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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 type of chart, generally speaking,

when we think of variable control chart, we are talking about measurement data.

Something that can be measured.

So, we are talking about characteristics of a product, or

a process such as a 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.

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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 at 34.8 inches.

It has meaning, rather than when you're talking about discrete distributions

where there are no decimal points, so

we're talking about continuous distributions here.

We're talking about measuring data.

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The difference with between attribute control charts and

variable control charts is that variable control charts, 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 X-bar, R chart that we are going to look at next.

We don't used the X-bar chart without the R chart or

the R chart without the X-bar chart.

They always go in pairs.

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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 may be different types of control charts using different aspects

of variability that they're measuring.

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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's a barista at a coffee house and

she is known for her Cold Americanos that she sells.

These are White Americanos, these are with cream and

milk in them and she prides herself on making these.

She builds each drink with a very elaborate process that involves

making the espresso in stainless steal cups that have been pre-cooled,

transferring them in to a glass cut,

adding the cold milk, which is maintained, in a refrigerator set

at a certain temperature at 34 degrees Fahrenheit, 0 degrees Celsius.

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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, in

terms of the variability that is in the process.

So, what she has done is she has 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 p.m hour each day.

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

In the rows, we have each of those samples, that's day one,

day two, day three, day four, day five.

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

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 has 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 of that, that there are five samples of size four.

Right, 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 X bar R chart, a mean and range chart, so the first thing

that we need to do is take each sample, take each row and calculate it's average.

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

And then, for the range for each of those rows, you want to 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 .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

center lines for both of those charts.

So, you've already go the center line for

the rain chart as well as the average chart.

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All right.

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

Now, if you're not comfortable with Greek symbols,

you might be intimidated by these but what these are basically saying is that,

the upper control limit for the arranged 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.

You take the average range, and you multiply with something called a D4.

The lower control limit for the range chart is based on a D3

number multiplied by the D 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 have taken, plus the A2 times R bar.

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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, to those formulas.

Now, where are these numbers coming from?

They're basically representing the idea of 3 standard deviations, so,

because we have a very small sample size,

it's 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 three standard deviations.

So, the 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 .074 multiplied by 2.28 to multiplier

that you saw on the chart on the previous slide.

So, upper control limit is 0.1689,

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

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Right. So, we get the upper and

lower control limits for the R chart.

Similar calculations for the X chart, the X bar chart.

Center line is based on mean of means.

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

The multiplier in this case is A2 value.

And for the lower control limit, you are using the same multiplier.

But you are subtracting, in this case.

So, you have mean minus 0.73 times the range.

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Now, what you've noticed over here.

Here or what you should have noticed here is that between these two charts,

between the range chart and the X chart in the X bar chart.

In 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 compute the upper and

lower control limits.

But 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 X bar R chart,

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

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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 a upper and lower control limit.

Five samples are not going to be enough.

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All right, taking those results for

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

Well, 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 a 1.68 degrees Fahrenheit, which is pretty good.

The range is pretty small, it's between 0 and 0.1688, the temperatures for

the actual icy cold Milky Americanos is between 35.0356 and

35.1434 degrees Fahrenheit.

So again,

a very small range of temperatures that you're 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's 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 customers expectation?

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

Whether the customer is gonna 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.