Standard deviation in size data – Stdev.S ?

Many of our software packages have the integrated capability to display statistics of data selections. For example, selecting a few size measurements then leads to a display of the mean, the standard deviation and the relative standard deviation. How can you calculate these values on your own? In other words, what is the mathematical formula for the determination?

Let’s take a look at this specifically for the Zetasizer. When we select (or highlight) a set of data in, for example, the latest Zetasizer Xplorer software, we can display the statistics of that selection. For any parameter in the statistics table window,  a relevant Mean, Standard Deviation, and RSD (= relative standard deviation) is available in the Zetasizer. In addition, we can see a column heading for Minimum and Maximum. Let’s start with the easiest: the minimum and maximum are the smallest and the largest values in the whole range of selected numbers. Please note, the number of selected records is not available in the Statistics Table.

Below is an example of a typical Statistics Table where the Mean of the Z-average is 330.7nm. In the example, the Standard Deviation of 3.9nm corresponds to a relative deviation of RSD = 3.9/330.7 = 0.0118 = 1.18% . As you can tell from the example, the units of RSD are percent. The units for Mean and Standard Deviation are the same as the units of the parameter. For instance, in the case of the Polydispersity Index, the value is unitless, and thus the Std deviation is also unitless. On the other hand, for zeta potential and zeta deviation, the units would be milliVolt mV.

How to calculate the mean?

The mean is just the standard average (arithmetic mean) of the selected parameters. The equation for this calculation is

Mean = Average() = sum of all selected values/number of selected items

In Microsoft Excel, this Mean is available as the function Average(). In order to transfer the numeric values from the software into Excel, simply highlight the desired records, Ctrl-C and Ctrl-V into an Excel spreadsheet.

How to obtain the standard deviation?

The standard deviation function in Excel is Stdev.S(). This estimates the std deviation based on a sample. i.e. the “non-biased” version, where the data represent a partial sampling of the whole population {as opposed to Stdev.P() which calculates the standard deviation based on the entire population}. If you want to reproduce the calculation on your own simply use the Stdev.S function:

Standard deviation = Stdev.S() = [ sum of all squared differences / (number of selected items – 1) ]^0.5

Previously

• How to find the Zetasizer Red Sensitivity
• To order consumables, here is How to get a quote for consumables.
• The easiest method to get Help for your instrument – one step away

Have any questions? Please email me ulf.nobbmann@malvern.com – Thanks! Opinions are those of the author. Our editorial team modifies them occasionally.