NM7_reliability_s02

NM7_reliability_s02 - CGN 3421 Computer Methods Gurley...

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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. µσ µ Z µ Z 'Z [] ± 0 --- Z K K ± ² 0 ²² σ Z σ Z 8#4 Z µ Z Ō () Z µ Z Ō ± 0 Z K µ Z Ō ³ K ± ² 0 ² σ Z ± 0 Z K ³ K ± ² 0    µ Z ³ Ō ²

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CGN 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 second-order 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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NM7_reliability_s02 - CGN 3421 Computer Methods Gurley...

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