for

you to come up with the upper and lower control limit?

And that's going to be context dependent in the sense that,

what are the different types of things you want to cover in the number of samples?

If it's many shifts in the day, many samples during many shifts in the day,

many days of the week, then obviously you have to have coverage for

those sorts of things in your data collection.

So then I would have samples that are collected over a couple of weeks to make

sure that every day of the week is represented at least twice.

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

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

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

separate different shifts.

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

in addition to the mechanics of these control charts.

11:08

So finally, when and how do you use the control chart,

the P chart, the proportion control chart?

You use it to compute control limits for attribute kind of data.

It's a discrete distribution.

It's data that's dealing with conforming and nonconforming items.

All it's telling you is, whether there was a product that was defective or

non-defective or it was defective or as expected.

11:37

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

the process with an attribute kind of control chart with a P chart.

What do I mean by that?

We used sample sizes of 15.

It was a constant sample size across.

It's possible for you to construct a P chart based on having different

sample sizes for each of those samples.

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

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

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

12:17

Recall that you're only dealing with two possible outcomes.

So this is based on, if you're familiar with the binomial distribution,

you're talking about a binary decision.

A binomial distribution is saying something is either good or bad.

It's a 01 kind of decision.

12:35

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

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

So you want to think about that.

And then in terms of their frequency of how you collect data whether it should be

every hour or once every day.

That's something that you also have to think about in terms

of designing this system of assessing your process over a longer period of time.