**There are two reasons why auditor only provides reasonable assurance and not perfect assurance: (1) Sampling risk is the chance that auditor’s conclusion will be wrong because only a portion of the population was examined. A sample may not be representative of the population, and (2) Non-sampling risk is the chance that auditor’s conclusion will be wrong for reasons that would happen even if every item had been tested, includes human errors such as the failure to recognize a misstatement and the misinterpretation of results. Judgment sampling estimates the amount of sampling risk that the auditor faces purely by human guess. Statistical sampling determines sampling risk mathematically Auditor sets an acceptable level of sampling risk before beginning a test and statistical sampling computes the number of items to be tested to reduce risk to that desired level. Auditor can also perform a test and then use statistical sampling to determine the amount of sampling risk that is present. **

**Sampling is used in virtually every aspect of auditing. The term refers to making a decision about a whole (a population) by testing only a part of that group (called a sample)**. **Sampling creates a problem**; because auditor does not look at every piece of evidence, a chance exists that a material misstatement could be missed. Thus, some risk always remains that the audit opinion will be wrong.

Through this post I am going to provides basic auditing technique in preparing statistical sampling. It is a step-by-step approach illustrated with some case examples. Enjoy!

### Types of Statistical Sampling Used by Auditors

**In general, there are two types of statistical sampling**:

**Attributes Sampling**– Attributes sampling estimates a percentage and is often used in the tests of controls because the auditor is interested in the error rate that has occurred in connection with a particular control activity. Auditor is attempting to determine if activity is functioning as intended.**Variables Sampling**– Variables sampling estimates a total and is often used in substantive tests where the auditor is attempting to corroborate a reported account balance.

### Steps to Take to Prepare Attributes Sampling Plan

**Auditor carries out several steps in following an attributes sampling plan:**

**Step-1. Anticipates the deviation rate for the control activity (or other application) being tested**. Expected rate is based on difficulty of activity, experience of person performing the control, results found in previous audits, etc. The more deviations that are anticipated the larger the sample must be.

**Step-2. Sets a tolerable deviation rate**. This is the maximum error rate that the auditor could tolerate and still believe the control activity was operating effectively and efficiently. As the rate that the auditor can tolerate gets smaller, sample size must get larger.

**Step-3. Sets allowable level of risk that the sample will be misleading**. For this particular testing, auditor is especially worried that the sample will look better than the actual population. In that case, auditor may think that the control activity is functioning properly and will set control risk too low and do less substantive testing than is needed. To reduce the level of risk that the sample will be misleading, the size of the sample must be increased.

**Step-4. Based on these three figures, a calculation or a chart is used to determine proper sample size**. Except with very small populations, the number of items in the population has little or no effect on the sample size.

**Step-5. The appropriate number of items is selected from the population**. Items must be picked randomly, each item should have an equal chance of selection.

**Step-6. Sample items are examined and the number of deviations (usually errors) is determined**. That number is restated as a percentage based on sample size.

**Step-7. Another chart is used to convert the actual deviation rate of the sample to the potential upper deviation rate of the population**. Difference between the actual rate and upper deviation rate is called the allowance for sampling risk.

**Step-8. Upper deviation rate is compared to maximum tolerable rate**. If upper deviation rate is lower, auditor assumes control activity is functioning as intended and will probably set control risk at a low level. If upper deviation rate is more, auditor assumes activity is not functioning as intended; control risk is high.

**Step-9. Auditor also examines both the cause and size of the errors that were found**. Even if rate is low, the type of deviation may indicate a serious problem.

Case ExamplesLet say you are an auditor, and are attempting to estimate a balance, you must estimate the amount of variability of the items in the population, aren’t you? For example, if one item is $12 and the next item is $989, there appears to be a high degree of variability among the items.

Should you select a smaller or larger sample?

When variability is high, you would then sample more items in order to get a representative sample.Let’s use another scenario; you are attempting to estimate a balance, if the variability of the items in the population is high, the sample size will be quite large.

In order to reduce this problem, should you choose to stratify the population so that the sample size will be lower?

Stratifying a population means that the larger items are pulled out and tested separately. The items that remain in the population will have a much smaller variability and, therefore, will necessitate a smaller sample size. The same effect can be achieved by using alternative methods of statistical sampling such as difference estimation, ratio estimation, and probability proportionate to size sampling.

### Steps to Carry Out in Variable Sampling Plan

**Auditor carries out several steps in a variables sampling plan. There are several variations but following is an example of classical mean-per-unit sampling**:

**Step-1. Sets the level of a tolerable misstatement**. This is the size of the largest misstatement in the account being examined that (when combined with misstatements in all other accounts) would still not cause the financial statements to be materially misstated. If auditor reduces the size of a misstatement that can be tolerated, a bigger sample will be required.

**Step-2. Sets allowable risk levels that sample will be misleading**. To reduce these risk levels, a larger sample size must be selected:

- A misleading sample can cause incorrect acceptance. There is a risk that a sample will substantiate client figure when the population is actually materially misstated. This problem leads to an unqualified opinion being given on statements that are not fairly presented.
- A misleading sample can cause incorrect rejection. There is a risk that a sample will not substantiate the client figure even though the population is not materially misstated. This problem leads to additional (unnecessary) testing being performed.

**Step-3. Estimates the amount of misstatement in the population**. This figure is usually affected by the efficiency of the accounting system and staff. The bigger the expected misstatement, the larger the sample size required. Again, population size has only a little impact on sample size.

**Step-5. Auditor takes a preliminary sample to estimate the variability of the items in the population**. If the items are all about the same amount, variability is low. Variability is measured by estimating the standard deviation. The higher the variability, the more items that have to be selected to get a representative sample.

**Step-6. The various factors are entered into mathematical formulas to determine the appropriate sample size**.

**Step-7. Auditor randomly selects items for the sample and measures the average value of these items**. The average of the sample is then extended to the entire population to give a total. Difference between this total and client figure is the projected misstatement. If projected misstatement is less than tolerable misstatement, the test has provided evidence that reported balance is fairly presented. If projected misstatement is more, the test has not provided evidence that balance is fairly presented.

**Case Examples**

**Assume you are an auditor and are performing a test to estimate an error rate**. The sample indicates that the error rate is 2% but statistical sampling indicates that the population rate could be as high as 3% (the upper deviation rate). You had previously established a tolerable error rate of 2.4%. **Because the sample error rate (2%) is below the tolerable rate (2.4%), may you assume that this particular control activity is operating effectively?**

**You may not!** **Why?** **Because the upper deviation rate (3%) is above the tolerable rate (2.4%), you should conclude that the control activity is not being performed effectively. The assessment of control risk will probably be set at a higher level**.

Let’s try another scenario…

Assume the sample indicates that the error rate is 2% but statistical sampling indicates that the population rate could be as high as 3% (the upper deviation rate). **Should the 1% difference between the sample rate (2%) and the upper rate estimated for the population as a whole (the 3% upper deviation rate) to be considered as the allowance for sampling risk?**

**Yes!** **Why?** **Because only a sample is being selected, a difference will exist between the rate determined for the sample and the upper deviation rate that is estimated for the population. That difference (1% in this case) is known as **“**the allowance for sampling risk**”.

In variables sampling, the variability of the items in the population can often be so great that sample size has to be large, causing the auditor to do extensive testing.

One alternative is to stratify the population. Items are divided into two or three separate populations based on size. Because they are grouped by size, the variability of each population will be relatively small and the required sample will be reduced. Two other alternatives are ratio and difference estimations. These methods do not estimate the average item in the population but rather the average of differences between book values and audited values or the ratio of book values to audited values.

### Using Proportional to Size Sampling [PPS Sampling]

**Probability proportional to size sampling (PPS sampling) is another way of estimating a total while keeping sample size small. The sampling unit is each dollar in the population and not each document. This approach is also called monetary unit sampling and dollar unit sampling**.

**Instead of selecting every -nth item (every fiftieth invoice, for example), auditor picks every -nth dollar (every $5,000th, for example, in a list of invoices)**. Although the nth dollar is selected, the entire document is tested. Here is the tumble of rules to follow:

- Bigger items have more dollars so they are more likely to get picked. Stratification and the degree of variability are not important.
- Sample size is determined mathematically based on tolerable misstatement, expected misstatement in population, allowable risk of incorrect acceptance, and expected number of errors.
- Total dollar figure of population is divided by sample size to get sampling interval (such as every $5,000th).

**If sample has no errors, population total is accepted. If misstatements are found, the size of a projected misstatement must be determined as follows**:

- For items that are larger than the interval [in this example, an invoice that is over $5,000] the amount of the misstatement is just used in arriving at the projected misstatement.
- For items that are smaller than interval, the percentage of the misstatement (called the tainting percentage) is determined and multiplied by the interval. If a $500 item has a 2% error, 2% is multiplied by sampling interval to get figure to include in projected misstatement.

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