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Understanding Standard Deviation and Control
Charts
Many people ask: "Why aren't my upper and lower control limits (UCL,
LCL) calculated as:
µ + 3sigma (where µ is the mean and sigma is the standard
deviation)?"
To answer this question, you have to understand some key, underlying
statistics: variation, standard deviation, sampling and populations.
Variance (stdev^2) is the average of the square of
the distance between each point in a total population (N) and the mean
(µ).
If your data is spread over a wider range, you have a larger variance
and standard deviation. If the data is centered around the average, you
have a smaller variance and standard deviation.
Standard deviation (stdev or sigma) is the square root of the
variance:
And it can be estimated using the average range (Rbar) between samples
(Rbar/d2) when the number of subgroups is 2-10, or using
standard deviation Sbar/c4 when n>10.
Rbar = Rave = ΣRi/n
Sampling: Early users of SPC found that it cost too much to evaluate
every item in the total population.. To reduce the cost of measuring everything,
they had to find a way to evaluate a small sample and make inferences
from it about the total population.
Understanding Control Chart Limits:
Ask yourself this question: "If a simple formula using the mean and
standard deviation would work for any data, why are there so many different
control charts?"
The short answer: to save money by measuring small samples, not
the entire population.
Long answer: When using small samples or varying populations, the
simple formula using the mean and standard deviation just doesn't work,
because you don't know the average, µ, or sigma of the total population,
only µ or sigma of your sample.
Why are there so many control charts? Because:
You have to estimate µ and sigma using the average and range
of your samples.
In variable charts, the XmR uses a sample size of 1, XbarR (2-10)
and XbarS (11-25). These small samples may be taken from lots of 1,000
or more.
In attribute charts, the c and np chart use small samples and
"fixed" populations; the u and p charts have varying populations.
So, you have to adjust the formulas to compensate for the varying samples
and populations.
To reduce the cost of inspection at Western Electric in the 1930s, Dr.
Walter S. Shewhart developed a set of formulas and constants to compensate
for these variations in sample size and population. That's why they are
sometimes called Shewhart Control Charts.
Reference: You can find these in any book on statistical
process control (e.g., Introduction to Statistical Process Control, Montgomery,
Wiley, 2001, pgs 207-265).
So stop worrying about the formulas.
Start monitoring your process using the charts.
Stability Analysis
XbarR Chart
Conforms with ANSI/ASQC B1, B2, B3 1996
The XR chart can help you evaluate the cycle time for almost any process:
making a widget, answering a customer call, seating a customer, delivering
a pizza, or servicing an appliance. This chart is especially useful when
you do this many times a day. Using a small sample (typically five and
as many as 25) you can effectively measure and evaluate the process.
XbarS Chart Conforms
with ANSI/ASQC B1, B2, B3 1996
The Average and Standard Deviation chart is especially useful when you
have more than five samples.
XMedianR Chart
Conforms with AIAG SPC 2nd Edition
The XMedianR works just like the XbarR except that it uses the median
instead of the average as a measure of central tendency.
XmR Chart Conforms
with ANSI/ASQC B1, B2, B3 1996
The XmR (Individuals and Moving Range) chart can help you evaluate a process
when there is only one measurement and they are farther apart: monthly
postage expense and so on.
Calculate, plot, and evaluate the range chart first. If it is "out
of control," so is the process. If the range chart looks okay, then
calculate, plot, and evaluate the X chart.
XmR Median R Chart
The XmR Median R chart uses the Median(R) to reduce the bias in individuals
charts..
Compare XmR and XmR Median R charts to see differences in control limits.
XmR Trend Chart
(Source: Statistical Methods for the Process Industries, W McNeese and
R Klein, ASQ Press, Milwaukee, pg. 280-290)
UCL and LCL are the same as for the XmR chart
The only difference is how the X center line (CL) is calculated using
linear regression to give you the slope of the trend and a y-intercept
value: "b", calculated as follows:
time(i)=ti=1,2,3,4....k for each X value
m=slope
b=intercept
The XmRtrend then calculates the linear correlation coefficient (Ryx)
for the degrees of freedom (df=k-2).
If Ryx is greater than the probability for this degree of freedom,
you have a "significant correlation" between x and y.
(Probability that you will conclude there is no correlation when there
is one = alpha = 0.05).
If Ryx2 is greater than 0.80, then the correlation
indicates a "useful fit."
What does this mean?
Significant correlation.
A measure of x versus y. Is the relationship between x and y statistically
significant? This is a measure of how well the trend line reflects the
relationship between x and y.
Usefit Fit
Even if there is a significant correlation above this asks the question
- Is it useful? Can I make an assumption or prediction about y based on
past history?
A measure of the variation in x vs the variation in y.
This is a measure of how the points vary within the control limits.
| df |
Probability |
df |
Probability |
df |
Probability |
df |
Probability |
| 1 |
.997 |
11 |
.553 |
21 |
.413 |
35 |
.325 |
| 2 |
.950 |
12 |
.532 |
22 |
.404 |
40 |
.304 |
| 3 |
.878 |
13 |
.514 |
23 |
.396 |
45 |
.288 |
| 4 |
.811 |
14 |
.497 |
24 |
.388 |
50 |
.273 |
| 5 |
.754 |
15 |
.482 |
25 |
.381 |
60 |
.250 |
| 6 |
.707 |
16 |
.468 |
26 |
.374 |
70 |
.232 |
| 7 |
.666 |
17 |
.456 |
27 |
.367 |
80 |
.217 |
| 8 |
.632 |
18 |
.444 |
28 |
.361 |
90 |
.205 |
| 9 |
.602 |
19 |
.433 |
29 |
.355 |
100 |
.195 |
| 10 |
.576 |
20 |
.423 |
30 |
.349 |
|
|
(Source: Statistical Methods for the Process Industries, W. McNeese, ASQ
Quality Press, Milwaukee)
p and np Charts
Conforms with ANSI/ASQC B1, B2, B3 1996
The p and np charts will help you evaluate process stability when counting
the number or fraction defective. Examples include the number of defective
products, meals in a restaurant, incorrect perscriptions, bills, invoices,
or paychecks.
c and u Charts
Conforms with ANSI/ASQC B1, B2, B3 1996
The c and u charts will help you evaluate process stability when there
can be more than one defect per unit. Examples include defects per product,
errors per invoice, patient falls in a hospital.
g Chart
The g chart will help evaluate process stability when tracking rare events:
- surgeries between infections
- days between accidents (safety)
- days between wrong site or wrong patient surgeries
Just count the number of days between events (g).
Then use g to calculate the UCL.
t Chart
The t chart will help evaluate time between rare events:
- wrong site or wrong patient surgeries
- cardiac arrests
- patient falls
Just count the time or number of units between events.
Transform the time into into a more normal distribution (y).
Calculate the range (R) between events.
Then use y and R to calculate the UCL and LCL (if any).
Control Chart Constants
Histograms - Number of bars
The most frequent question we get is: "How do you select the number
of bars on the histogram?"
The simple answer is we round the square root of the number of data points.
For example:
- 25 data points = 5 bars
- 100 data points = 10 bars
If there are too many bars (e.g., more than 50) to display nicely on
the page, we limit the number of bars.
Juran's Quality Control Handbook provides these guidelines for the
number of bars and states that they are not "rigid" and should
be adjusted when necessary.
| Number of Data Points |
Number of Bars |
|
20-50
|
6
|
| 51-100 |
7 |
| 101-200 |
8 |
| 201-500 |
9 |
| 501-1000 |
10 |
| 1000+ |
11-20 |
You won't always get exactly 5 bars or 10. Why? Because we're trying
to fit the graph to the page between varying specification limits within
the constraints of Excel while still making it as readable as possible.
Compare the QI Macros histogram output to Minitab and you'll see that
they are similar:
 
 
 
 
Download a free pdf of histogram formulas and sample calculations at
histogram-manual-calcs.pdf
If you still need more help after reading the following information and
our free pdf, consider our Histogram Whitepaper.
Histograms - Process Capability Metrics
- Cp measures how well the data fits within
the spec limits (USL, LSL)
- Cpk measures how centered the
data is between the spec limits.
A Cpk of 1.33 is considered to be at 4-Sigma.
Use Cp when you have a sample, not the population.
Cp and Cpk use Sigma estimator.
Sigma estimator =
d2 is a constant based on subgroup size
c4 is a constant based on subgroup size
Xbar = Σ(Xi)/n
Rbar = Average(Ri) (Average of the Ranges in samples)
Cp = (USL-LSL)/(6*sigest)
CpU (upper) = (USL-Xbar)/(3*sigest)
CpL (lower) = (Xbar-LSL)/(3*sigest)
Cpk = Min(CpU,CpL)
ZT (target) = CpkT = (Xbar-Target)/(3*sigest)
Download a free pdf of histogram formulas and sample calculations at
histogram-manual-calcs.pdf
One-Sided Specifications
These equations assume that the process has both upper and lower specification
limits.
Until 10/2008, the macros used either CpU (USL) or CpL
(LSL) for both Cp and Cpk.
After 10/2008 Cp, Cpk, Cpm and other measures relying on USL/LSL will
show "*".
Cpm = (USL-LSL)/(6√(sigma2+(Xbar-Target)2))
(Cpm can be used when you have a target value.)
Process Performance
Use Pp when you have the total population.
stdev = stdev(Xi)
Pp(sigma) = (USL-LSL)/(6*stdev)
Ppu (upper) = (USL-Xbar)/(3*stdev)
Ppl (lower) = (Xbar-LSL)/(3*stdev)
Ppk(sigma) = Min(Ppu,Ppl)
Cp, Cpk in Older Versions of the QI Macros
In older versions of the QI Macros:
- Cp Rd2 and Cpk Rd2 are used when you have a sample. These calculations
are the same as Cp and Cpk in the current version of the software.
- Cp sigma and Cpk sigma are the same as Pp and Ppk. Use these when
your data represents the total population. Since this was confusing
to users, Cp sigma and Cpk sigma are NOT included in the current version
of the software. Pp and Ppk are included and provide the same values.
- Sigma estimator =
Rbar/d2 when number of subgroups <=10
Sbar/c4 when number of subgroups > 10
Parts Per Million
Min-Max
Min = Min(Xi)
Max = Max(Xi)
PPM
defects = number of points outside USL-LSL
% Total Defects = (defects100)/(Total points)
PPM = % Total Defects 10000
Expected PPM = (Normsdist(Zlower) + (1-Normsdist(Zupper)))1,000,000
Z Score
Z scores help estimate the non-conforming PPM.
Z scores standardize +/-3*stdev values into +/-3.
Zlower = (Xbar-LSL)/stdev
Zupper = (USL-Xbar)/stdev
Zbench = normsinv(1-(Expected PPM/1,000,000)) Zbench is the Z score
for the Expected PPM
ZT (target) = CpkT = (Xbar-Target)/(3*sigest)
Ztarget = Cpk for a target value instead of the USL or LSL. To calculate,
go into the Histdata sheet and input the target value to the right of
the cell marked "Target".
Variable
Sample Size (SS) = (ZStdev)/Confidence)2
Where:
Z = 1.96 for 95% confidence level
Stdev = standard deviation (or 0.5 Default)
Confidence interval expressed as decimal (e.g., .05 = ±5%)
Attribute
Sample Size (SS) = (Z2p(1-p)/Confidence2
Where:
Z = 1.96 for 95% confidence level
p = percent defects (0-0.5 = 0-50%)
Confidence interval expressed as decimal (e.g., .05 = ±5%)
Correction for Finite Population
new sample size = SS/(1+ (SS-1)/population)
Sigma
If observed PPM > 0, it uses the observed value to calculate Sigma.
If observed PPM = 0, it uses Expected PPM to calculate Sigma.
If both are zero, it defaults to 6 sigma.
If there are no USL/LSL, it leaves it blank.
Scatter Diagram
Used to evaluate the correlation between two variables.
-
R2 close to 1.0 means a perfect fit
-
R2 greater then .8 means that 80% of the variability in
the data is accounted for by the equation. Most statistics books imply
that this means that you have a strong correlation.
Reference Books
If you want a good Lean Six Sigma reference, consider my DeMystified
book:
If you don't have a good statistical reference book, I can recommend:
 Juran's
Quality Control Handbook (4th), McGraw Hill, NY, 1988.
 Douglas
Montgomery, Introduction to Statistical Process Control (4th), John
Wiley & Sons, NY, 2001.- Montgomery Instructor Companion Site includes Data
Sets Referenced in Montgomery's book
 Douglas
Montgomery, Design and Analysis of Experiments (5th), John Wiley &
Sons, NY, 2001.
- Measurement Systems Analysis (Version
3), AIAG, 2002.
- Engineering
Statistics On Line Handbook
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