Question 4 of 20: Define Standard Deviation

Question 4: Define Standard Deviation (2 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

The standard deviation, σ (population) or s (sample), is a measure of how widely a set of data values is spread around its mean. It is defined as the square root of the variance — the average of the squared deviations of each value from the mean: $s = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}$ for a sample of n observations (the $n-1$ divisor, Bessel's correction, gives an unbiased estimator of the population variance). Because it is expressed in the same units as the original data (unlike variance, which is in squared units), the standard deviation is the practical measure engineers quote for reporting data spread, setting statistical process-control limits, and computing confidence intervals; a small σ indicates values cluster tightly around the mean, while a large σ indicates wide scatter.

In a treatment-plant setting, standard deviation is what turns a raw compliance record into a design and operating decision: a high σ in influent BOD, for example, signals that equalization (Question 10) or a larger safety factor on treatment capacity is warranted, whereas a low σ supports designing closer to the mean loading without as much reserve capacity. Reporting a mean without its accompanying standard deviation is, for this reason, an incomplete engineering description of a waste stream — two streams can share an identical average BOD yet require very different treatment-system design margins if one is tightly clustered and the other widely scattered. Both numbers, together, are the minimum needed to make a defensible design decision. Quoting a mean without a measure of spread is one of the most common ways a data summary understates the real engineering risk in a variable waste stream.