Very basic review of some statistics terms

Very basic review of some statistics terms#

Engineering Testing and Measurements

Review of basic statistics

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.
Sample – a subset of the population that is analyzed statistically. A sample consists of $n$ measurements: $x_1,x_2, x_3,...,x_n$
Statistic – a numerical attribute of the sample (e.g., mean, median, standard deviation).
Population mean is denoted as: $\mu$
Sample mean is an arithmetic average:

\[ x_{\mathrm{avg}}=\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}\]

Deviation is the difference between a particular measurement and the mean, $ d = x_i - \overline{x} $
Deviation of one particular measurement is the same as the precision error or random error of that measurement.
Deviation is not the same as inaccuracy - difference between a particular measurement and the true value
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.
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 $\overline{x}$