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This lesson is about variable data control charts,

specifically about one type of variable data control chart.

Before we get to that particular kind of chart, generally speaking,

when we think about variable control charts,

we're talking about measurement data, something that can be measured,

so we're talking about characteristics of a product or

a process such as the weight of a product, the height, the length,

the viscosity of a particular liquid, the density, those kinds of things.

So it's something that can be measured.

To put it in simpler terms, it's where decimal points have a meaning.

So if we talk about 22.8 degrees or

we talk about 34.8 inches it has meaning, rather than when you're

talking about discreet distributions where there are no decimal points.

So, we're talking about continuous distributions here, right?

We're talking about measurement data.

The difference between attribute control charts and

variable control charts is that variable control charts, is they are used in pairs.

So you're always looking at the variability in some kind of

a measurement of range or measurement of standard deviation

in addition to looking at the variation in the mean.

So, they're always going to be in pairs so

that's why we call this the Xbar-R chart that we're going to look at next.

We don't use the Xbar chart without the R chart or

the R chart without the Xbar chart.

They always go in pairs.

They study process averages and they study the process variability.

And that variability can be measured in our case, we'll be looking at range.

It can be measured as standard deviation.

So there maybe different types of control charts,

using different aspects of variability that they're measuring.

Now, let's take an example here and

work through it to get a sense of the Xbar-R chart.

So we have Holly, who is a barista at a coffee house, and

she is known for the cold Americanos that she sells.

These are White americanos, these are whipped cream and

milk in them, and she prides herself in making these, right?

She builds each drink with a very elaborate process that involves

making the espresso in stainless steel cups that have been precooled,

transferring them into a glass cup, adding the cold

milk which is maintained in a refrigerator at a certain temperature.

Set at 34 degrees Fahrenheit, 0 degrees Celsius.

Right? Now Holly wants to assess the consistency

of temperature that she gets from making these Americanos.

So she wants to see whether this works out to be within a certain range.

So she wants to see how good this is in terms of

the variability that is in the process.

So, what she has done is she's collected data over the last five days

using samples of the first four Americanos.

So each of those five days, she's taken the first four Americanos that

were made between the 12 noon to 1 pm hour each day.

So what you have on this slide is the data that she has.

So in the rows, we have each of those samples.

So that's day 1, day 2, day 3, day 4, day 5.

And then what you have is, in terms of the temperature for

each Americano, four Americanos that are taken on each of those five days.

So what is the sample size here?

The sample size is four and the number of samples that she's taken is five.

Five samples of size four.

This is going to have some meaning when we do some calculations.

So it's worthwhile for you to make a note that there are five samples of size four.

So let's get into some basic calculations of this.

So what can we see in terms of the basic averages and

the ranges that we can get from this?

So we're moving towards a Xbar-R chart, a mean and range chart.

So the first thing we need to do is take each sample, take each role and

calculate its average.

You add them up, you divide by four, you get an average, right?

And then for the range for each of those rows, you wanna calculate,

take the maximum, subtract from that the minimum and you get a range.

So if you do that for all five samples, you can get the ranges and

the averages for all samples.

And what you have in the last row is the average of averages.

So it's a mean of means.

So the 35.08 is representing the mean of means for all of the samples.

And then you have a range average of 0.074.

Right?

So, that's what we get from simply looking at the averages and

the ranges, and what you're also seeing over here

is these averages are going to be used as a central line for both of those charts.

So, you've already got the central line for

the range chart as well as the average chart.

All right.

Now let's look at the computations for the upper and lower control limits.

Now if you're not comfortable with symbols,

with the Greek symbols, you might be intimidated by these.

But what these are basically saying is that, the upper control limit for

the range control chart is going to be based on the average range

that you already got, so what you're looking at,

sigma R divided by K, is simply the average of all the ranges.

So you take the average range and you multiply with something called the D4.

The lower control limit for the rain chart is based on a D3 number.

Multiply that by the average range and then when you look at the upper and

lower control limit formulas for the means chart, you're looking at the mean of

means and that's why you have the double bar on top of the X.

It's saying that it's the average of the five averages that you had taken

plus the A2 times R bar.

Lower control limit is x double bar minus A2 times R bar.

Now the question that you should be asking or

what you should be wondering about at this point is, what is this D3, D4, and A2?

What are these things and where do they come from?

So, where they come from is this chart that we can use to pick out these values.

So what is this chart?

This chart is taking the different sample sizes that you might use, and giving you

the different A2, D3, and D4 values that you would plug in into those formulas.

Now where are the numbers coming from?

They're basically representing the idea of three standard deviations.

So because we have a very small sample size,

its not appropriate for us to use standard deviations.

We are using the idea of three standard deviations by

substituting with these multipliers.

So the A2, D3 and D4 are multipliers that help us replicate the idea of plus or

minus 3 standard deviations.

So, that one that we are going to use here is based on our sample size of,

now you may recall that I said earlier, we have five samples of size four.

So, we go to the row that says, sample size of four and

it tells us 0.729 is the A2 value that we need to use and

then 0 and 2.282 are the D3 and D4 values.

So we're simply gonna take these and plug it into the formulas.

The center line for the range chart is based on the mean of the ranges.

We already got that earlier as 0.074 from that table that we had.

The upper control limit is going to take that 0.074, multiply it by

the 2.282 mutliplier that you saw on the chart on the previous slide.

So upper control is 0.1689.

Lower control limit based on a multiplier of 0 is going to be 0.

Right?

So we get the upper and lower control limits for the R charts,

similar calculations for the x chart, the x bar chart.

Center line is based on means.

Upper control limit is mean of means plus the multiplier 0.73.

Multiplier in this case is the A2 value and for the lower control limit,

you're using the same multiplier but you're subtracting in this case.

So you have mean minus 0.73 times the range.

Now, what you've noticed over what you should have noticed over here,

is that between these two charts, between the range chart and the x chart,

in the x bar chart, between the r bar chart and the x bar chart, we are using

the range to come up with the upper and lower control limits for the x bar chart.

Right? So this seems kinda strange that we're

using something from a different chart to control the upper and

lower control limits.

The reason I bring this up is because it's important for the range to be

in statistical control, if you are going to use that range to compute the X chart.

In other words, you need both of them to be in statistical control

to call a process as being in statistical control or

to come up with the inherent capability of the process.

You need both of them to be within the statistical control limits, right?

All right, so let's take a look at the interpretations

of the chart by plotting the points on each of these charts.

So once again, like you had earlier for

the other kinds of charts, here, for the Xbar-R Chart,

you have the points plotted on the chart of upper and lower control limits.

What do you see here?

You see a nice level of consistency.

Right? Now once again we've used,

as we talked about in the case of the proportion chart and

the count chart that we looked at earlier,

we've used a very small number of samples to come up with these values.

So, if you were to do this problem in reality

you would want to get a larger sample and use that.

What I'm talking about is the number of samples.

Your sample size may remain small, but you definitely want a larger number of samples

to come up with an upper and lower control limit.

Five samples is not going to be enough.

All right, taking these results for

what they are, let's take a look at what this implies.

What this is showing us that Holly's process is pretty consistent.

Right? She's

giving a pretty consistent temperature.

The maximum that it varies is 0.1688 degrees Fahrenheit.

The maximum of that range chart was 8.1688 degrees Fahrenheit.

Which is pretty good.

The range is pretty small.

It's between 0 and 0.1688, right?

The temperatures for the actual IC called Milky Americanos

is between 35.0356 and 35.1434 degrees Fahrenheit.

Again a very small range of temperature that you are getting from this so,

it seems to be a pretty tightly controlled process, she's able to achieve

that consistency in the coffee that she is serving her customers.

Now the question that you have not addressed, by looking at

whether the process is a statistical process control and even focusing on

the inherent capability of the process, is what is the customer's expectation, right?

We don't know how this temperature compares to what the customer expects.

Whether the customer is going to be happy with this particular temperature or

not, that's something that you don't know from doing a statistical process control

analysis.

So, keep a note of that.

4-2.5 Variable Data Control Charts

Operations Management

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