We're looking at chapter seventeen,
slide seven,
and looking at X bar and R charts.
An experienced operator
has operated a machine over several days being careful to keep
assignable causes of variation absent from the process.
Twenty-five random samples of 'n' equals six were taken during this time,
and the sum of your X-bars
was seventy five point one eight six (75.186) centimeters, and the sum of your R's
were four point two seven zero (4.270) centimeters.
Compute control limits for your X-bar and R charts
that are three standard deviations from their expected values.
Assume the process mean was correctly set.
Let's take a look at the information here.
In this, we see,
'n' is equal to six.
That means you have six observations in each sample set.
How many sample sets were taken?
Twenty five.
Twenty-five sample sets were taken.
We found that the mean of
the samples sets,
add it all up, the twenty five sample sets each had a different mean from the six
observations within each set, but if we add them all up it was seventy five
point one eight six (75.186).
If we add the ranges of each one of those sample sets, it's four point two
seven zero (4.270) cm.
So, looking at that information,
which I've written here,
we have our sum of our X-bars, and the centimeters that that is, the
sum of our 'R's'
We have our 'n,' but we have this twenty-five. What is a twenty five?
Twenty-five is actually
'K'
K is the
number of samples sets, where 'n' is the number within each
sample set.
So twenty five samples sets of six observations in each set were taken.
We want
to compute here
the upper and lower uh...
control limits
which is
upper control limit and lower control limit
for the X-bar chart,
and then for the R chart, let's look at the X-bar chart first.
This
is going to be equal to what's called X double bar,
or the grand
mean, or the mean of means,
plus
A-2-R-bar
R bar is actually our mean
of our ranges and we'll compute that in just a minute.
The A2 is a number we find on a chart that we'll go over in just
a minute.
The lower control limit here, is the grand mean, or the mean of means, or the
X double bar.
minus
A 2 R
bar.
So, we have this mean of means
in this formula, and we have a mean of ranges in the formula.
How do we find that out?
Well,
X double bar is actually equal to
the sum of our X bars
divided by the number of samples sets that were taken.
So that's K.
So, here, our X double bar
is going to be
seventy five point one eight six (75.186)
divided by the number of sample sets that were taken, K, which is twenty five (25),
and if I did my math right I got a three point zero zero seven
four four (3.0744).
R bar,
which is our range, our
mean of our ranges, is actually the sum of the ranges of each one of those, from
those sample sets,
again divided by the number of sample sets that were taken, or K.
Ao, that is four point two seven zero (4.270) divided by twenty five (25),
and, again, if I did my math right, that is zero point one seven
zero eight (0.1708).
So, here are the numbers that we are looking at.
We need to find for, first, our
X-bar chart,
the upper and lower control limits.
So, using these numbers,
I'm going to again
copy down what we have for the information here,
we have the sum our X-bars,
being
seventy five point one eighty six (75.186)
the sum of our R's being four point two seven zero (4.270),
our 'n' is equal to six, our K is equal
to
twenty five,
our
grand mean,
or mean of means,
or X double bar,
we figured out to be three point zero zero seven four four (3.00744)
and our
mean of our ranges was is zero point one seven zero eight (0.1708).
Looking
at what we have here, we want to start with the upper control limit for the
X bar chart. Now,
we said that it was
the grand mean
plus
A 2 R bar, well we have the grand mean,
it's three point zero zero seven four four (3.00744) plus some number
times
point one seven oh eight (0.1708), our R bar. But what is that number?
Well on page seven forty (740),
of your book
is
appendix five
and table two is 'factors for determining
from R the three similar control limits for X bar and R charts.
In other words, this is a chart you need to take a look at.
We said
our 'n' was six and if you look at this first column,
it's 'n',
the number of observations in subgroups.
We're going to use
that number all the way across, its going to determine for us...
what is it going to determine... well...
I love this chart because it actually gives you all the answers,
A2, the factor for the X bar chart,
which is point four eight (0.48), and in case you forget
the formula,
this chart
is nice enough...
and it's not for some reason letting me scroll down...
on it... but it's nice enough if you look look down at the bottom of the
chart, you'll find
the different uh...
formulas actually on the chart.
So our A2
for six...
for six n's
is point four eight (0.48).
So, for here we said my A2 was point
four eight (0.48),
so, now, I can just do the math,
and if I take
three point zero zero seven four four (3.00744),
I add it to point four eight (0.48),
times point one seven zero eight (0.1708),
that multiplication is point zero eight one nine eight four (0.081984),
it's going to give me my upper control limit of three point zero eight nine
four two four (3.089424).
My lower control limit for X bar
is the grand mean, or the mean of means, or the X double bar minus
A2
R bar.
We're actually using the same numbers, three point zero zero seven four (3.00744)
but subtracting
point fort eight (0.48)
times point one seven zero eight (0.1708)
which is going to give us
two point nine two five four five six (2.925456).
This is my upper and lower control limits.
What does this mean?
Literally we're looking at three standard deviations,
and it's literally
just that normal curve turned on its side,
where the dotted line now
is our grand mean
our upper control limit is going to be up here,
which is
equal to
three point
zero
eight nine four two four (3.089424)
and our lower control limit for X bar is down here,
two point nine two five four five six (2.925456).
So, I should have three standard deviations on both sides of that grand
means,
from the different sample sets that I have taken.
How many sample sets? That was our K. That was twenty five (25).
Using these same numbers, we want to find
our upper control limit
for R chart and our lower control limit for R chart
The upper control lmit
is found by D 4 R bar. Now, if we go back to
that chart that we looked at,
remember we were looking at
six n... excuse me...
where 'n' equals six,
and we're looking at D 4.
I love this chart because, look, upper contorl limit
underneath what it says, factor for R chart,
upper control limit D 4.
Here, it is two point zero (2.0).
So, for our upper control limit,
D 4, we're going to have a two
point zero zero (2.00)
times our R bar
which remember we said was point one seven zero eight (0.1708),
and if we do the math
on this
it should equal
point three
four one six (0.3416).
Our lower
control limit
is found by
D 3 R bar,
which if you look at that chart again, I just love this chart,
again factors for R chart, lower control limit D 3, we just have to
find what our 'n' is
six which means our D 3,
for finding our lower control limit for the R chart is zero point zero zero (0.00).
I'm real good, I can take
zero
point zero zero (0.00) and multiply it times point one seven zero eight (0.1708) in my head
because that's zero.
If you notice
on this chart
you had that lower control limit being zero.
Remember R is range, we're looking at ranges here.
Can a range ever be less then zero?
No, because it's literally the number of clicks on a number line, is all it is,
from one point to the next.
It can never be less than zero.
So, you can take that in and do what I call my own personal dummy
checks,
to make sure that I'm doing it right,
and literally this right here again,
we're looking at
what looks like the roadway,
but we're looking at three standard
deviations on each side
to get our upper control limit for our R chart and our lower control limit for our
R chart.
And literally zero is here,
and point three
four one six, yeah, this right here has a very small
range,
the standard deviations are very small.
This could be a medication,
You know when we're cutting
lengths of wood
the deviations can be a little bit bigger
but we're looking
at something like... pediatric medication, you want
your deviations to be very small,
and that ends chapter seventeen's
problem.