Probability proportional to size sampling
Probability proportional to size sampling is also know as monetary unit sampling or dollar unit sampling. It is a method of sampling that takes the varying size of each item within the population into account when selecting the audit sample.
The audit sample size is calculated based on the population itself and risk factors such as materiality, expected error and required reliability level (these are judgemental factors that the auditor sets based on the engagement and specific risks). The sample size is then used to calculate the Sampling Interval, which is Population/Sample Size. Note that the population is the sum (total) of the “Amount” column, not the number of items in the population.
Each Dollar/Pound/Euro/Monetary Unit within the population then has an equal chance of being selected in the audit sample. There are various different ways of selecting the sample, and the 2 most common are Fixed Interval and Cell Selection Method.
In Fixed Interval, a random starting point is selected (between 1 and the sampling interval), and this is the first “Dollar” selected. The count is then incremented by the sampling interval to select the next and subsequent items for the audit sample. This is best explained with an example. Here we have a population of 10 items, with a total value of 1200, and we have a sample size of 3:
Value 
Cumulative Total 

Invoice 1 
100 
100 
Invoice 2 
34 
134 
Invoice 3 
23 
157 
Invoice 4 
403 
560 
Invoice 5 
108 
668 
Invoice 6 
78 
746 
Invoice 7 
61 
807 
Invoice 8 
19 
826 
Invoice 9 
285 
1111 
Invoice 10 
89 
1200 
The sampling interval is 400 (1200/3), so we’d pick a random start point between 1 and 400, lets say 263 (in reality you would use a random number generator to pick this start point). So our first item selected would be Invoice 4, as the cumulative total for 263 lies within item 4. The second sample item would be at 663 (263+400), or Invoice 5. Our final item would be at 1063, so Invoice 9.
When the Cell Selection Method is used the population is split into “Cells”, which are the same size as the sampling interval, and a random Dollar is picked from within each cell. In the above example a random Dollar would be selected from the first cell (1400), another from the second (401800) and the final from 8011200. So you might for example select Dollar 102 (Invoice 2), Dollar 799 (Invoice 7) and Dollar 820 (Invoice 8).
When there are large items in the population (larger than the sampling interval), these are usually extracted first and will definitely be selected for testing. The remainder of the audit sample is then selected using either cell selection or fixed interval. In the above example, Invoice 4 would be considered a large item and would be stripped out.
Once the audit sample has been selected, the auditor then tests the items, and records any difference between the book (recorded) value, and the actual value. These differences are then extrapolated over the population to provide the auditor with estimated error levels.
Due to the complexity of the maths involved in probability proportional to size sampling, it is virtually impossible to carry out without the use of specialized software that will calculate the sample size, extract the audit sample and then evaluate and extrapolate the results. However, with the use of software it can be incredibly easy to perform probability proportional to size sampling – have a look at this video, which shows just how easy it can be using the Monetary Unit Sampling routine in the incredible Excel addin TopCAATs, designed specifically to help auditors, and it includes many sampling routines!