0:14
This lesson is about control charts for count kind of data.
So count kind of data is different from proportion kind of data
in the sense that when you're looking at proportion defective,
you're only looking at whether a product is defective or non-defective.
You're getting a yes-no answer.
You're getting a binary answer and it's based on a binomial distribution.
Whereas what we're doing here is we're looking at count data, so
we're looking at the count of defects.
We're looking at a sample and seeing how many defects did we find in that data?
So a single product or a single unit of a product can have multiple defects,
and that's going to have meaning in count kind of data,
while it did not have any meaning in proportion kind of data.
So count kind of data is still dealing with attribute kind of measurement,
with discrete distribution, so we're getting a count.
You cannot have 2.5 defects, for example, so it can be either two defects or
three defects.
So it's still a discrete distribution that we're dealing with.
So let's take a look at an example here to get a sense of getting the upper and
lower control limits and then using a control chart based on count data.
So we have a textile manufacturer who wants to determine the quality level
of their weaving process.
They take daily samples of five linear yards of material and
count the number of flaws.
So each of the samples is a sample of five linear yards.
They've decided to take one sample per day, and
they're going to count the number of flaws.
So if you think about this for a minute, in a daily sample of five linear yards,
what is the upper limit of the number of defects that you can find?
And we treat that as infinity.
We say that the upper limit for
the number of defects that you can find is going to be infinity.
Because we're simply not defining what a defect is.
It could be many different things, and we're going to say that you can find many,
many different defects in every sample of five linear yards.
Why is this important?
It's because we are going to use what is known as the Poisson
distribution as the underlying distribution for this kind of data.
So the count data is coming from a Poisson distribution, and
that has meaning in terms of computing the upper and lower control limits for
this type of control chart.
So moving on with the problem, 5 linear yards of data every day,
collect the data for 20 days, so what we'll have
is the number of defects that you can see over a 20-day period.
So here's the data that we have.
We can see that we have 20 days' worth.
And how do we use this to compute the control limits for a control chart?
So let's take a look at the numbers and the calculations for that.
3:12
So what you have here is for the C chart, the center line is simply going
to be the mean of all the defects that you found in all of the samples.
So you're going to take an average, and if you remember how many samples we had,
we had 20 samples so it's gonna be total number of defects divided by 20 is going
to give us our center line.
And the nice part about this kind of distribution is that
the standard deviation is simply the square root of the mean.
So you have the center line that's being C bar,
that's the mean of the count of defects that you've got,
and the center deviation is simply the square root of that.
So the upper and lower control limits are going to be based on the mean plus or
minus 3 times the standard deviation which is the square root of the mean.
So in that sense the calculations are going to be much more easier here
when we are looking at the standard deviation.
The center line for the control chart works out to be 10.2
based on 204 defects in 20 samples.
And the upper control limit is going to be based on the square root of 10.2 and
you take plus or minus 3 times the standard deviation.
And that gives us our upper and lower control limits.
So we take this and we can compare it with the number
of defects that we saw in each of the samples and
if it is below 1 or above 20.
In fact, if it's going to be above twenty, we are going to call it out of control.
So we can see that right from here without even plotting on the control chart
in that it needs to be between two numbers, right?
5:02
Now one point of caution before we move on to look at the chart
is that although we did not get a negative value for the lower control limit here,
based on the calculations, you might get a negative value for a different problem.
As we cannot have a negative count of defects, it's not going to make any sense.
We'll do the same thing that we did in the case of the P chart,
in the case of the proportions, we'll bump it up to a 0.
So we change it to 0 if we do find the lower control
limit to be a negative number based on the calculations.
Here that's not the case, so we don't need to make any change.
We have a nice symmetrical control chart here.
The upper and lower control limit are equidistant from the center line, and
then when we look at the plot, everything seems to be doing fine.
So what can we say from this control chart?
Based on the idea that we were calibrating the upper and lower control limits
based on data that we got from the process based on these 20 samples,
we can say that the current process will produce between 1 and 19 defects, right?
So our inherent capability of the process
is to produce as many as 19 defects the way it's running right now, right?
Less than 1 would be something that's extraordinary, and
greater than 19 would be something that's extraordinary.
That's what we said earlier.
That's exactly what we're saying here.
6:28
If we do get numbers that are above or
below, we need to figure out what happened.
It could be root causes of something that has gone wrong.
Or if it's on the lower side,
it's how did we manage to get points that were lower than the lower control limit?
Because we're talking about count of defects,
lower numbers are going to be better.
7:12
The condition here is that each sample has to be of an equal size.
So in this particular example, we talked about five linear yards of cloth.
If you're talking about a sample that you take of any kind of a product.
Let's say you're looking at cell phones, if you're taking a sample of
ten cell phones and you're counting the number of defects in the ten cell phones,
the sample size has to be constant when you're counting the number of defects.
Just simply to give them the same opportunity for having defects,
to be fair in that sense to the sample.
Gopesh AnandAssociate Professor of Business Administration
