Material Variability…
… or
“how do we know what we have?”
Why are materials and
material properties variable?
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Metals
Concrete
Asphalt
Wood
Plastic
Types of Variance
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Material
Sampling
Testing
Errors vs. Blunders
Cumulative
Precision and Accuracy
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Precision – “variability of repeat
measurements under carefully controlled
conditions”
Accuracy – “conformity of results to the true
value”
Bias – “tendency of an estimate to deviate in
one direction”
Addressed in test methods and specifications in
standards
Accuracy vs. Precision
Bias
Precision
without
Accuracy
Accuracy
without
Precision
Precision
and
Accuracy
Repeatibility vs. Reproducibility
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Repeatability
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Within laboratory
Reproducibility
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Between laboratory
Bias
Sampling
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Representative random samples are
used to estimate the properties of the
entire lot or population.
These samples must be subjected to
statistical analysis
Sampling - Stratified Random
Day 1
Day 2
Day 3
Sampling
Lot #1
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Lot # 2
Lot # 2
Need concept of random samples
Example of highway paving
 Consider each day of production as sublot
 Randomly assign sample points in pavement
 Use random number table to assign positions
 Each sample must have an equal chance of
being selected, “representive sample”
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Parameters of variability
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Average value
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Measures of variability
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Central tendency or mean
Called dispersion
Range - highest minus lowest
Standard deviation, s
Coefficient of variation, CV%
(100%) (s) / Mean
Population vs. sample
Basic Statistics
n
x
x
i 1
i
n
Arithmetic Mean
“average”
2
 n

  xi  x  
 i 1

s
n 1 






1
2
Standard Deviation
“spread”
Basic Statistics
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Need both average and mean to properly
quantify material variability
For example:
mean = 40,000 psi and st dev = 300
vs.
mean = 1,200 psi and st. dev. = 300 psi
Coefficient of Variation
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A way to combine
‘mean’ and ‘standard
deviation’ to give a
more useful
description of the
material variability
s
n%   100
x
Population vs. Lot and Sublot
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Population - all that exists
Lot – unit of material produced by
same means and materials
Sublot – partition within a lot
Large spread
Normal Distribution
Small spread
Frequency
m= mean
34.1%
2.2%
2.2%
-3s
34.1%
13.6%
-2s
-1s
13.6%
+1s
+2s
+3s
LRFD(Load and resistance factor design method)
for Instance…
A very small probability that the load
will be greater than the resistance
Load
Mean load
Resistance
Mean resistance
Control Charts
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Quality control tools
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Variability documentation
Efficiency
Troubleshooting aids
Types of control charts
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Single tests
X-bar chart (Moving means of several tests)
R chart (Moving ranges of several tests)
Control Charts
(X-bar chart for example)
Moving mean of 3 consecutive tests
Mean of 2nd 3 tests
UCL
Result
Target
LCL
Mean of 1st 3 tests
Sample Number
Data has shifted
Data is spreading
Use of Control Charts
Refer to the text for other examples of trends
Example
A structure requires steel bolts with a strength of 80 ksi. The standard
deviation for the manufacturer’s production is 2 ksi. A statistically sound
set of representative random samples will be drawn from the lot and
tested. What must the average value of the production be to ensure that
no more than 0.13% of the samples are below 80 ksi? What about no more
than 10%?
Req’d mean = ??
1.
2.
3.
4.
80 ksi
-3s -2s -1s
1. Solution to 1.
+1s +2s +3s
z ~ -3  -3s
m – 3s  80 ksi
Required mean = 86 ksi
What does it mean?
2. Solution to 2.
1. z~ -1.2817  -1.2817s
2. m – 1.2817s = 80 ksi
3. Required mean = 82.6 ksi
4. What is the difference between 1
and 2
Control Charts
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Quality control tools
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Variability documentation
Efficiency
Troubleshooting aids
Types of control charts
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Single tests
X-bar chart (Moving means of several tests)
R chart (Moving ranges of several tests)
Control Charts
(X-bar chart for example)
Moving mean of 3 consecutive tests
Mean of 2nd 3 tests
UCL
Result
Target
LCL
Mean of 1st 3 tests
Sample Number
Data has shifted
Data is spreading
Use of Control Charts
Refer to the text for other examples of trends
Other Useful Statistics in CE
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Regression analysis
Hypothesis testing
Etc.