Material Variability…

… or “how do we know what we have?”

Why are materials and material properties variable?

Metals Concrete Asphalt Wood Plastic

Types of Variance

Material Sampling Testing

Cumulative

Errors vs. Blunders

Precision and Accuracy 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

Precision without Accuracy

Accuracywithout Precision

Precisionand

Accuracy

Bias

Repeatibility vs. Reproducibility

Repeatability Within laboratory

Reproducibility Between laboratory Bias

Sampling

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 Sampling

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”

Day 1 Day 2 Day 3

Lot #1 Lot # 2 Lot # 2

Parameters of variability Average value

Central tendency or mean Measures of variability

Called dispersion Range - highest minus lowest Standard deviation, s Coefficient of variation, CV%

(100%) (s) / Mean Population vs. sample

Basic Statistics

n

xx

n

ii

1

2

12

1

1

n

xxs

n

ii

Arithmetic Mean“average”

Standard Deviation“spread”

Basic Statistics 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 A way to combine

‘mean’ and ‘standard deviation’ to give a more useful description of the material variability

100% x

sn

Population vs. Lot and Sublot

Population - all that exists Lot – unit of material produced by

same means and materials Sublot – partition within a lot

Normal DistributionFr

equ

en

cy

34.1% 34.1%

13.6% 13.6%2.2% 2.2%

= mean

-3 -2 -1 +1 +2 +3

Small spread

Large spread

LRFD(Load and resistance factor design method) for Instance…

Load Resistance

A very small probability that the load will be greater than the resistance

Mean load Mean resistance

Control Charts Quality control tools

Variability documentation Efficiency Troubleshooting aids

Types of control charts 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

Sample Number

Resu

lt

Mean of 1st 3 tests

Mean of 2nd 3 tests

Target

UCL

LCL

Use of Control ChartsD

ata

is

spre

adin

g

Data

has

shifte

dRefer to the text for other examples of trends

ExampleA 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%?

-3 -2 -1 +1 +2 +3

1. Solution to 1.1. z ~ -3 -32. – 3ksi3. Required mean = 86 ksi4. What does it mean?

2. Solution to 2.1. z~ -1.2817 -1.28172. – 1.2817 = 80 ksi3. Required mean = 82.6 ksi4. What is the difference

between 1 and 2

Req’d mean = ??

80 ksi

Control Charts Quality control tools

Variability documentation Efficiency Troubleshooting aids

Types of control charts 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

Sample Number

Resu

lt

Mean of 1st 3 tests

Mean of 2nd 3 tests

Target

UCL

LCL

Use of Control ChartsD

ata

is

spre

adin

g

Data

has

shifte

dRefer to the text for other examples of trends

Other Useful Statistics in CE

Regression analysis Hypothesis testing Etc.