Very basic review of some statistics terms

Engineering Testing and Measurements

Review of basic statistics

1. 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.

2. Sample -- a subset of the population that is analyzed statistically. A sample consists of $n$ measurements: $x_1,x_2, x_3,...,x_n$

3. Statistic -- a numerical attribute of the sample (e.g., mean, median, standard deviation).

4. Population mean is denoted as: $\mu$

5. Sample mean is an arithmetic average:

Deviations¶

• 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.

  $$\left|\overline{d}\right|=\frac{1}{n}\sum_{i=1}^{n}\left|d_{i}\right|$$

  (2)

• 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:

  $$S=\sqrt{\frac{\sum_{i=1}^{n}d_{i}^{2}}{n-1}}=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}{n-1}}$$

  (3)

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• Notice the use of: $n - 1$, it means we have only $n-1$ degrees of freedom as one degree is taken by the $\overline{x}$