Module 4. Introduction to Statistical Process Control

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Transcript Module 4. Introduction to Statistical Process Control

Introduction to Statistical
Process Control
Module 4
History of Statistical
Process Control
• Quality Control in Industry
– Shewhart and Bell Telephones
• Deming & Japan after WWII
• Use in Health Care & Public Health
The Run Chart
The Count
Cups of Coffee
Day
Sunday
12
Monday
2
Tuesday
4
Wednesday
3
Thursday
5
Friday
4
Saturday
2
The Mean
The mean of 4, 7, 8 , and 2 is equal to:
4+7+8+2
4
The Median-Odd Numbers
• = the middle value in an ordered series of
numbers.
• To take the median of 1, 7, 3, 10, 19, 4, 8
• Order these numbers: 1,3,4,7,8,10,19
• The median is zth number up the series
where z=(k+1)/2 and k=number of
numbers.
• What is the median in this case?
The Median-Even k
• Order the numbers, i.e., 1,7,10,14,15, 17.
• Find the middle values, i.e., 10 and 14.
• Take the average between these two
values.
• What is the median?
The Proportion
You have these 10 values representing 10 people:
0,0,0,1,0,1,0,0,1,0.
Zero means person did not get sick.
One means person did get sick.
What is the mean of these 10 values?
(0+0+0+1+0+1+0+0+1+0)/10 = .333
Proportion= n/N, where n=number of people who got sick and
N=total number of people. n=numerator, N=denominator.
What is a population?
•
•
•
•
•
•
•
•
A group of people?
A group of people over time?
Hospital visits?
Motor vehicle crashes?
Ambulance Calls?
Vehicle-Miles?
X-Rays Read?
Other?
Populations take on
Distributions
In simple statistical process control, we deal
with 4 distributions.
From central tendency to variation.
The Normal Distribution
How do we describe variation about the red
line in the normal curve?
• In other words, how fat is that distribution?
• How about the average difference between each
observation and the mean?
– Oops, can’t add those differences, some are positive
and some are negative.
• How about adding up the absolute values of
those differences?
– Bad statistical properties.
• How about the average squared difference?
– Now we are talking! 
Population Variance
N
Population Variance =
1
N
∑ (x - µ)
i
i=1
The average squared deviation!
2
Population Standard Deviation
Population Standard
Deviation =
N
√
1
N
∑ (x - µ)
i
2
i=1
The square root of the average squared deviation!
Standard Deviation has
Nice Properties!
025
.025
It’s Time to Dance
How do I estimate the standard deviation of the
means of repeated samples?
• Estimate the standard deviation of the
population with your sample using the
sample standard deviation.
• Estimate the standard deviation of the
mean of repeated samples by calculating
the standard error.
Sample Standard Deviation
N
S
=
√
1
N-1
∑ (x - x)
i
2
i=1
How is this different from the Population Standard
Deviation?
Standard Error
s
SE
=
√
n
How is this different from the Sample Standard
Deviation?
Z-Score for Distribution of Sample
Means
Z=
x- μ
SE
X = mean observed in your sample
μ = is the population mean you believe in.
Z = number of standard errors x is away from μ,
You can convert any group of numbers to z-scores.
If we kept dancing for hundreds of times
Here is the distribution of our sample means
(standardized)
Wait a minute!
• When you do a survey, you only have one
sample, not hundreds of repeated
samples.
• How confident can you be that the mean
of your one sample represents the mean
of the population?
• If you think reality is a normal distribution
with mean y and standard deviation s, how
likely is your observed mean of x?
Welcome to
• Confidence Intervals
• P-Values
• Let’s focus on p-values for now.
Here is our distribution of sample means—WE BELIEVE
Area under curve
is probability and
it adds up to one.
.025
What is the
probability of
observing a mean
at least as far
away as zero (on
either side) as
1.96 standard
errors? 2.72?
.025
Here is the mean we observed
(1.96 ≈ 2)
Remember the p-value question?
• If you think reality is a normal distribution
with mean y and standard deviation s, how
likely is your observed mean of x?
Let’s ask it again.
• We have systolic blood pressure measurements on a sample of 50
patients for each of 25 months. For each of those months, we a
mean blood pressure and a sample standard deviation.
• “You think reality for each month should be a normal distribution with
a mean blood pressure that equals the average of the 25 mean
blood pressures.
• You also think that for each month, this normal distribution should
have a standard error based on the average sample standard
deviation across the 25 months.
• How likely is your observed mean in month 4 of 220 if the average
mean across the 25 months was 120?
• How many standard errors is 220 away from 120? What is the
probability of being at least that many standard errors away from
120?
Welcome to the Shewart Control
Chart
1
2
3
4
5
6
7
.
.
.
25
120
Anatomy of the control chart:
From Amin, 2001
Indian Health Service, DHHS