CGN 3421  Computer Methods
Gurley
Numerical Methods Lecture 7  Statistics, Probability and Reliability
page 112 of 125
Numerical Methods Lecture 7  Statistics, Probability and Reliability
Topics
A summary of statistical analysis
A summary of probability methods
A summary of reliability analysis concepts
Statistical Analysis
The value of a measured quantity can often vary from one measurement to the next, and from one sample to the
next (e.g. student grades on an exam, strength of concrete cylinders). We will refer to such a changing quantity
as a ‘random variable’. Statistical analysis allows us to view important characteristics of the random variable
without having full information. That is, we won’t know what the exact strength of the next concrete cylinder
to be tested is, but we can take a good guess based on previous measurements and statistical analysis.
Mean and Standard Deviation of a Single Variable
The most fundamental statistics are the mean
and standard deviation
.
Given:
A single random variable ‘X’ sampled ‘N’ times
The
mean
of X  denoted
: average value of the measured quantity
The
standard deviation
 denoted
: the average distance from the mean, or the average spread
‘var
x
’ is the variance of x. The standard deviation is the square root of the variance.
an equivalent expression is
A higher standard deviation increases the odds of being far away from the mean.
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View Full DocumentCGN 3421  Computer Methods
Gurley
Numerical Methods Lecture 7  Statistics, Probability and Reliability
page 113 of 125
Example: Two different sets of exam grades
Class #1 and class #2 have about the same mean value
(red line)
Class #1 has a small standard deviation: most students are near the mean
(blue line borders)
Class #2 has a larger standard deviation, so students have a higher probability of being well over or well
under the class average grade.
We can use the mean and standard deviation to estimate the likelihood of deviating from the mean value.
Higher
= higher probability of being further from the mean. We will get into quantifying this probabil
ity in a few pages.
The mean and standard deviation are classified as first and secondorder
statistics (involving the mean of
X, and mean of X
2
, respectively). If we stick with using these two stats to describe data, we are making
assumptions about the form of its probability. We assume the fluctuations about the mean are equally
likely to be above or below the mean. That is, the probability behavior is
SYMMETRIC
about the mean.
This will not always be realistic. For example, if I give an easy test, the class average may be 100, but the
standard deviation may be 15. If we assume the distribution of grades is symmetric about the mean, that
would result in scores above 100, which is out of bounds. So there are cases when just the mean and star
dard deviation are not enough.
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 Spring '08
 Long
 Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function, Gurley

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