# Variable sampling mean per unit estimation ratio estimation difference estimation CPA exam AUD

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Variables sampling, like MUS, is a statistical method that auditors use. Variables sampling and nonstatistical sampling for tests of details of balances have the same objective—to measure the misstatement in an account balance.  As with nonstatistical sampling, when auditors determine that the misstatement amount exceeds the tolerable amount, they reject the population and take additional actions.
Several sampling techniques make up the general class of methods called variables sampling: difference estimation, ratio estimation, and mean-per-unit estimation.
Differences Between Variables and Nonstatistical Sampling
The use of variables methods shares many similarities with nonstatistical sampling. All 14 steps we discussed for nonstatistical sampling must be performed for variables methods, and most are identical. Some of the differences between variables and nonstatistical sampling are examined after we discuss sampling distributions.
Statistical Inference
Naturally, when samples are taken from a population in an actual audit situation, the auditor does not know the population’s characteristics and, ordinarily, only one sample is taken from the population. But the knowledge of sampling distributions enables auditors to draw statistical conclusions, or statistical inferences, about the population.

Difference Estimation
Auditors use difference estimation to measure the estimated total misstatement amount in a population when both a recorded value and an audited value exist for each item in the sample, which is almost always the case in audits. For example, an auditor might confirm a sample of accounts receivable and determine the difference (misstatement) between the client’s recorded amount and the amount the auditor considers correct for each selected account. The auditor makes an estimate of the population misstatement based on the number of misstatements in the sample, average misstatement size, individual misstatements in the sample, and sample size. The result is stated as a point estimate of the population misstatement plus or minus a computed precision interval at a stated confidence level.
Difference estimation frequently results in smaller sample sizes than any other method, and it is relatively easy to use. For that reason, difference estimation is often the preferred variables method.
Ratio Estimation
Ratio estimation is similar to difference estimation except the auditor calculates the ratio between the misstatements and their recorded value and projects this to the population to estimate the total population misstatement.
Ratio estimation can result in sample sizes even smaller than difference estimation if the size of the misstatements in the population is proportionate to the recorded value of the population items. If the size of the individual misstatements is independent of the recorded value, difference estimation results in smaller sample sizes. Most auditors prefer difference estimation because it is somewhat simpler to calculate confidence intervals.
Mean-per-Unit Estimation
In mean-per-unit estimation, the auditor focuses on the audited value rather than the misstatement amount of each item in the sample. Except for the definition of what is being measured, the mean-per-unit estimate is calculated in exactly the same manner as the difference estimate. The point estimate of the audited value equals the average audited value of items in the sample times the population size. The computed precision interval is calculated on the basis of the audited value of the sample items rather than the misstatements.
Stratified Statistical Methods
As we discussed earlier in this chapter, stratified sampling is a method of sampling in which all the elements in the total population are divided into two or more subpopulations. Each subpopulation is then independently tested. The calculations are made for each stratum and then combined into one overall population estimate for a confidence interval of the entire population. The results are measured statistically. Stratification is applicable to difference, ratio, and mean-per-unit estimation, but is most commonly used with mean-per-unit estimation.
ARIA is the statistical risk that the auditor has accepted a population that is, in fact, materially misstated. ARIA is a serious concern to auditors because of the potential legal implications of concluding that an account balance is fairly stated when it is misstated by a material amount.
Acceptable risk of incorrect rejection (ARIR) is the statistical risk that the auditor has concluded that a population is materially misstated when it is not.