Audit Sample
The advances of computer aided audit tools have made sampling redundant in many situations, as it becomes possible to test 100% of a population. However, there are still many situations where the only option is to pick an audit sample, and test these items to gain some comfort over the population as a whole. Despite sampling being unavoidable during most audits, it is one area that many auditors struggle with.
When it comes to picking the audit sample there are many factors to consider, such as what sampling method to use (statistical vs. non statistical sampling), how big the sample should be, what the required comfort level is, etc. There can also be some fairly complex maths involved if you don’t have access to sample size calculators or tables.
This website is all about sampling in the audit. We’ll talk about choosing the most appropriate sampling method, picking the audit sample, testing the audit sample and then evaluating the results of the audit sample. Our goal is to help auditors understand sampling within the audit process, to ensure that the work they are performing is valid.
There is no point in spending time picking an audit sample, testing the sample and not having sufficient appropriate audit evidence (e.g. the audit sample may not be representative of the population).
The biggest risk is that the auditor doesn’t know that the results of their testing are not valid, and they rely on those results to form an audit opinion. There is also a risk of “over-auditing” by picking an audit sample that is too large – the goal should be to pick and test the smallest audit sample possible that will lead to statistically valid results.
Although geared up towards financial statement audits and internal audits much of the information contained in this site is appropriate to sampling in general. If you’re interested in sampling please feel free to browse this site and we hope that you find its contents useful!
Selecting the Audit Sample using Systematic Sampling
Systematic sampling is a statistical sampling method where every nth item is selected from the population to form the audit sample. The sampling frame (or population) must be ordered in some manner to use systematic sampling. The steps to produce a systematic audit sample are:
- Calculate the sampling interval – n = no of items in population / required sample size
- Select a random starting point within the first interval
- Extract every nth item
This may be easier to explain with an example. Lets say we have a population containing 360 items, and we need to sample 30 items. This gives a sampling interval of 12. Therefore we would pick a random start point between 1 and 12 (e.g. 7) and then extract every 12th item.
In this example, we would extract the following items… 7, 19, 31, 43, 55, 67… … 331, 343, 355.
From a practical perspective, it is relatively easy to construct formulas in Excel to extract a systematic sample – you can use the following formula
=IF(MOD(ROW()-[start point],[interval])=0,1,0)
=IF(MOD(ROW()-7,12)=0,1,0) using the above example
This formula works by looking at the row number, subtracting the starting point, and then checking to see if it is visible by the sampling interval. If it is divisible (without leaving a remainder) then it returns a 1, and if it is not divisible it returns a 0. You can then apply filters to only show those records with a 1, and this is your audit sample.
Alternatively you can use software such as TopCAATs for you, which will extract the audit sample to a new worksheet.
When using systematic sampling, you have to be careful to ensure that there is not bias in the population, or that your sampling interval does not introduce bias in the audit sample. For example, imagine you were selecting a sample of stock locations, and the warehouse was laid out so that all the even locations were on one side of the aisle and all the odd locations on the other. If you selected every 50th item and started at item 20, you would never pick an odd location, and therefore there is no chance of ever selecting items on one side of the aisles. In this scenario, a random sampling approach would be more suitable.
Therefore systematic sampling is most appropriate when looking over a period of time, as it ensures an even spread of items being selected in the audit sample over the period under review.
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 (1-400), another from the second (401-800) and the final from 801-1200. 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 add-in TopCAATs, designed specifically to help auditors, and it includes many sampling routines!
Selecting an Audit Sample using Random Sampling
When using a random sampling approach (also called simple random sampling or SRS) each item within the sampling frame is given an equal chance of being selected in the audit sample. The sampling frame is not split or divided into groups. In addition, an 2 items in the population have the same chance of being selected as any other 2 items (and the same for any 3 items, 4 items, etc). I.e. the selection of one item has no impact on the probability of any other item being selected.
Random sampling can be vulnerable to detection (sampling) error. This is where the audit sample is not representative of the population. For example, if you pulled 10 balls out of a bag that contained 50 red and 50 blue balls, there is a chance that you could pull out 10 blue balls, and this could lead to a conclusion that the bag contains only blue balls.
In an audit scenario this means that the audit sample selected may not contain any erroneous items, but that doesn’t guarantee there are not errors in the population. Auditors minimise this risk by using statistically determined audit sample sizes. By specifying the level of acceptable detection risk, and knowing the size of the population and the expected error rate (the estimated number of erroneous items in the sample) the sample size can be calculated.
Once the audit sample size has been calculated, items can then be selected from the population at random. One way to do this is to generate random numbers (for example using the “=RAND()” function in Excel) for each item, then sorting these by random number and taking the top X or bottom X items (where X is the sample size).
Alternatively, software can be used to select the audit sample at random from the population, such as the random sampling routine in TopCAATs. Click here to see a video of random sampling in action
What is sampling and why is it appropriate to auditing?
There are many definitions of sampling out there, but in general sampling involves selecting individual items from a population, to enable you to draw a conclusion on the population as a whole.
In an audit, sampling procedures are used because it is not practical to examine every single item in a population. For example, the auditor may select an audit sample of non-current assets, and verify their existence, condition and value. It would not be practical for the auditor to track down every single asset on the books. But, if all the items in the audit sample are verified then it may be appropriate to draw the conclusion that all the assets are correctly recorded in the books (assuming the audit sample has been selected correctly and is of sufficient size).
Before you can begin a sampling procedure you must define the population that you wish to test. The population is all of the items that contain the characteristic that you wish to understand. Thankfully, in the case of audit sampling the population is usually easy to define, e.g. all the assets on the asset register, all the sales during the period under review, etc. However, there may be occasions where we need to sample over time or space, and the population may be more difficult to define, for example an investigation into call centre waiting times at different times of the day. In these cases the focus may be on discrete observations or periods.
The basis for selecting an audit sample is the list of items from which the audit sample is being chosen, which is called the sampling frame. The sampling frame may be identical to the population, but this is not always the case. For example, if an internal audit department were investigating customer satisfaction levels in a shop, it would be impossible to identify all the customers who have purchased from that shop. Therefore, an appropriate sampling frame may be all the customers on a specific date.
It is vital that the sampling frame is representative of the population. In the above example, the shop may be considerably busier at weekends than on a Monday. Therefore if the date chosen to sample customers is a Monday, when staff have more time to spend with each customer, the results of the testing may indicate that customers are very satisfied. However, in reality the majority of customers (who shop on weekends) may have a much lower satisfaction level.
Selecting an appropriate sampling frame that is representative of the population is a matter of auditor judgement.
The audit sample must also be representative of the population for the results to be valid, and this is often dependant on the sampling method chosen. For example, in the above scenario, if the auditor is left to select customers at random (haphazard sampling) they may be inclined to approach more attractive customers, who may also receive preferential treatment from staff based on their physical appearance. In this case a different method of selecting the audit sample would be more appropriate, for example approaching every 10th customer (systematic sampling).
FREE report: A guide to sampling for Auditors
