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Statistical Quality Control
Lab 202
By
S. O. Duffuaa
Systems Engineering
Department
Salih Duffuaa
Dr. Duffuaa is a Professor of Industrial and Systems Engineering at the •
Department of Systems Engineering at King Fahd University of Petroleum
and Minerals, Dhahran, Saudi Arabia. He received his PhD in Operations
Research from the University of Texas at Austin, USA. His research
interests are in the areas of Operations research, Optimization, quality
control, process improvement and maintenance engineering and
management.
He teaches course in the areas of Statistics, Quality control, Production
and inventory control, Maintenance and reliability engineering and
Operations Management. He consulted to industry on maintenance , quality
control and facility planning. He authored a book on maintenance planning
and control published by John Wiley and Sons and edited a book on
maintenance optimization and control. He is the Editor of the Journal of
Quality in Maintenance Engineering, published by Emerald in the United
Kingdom.
King Fahd University of Petroleum & Minerals
Department of Systems Engineering
Errors and Test of Significance
On
November 17- 21, 2007
Objective of Session
• Differentiate between Accuracy
precision and bias.
• Explain and measure precision and
accuracy.
• Identify and test outliers.
• Formulate hypothesis about measured
values.
• Conduct simple significance tests.
Accuracy
• Accuracy - The degree of agreement of a
measured value with the true or expected value of
the quantity of concern. Accuracy is measured by
absolute error (E) or relative error Er
• E = l Measured Value – True Value I.
• Er = E/True value (100) percentage.
• Measured value 10.2 , true value 10
• E = l 10.2 – 10.0 l = 0.2, Er = (0.2 /10) * 100 = 2%
Precision
• Precision - The degree of mutual agreement
characteristic of independent measurements as
the result of repeated application of the process
under specified condition.
• Precision is the agreement between two or
more measurements that have been made in
exactly the same way. Precision is determined
by replicating measurements. Precision can be
measured by using the measures of dispersion:
Precision
• Precision can be measured by using the
measures of dispersion:
– Range
– Variance
– Standard deviation
– Coefficient of variation.
Precision ( Example)
• Lab A made five measurements for the same
property with the same instrument and
procedures. The results are 9.80, 9.88, 10.02,
10.14, 10.21.
–
–
–
–
–
–
Mean = 10.01
Standard deviation = 0.17
Range = 0.41
Variance = 0.0289
COV = 1.72 %
E= .01. Er = 0.10%
Bias
• Bias - A systematic error inherent in a
method or caused by some artifact or
idiosyncrasy or the measurement system.
Bias may be both positive and negative,
and several kinds can exist concurrently
so that net bias is all that can be
evaluated.
• How to detect systematic errors?
Unbiased Measurement Process
•
Unbiased Measurement Process
•
Characteristic of A Measurement Process
• Accurate when the value reported does not
differ from the true value.
• Biased when the error of the limiting mean is not
zero, influenced by systematic error.
• An accurate method is one capable of providing
precise and unbiased results. In practice, we
evaluate inaccuracy. Likewise, we evaluate
imprecision, namely, the deviations of
measurements.
Types of Errors
• Random errors or chance errors are irregular
and unpredictable. Random errors result in
variability. In the determination of vanadium in
crude, Ali has made six repeat determinations,
and the results in ppm are:
20.2 19.9 20.1 20.4 20.2 20.4
• The determinations differ from each other
because of random errors. If the test method did
not introduce random error then the six
determinations would be identical (assuming
that gross errors are absent
Types Errors
• Systematic errors: Determinate or systematic
errors have a source that can usually be
identified. They affect a sequence of
determinations equally. They cause all the
results from replicate measures to either be high
or low.
• Sources of systematic errors
– Instrument
– Methods
– Operator or personal errors.
Types Errors
• Gross errors are defined as errors so serious
that there may be no alternative but to make a
completely new start, including new samples
and lab tests. Examples of gross error include a
complete instrument breakdown, power
outage, loss or accidental discarding of data,
and so on.
Example 16
•
Analyst
Ali
Ahmed
Mohammad
Emad
Table 15. Repeat Determinations by Four Analysts
Determination of Vanadium in Crude Oil (ppm) Mean
20.2
19.9
20.6
20.1
19.9
20.2
20.5
19.9
20.1
19.5
20.7
20.2
20.4
20.4
20.6
19.9
20.2
20.6
20.8
21.1
20.4
19.4
21.0
20.0
20.20
20.00
20.70
20.20
SD
0.190
0.486
0.179
0.456
Example 16
Problem 1
• Analyze the accuracy and precision of each of the
following labs and identify which lab has a random
systematic or gross errors. ( Explain if the lab is not
precise or accurate how did you reach to the
conclusion.
Lab A
10.08
10.09
10.10
10.11
10.12
Lab B
9.80
9.88
10.02
10.14
10.21
Lab C
9.69
9.78
9.79
10.05
10.19
Lab D
9.97
9.98
10.02
10.04
10.04
Example Page 4 Section 2
Lab A
10.08
•
10.09
10.10
10.11
10.12
Lab B
9.80
9.88
10.02
10.14
10.21
Lab C
9.69
9.78
9.79
10.05
10.19
Lab D
9.97
9.98
10.02
10.04
10.04
10.10
0.02
0.16%
10.01
0.17
1.72%
9.90
0.21
2.13%
10.01
0.03
0.33%
0.10
1.00%
0.01
0.10%
-0.10
-1.00%
0.01
0.10%
Mean
SD
Coefficient
of Variat ion
Error
Relat ive Error
Figure 1. Results of Analysis with a True Value of 10
Example Page 4 Section 2
• Laboratory A— The random error is small because the
measurements all go in one direction. The standard deviation and
coefficient of variation are small, therefore the results are precise.
Second, Lab A's results also include systematic error. Because all
the results are in error in the same sense—too high. Systematic
errors affect accuracy, or proximity to the true value. The error and
relative error are high indicating a lack of accuracy.
• Laboratory B—Laboratory B obtained an average 10.01. This
result is in direct contrast to that of Lab. A. The average 10.01 is
very close to our known true value of 10.00. We therefore can
characterize the data as accurate and without substantial systematic
error. However, the spread of the results is very large, indicating
poor precision and the presence of substantial random error.
• It should be noted that a comparison of Lab. A and Lab. B results
clearly indicates that random and systematic errors can occur
independently of each other.
Example Page 4 Section 2
• Laboratory C—Laboratory C's average of 9.90 and
with a standard deviation of 0.21 and a coefficient of
variation of 2.13%. The relative error is also high.
This indicates that the data is neither precise nor
accurate.
• Laboratory D—Laboratory D has achieved an
accurate mean and precise measures of dispersion and
error.
Significance Tests/ Test of hypothesis
• Purpose: To draw a conclusion about a
population using data from a sample.
• Test that the analytical procedure is not
subject to systematic errors.
• Test that the error is not significant.
• Test that the population mean is equal to the
sample mean.
• Test that one lab results is more accurate
than another lab.
Significance Tests/ Test of hypothesis
• Test the precision of an instrument.
• Compare the precision of two instruments.
• Test that Arabs are taller than European.
Significance Tests/ Test of hypothesis
• Significant tests are widely used in the
evaluation of experimental results. In making a
significance test we are testing the truth of a
hypothesis, which is known as null hypothesis.
• In all analytical procedures we adopt the null
hypothesis that the analytical method employed
is not subject to systematic error. The term null
is used to imply that there is no difference
between the observed and known values other
than that which can be attributed to random
variation
Significance Tests/ Test of hypothesis
• Assuming that null hypothesis is true, statistical theory
can be used to calculate the probability (i.e. the chance)
that the observed difference between the sample mean
and the true value μ, arises solely as a result of random
error.
• The lower the probability that the observed
difference occurs by chance, the less likely it is that
the null hypothesis is true.
• Usually the null hypothesis is rejected if the probability of
the observed difference occurring by chance is less than
1 in 20 (i.e. 0.05 or 5%) and in such a case the
difference is said to be significant at the 0.05 (or 5%)
level.
Demonstration
•
In this vanadium determination, it was believed that the true
vanadium concentration was 20.1 ppm, but Mohammad's
determinations had a mean of 20.7 ppm and a standard
deviation of 0.179. Can we conclude Mohammed is biased?
Table 15a. Repeated Determinations by Four Analysts
Analyst
Ali
Ahmed
Mohammad
Emad
Determination of Vanadium in Crude Oil (ppm)
20.2
19.9
20.6
20.1
19.9
20.2
20.5
19.9
20.1
19.5
20.7
20.2
20.4
20.4
20.6
19.9
20.2
20.6
20.8
21.1
20.4
19.4
21.0
20.0
Mean
SD
20.20
20.00
20.70
20.20
0.190
0.486
0.179
0.456
One Sample t-test
• Step 1: Null hypothesis –
Mohammad is not biased (i.e his
population mean μ is equal to 20.1).
H 0: = μ
• Step 2: Alternative hypothesis –
Mohammad is biased (i.e his
population mean μ is not equal to
20.1).
H 1: ≠ μ
One Sample t-test
• Step 3: Test statistic
• Step 4: Critical values – from the t-table for a two
sided test with 5 degree of freedom.
2.57 at the 5% significance level 4.03 at
the 1% significance level
• Step 5: Decision – We reject the null hypothesis.
• Step 6: Conclusion – We concluded that
Mohammad is biased.
Test Statistics
When calculating the value of the t-test we took into
account:
• The deviation of the sample mean from the
hypothesized population mean (μ).
• The variability between repeat determinations that
we would expect to find in the test method. This is
quantified by the sample standard deviation.
• The number of repeated determinations.
• A large value of test statistic is an indication of bias.
• In all significance tests, a large test statistic implies
that the null hypothesis is false.
Hypothesis Testing
• Hypothesis are suppositions, presumed to be
true for subsequent testing.
• Statistical significance is the level of
probability selected to determine if a set of
sample data is attributable to chance causes
alone.
• Chance causes are unknown factors that
contribute to variation. They are generally
numerous and individually of small
magnitude. They are not readily detectable
or identifiable
Alternative Way To test hypothesis
Example 15. Another example for large •
sample size
• The strength specification of a filament is
claimed to be 21.4 ounces. The strength
of 120 filaments was randomly sampled.
These are the results:
• = 20.5 oz.
• S = 1.1 oz.
• n = 120
• Is the manufacturer's claim substantiated?
Alternative Way To test hypothesis
• A common method of answering this question is to set
up confidence limits for the mean. The confidence
coefficient chosen was 95%.
• From Equation 5: CL= ± 1.96(SEM)
Therefore:
• CL = 20.5 ± 1.96
20.5 ± 1.96 (0.10) = 20.5 ± 0.20
i.e. 20.30 to 20.70 ounces
• The manufacturer's claim of 21.4 ounces is not within the
95% confidence limits for the mean of the sample. The
claim seems to be incorrect on the basis of this batch of
samples. We have just drawn a statistical inference on a
population based on a sample.
Graphical Representation
•
X 
S
n
Test of Significance
•
•
•
•
•
Ho = 21.4
H1 not = 21.4
 = .05
Test statistics Z = (X-bar - µ/ (/n)
Compute the value of test statistics and
compare it to Z-table values
• If Z-computes larger than table value reject null
hypothesis.
Example 18
•
The concentration (ppm) of lead in gasoline was determined by
two different methods for each of the four samples.
Table 16. Lead in Gasoline Sample Data
Sample
Wet
Oxidation
Extraction
Colorimetry
1
71
76
2
61
68
3
50
48
4
60
57
Question: Do the two methods give value for the mean lead
concentration, which differ significantly?
Solution Example 18
•
Step 1. Ho:
=0
μd
H1:
not = 0
μ dthe pairs of values above, the differences
Compare
• Step 2.
are:
Sample
Difference
1
(71 - 76) = -5
2
(61 - 68) = -7
3
(50 - 48) = 2
4
(60 - 57) = 3
Mean of the difference, md, is
Solution Example 18
• t =
= 0.7
• Compare this to table value with 3 degrees of
freedom t-table 3.16 at .05 level of significance
• Do not reject Ho.
Problem
• It is suspected that an acid-base titrimetric method has a
significant indicator error and thus tends to give results
with a positive systematic error (i.e. positive bias). To
test this, an exact 0.1M solution of acid is used to titrate
25.00 ml of an exactly 0.1M solution of alkali, with the
following results.
•
25.06 25.18 24.87 25.51 25.34 25.41
From this data we have: Mean = 25.228ml
Standard deviation = 0.238
Test the hypothesis that the method has a bias.
Outliers ( Dixon Q )
• An outlier is one or more values that appear to differ
markedly from the other values in the distribution. The
presence of an outlier may distort the true value of a set of
data. It therefore becomes necessary to identify outliers and
determine whether they belong in the data distribution.
•
Example Outliers
• Note: Dixon’s Q is calculated without regard to
sign.
• For example, the following values were obtained
for nitrate concentration in four water samplings:
• 0.403 0.410 0.401 0.380
• The last measurement (0.380) is suspect.
Should it be rejected? We find out by first
solving Dixon's Q at the 95% confidence level.
• Dixon's Q was found to be 0.7.
Steps to Reject an outlier
• Decide the probability level to set for
rejection (5% is standard).
• Consult the standard table of critical values
(Table 17 & 35) to determine if our Q of 0.7
should be accepted or rejected.
• The table of critical values lists for each
sample size a corresponding critical value. If
Q exceeds the critical value for the given
sample size, then Q is rejected.
Table 17. Critical Values Of Dixon's Q
•
Sample
Size
Critical
Value
Critical
Value
5%
1%
3
0.970
0.994
4
0.829
0.926
5
0.717
0.821
6
0.621
0.740
7
0.569
0.717
8
0.524
0.680
9
0.492
0.672
10
0.464
0.635