Very basic review of some statistics terms¶
Engineering Testing and Measurements
Review of basic statistics
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Population -- the entire collection of measurements, not all of which will be analyzed statistically. Some variable $ x$, anything that is measurable, such as a length, time, voltage, current, resistance, etc.
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Sample -- a subset of the population that is analyzed statistically. A sample consists of \(n\) measurements: \(x_1,x_2, x_3,...,x_n\)
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Statistic -- a numerical attribute of the sample (e.g., mean, median, standard deviation).
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Population mean is denoted as: \(\mu\)
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Sample mean is an arithmetic average:
Deviations¶
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Deviation is the difference between a particular measurement and the mean, $ d = x_i - \overline{x} $
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Deviation of one particular measurement is the same as the precision error or random error of that measurement.
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Deviation is not the same as inaccuracy - difference between a particular measurement and the true value
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Because of bias (systematic) error, $ x_{true}$ is often not even known, and the mean is not equal to \(x_{true}\) as there are bias errors.
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Average deviation -- averaging all the deviations is zero, $ \overline{d} = 0$
plainSample standard deviation * Average absolute deviation -- a better measure of deviation is the average absolute deviation (also called the average positive error), defined as the average of the absolute value of each deviation. $$ \left|\overline{d}\right|=\frac{1}{n}\sum_{i=1}^{n}\left|d_{i}\right|$$
- Sample standard deviation -- measure of how much deviation or scatter is in the data is obtained by calculating the sample standard deviation. For $ n measurements: $$ S=\sqrt{\frac{\sum_{i=1}^{n}d_{i}^{2}}{n-1}}=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}{n-1}}$$ \par
- Notice the use of: \(n - 1\), it means we have only \(n-1\) degrees of freedom as one degree is taken by the \(\overline{x}\)