Classical variables sampling: mean-per-unit, difference and ratio

In this lesson you will

Choose between classical variables sampling and MUS.
Compute a mean-per-unit sample size from population size, standard deviation and allowance.
Evaluate an estimate against book value using the achieved precision interval.

What classical variables sampling is

Classical variables sampling (CVS) uses normal distribution theory to estimate a population value or misstatement from a random sample. Each item, not each dollar, is the sampling unit. It produces a point estimate and an interval, the precision, at a chosen confidence level.

Three common estimators:

Mean-per-unit (MPU). Estimate the population's audited value as the sample mean audited value times the number of items.
Difference estimation. Estimate total misstatement as the average difference between audited and book values times the number of items.
Ratio estimation. Estimate the population's audited value as the ratio of total audited to total book value in the sample, times the population book value. The difference from book value is the estimated misstatement.

AU-C 530.05 classifies any approach with random selection and a statistical evaluation that measures sampling risk as statistical sampling. CVS meets both.

When to choose CVS over MUS

Situation	Better fit
Testing understatement or both directions	CVS
Many zero or negative balances	CVS
Many misstatements expected	CVS (difference or ratio)
Few misstatements expected, overstatement focus	MUS
Highly skewed population with no stratification	MUS, or CVS with stratification

CVS needs an estimate of the population's variability. High variability drives large samples, so stratification by size is common. AU-C 530.05 defines stratification as dividing a population into subpopulations of similar units. The stratified sample selector splits key items from the remainder.

Sample size for mean-per-unit

The unstratified MPU formula, ignoring the finite population correction, is:

n = (N x Z x SD / A) squared

where N is the number of items, Z is the confidence coefficient (1.96 for 95% two-sided), SD is the estimated standard deviation of item values, and A is the desired allowance for sampling risk. The allowance is set from tolerable misstatement, the risk of incorrect acceptance and the risk of incorrect rejection; sampling texts, including the AICPA Audit Guide Audit Sampling, show the relationship. Get SD from a pilot sample or prior-year data.

Evaluation

After testing:

Compute the sample mean audited value and the sample standard deviation.
Point estimate = N x sample mean.
Achieved precision = Z x N x sample SD / square root of n (with a finite population correction when n is a large share of N).
Interval = point estimate plus or minus achieved precision.

If book value falls inside the interval and the interval's limits are within tolerable misstatement of book value, you accept. If book value falls outside, the projected misstatement is the difference between book value and the point estimate, and you consider whether the population is materially misstated; see projecting and evaluating results.

Practical points

Use the classical variables sampling calculator rather than hand formulas once you understand them.
Difference and ratio estimation need enough differences in the sample to be reliable; with very few misstatements, they break down and MPU or MUS is better.
Document the estimator, confidence level, SD source, and the computation of the interval.

Common findings

Sample size computed with an SD from an unrelated population.
Precision not recomputed with the actual sample SD.
Book value compared only to the point estimate, ignoring the interval.

Worked example

Mean-per-unit on inventory

Granite Falls Supply Co. has 4,000 inventory items with a book value of $3,200,000. A pilot sample suggests a standard deviation of $150 per item. Tolerable misstatement is $90,000. The team wants 95% confidence (Z = 1.96) and an allowance for sampling risk of $40,000.

Sample size: n = (4,000 x 1.96 x 150 / 40,000) squared = (29.4) squared = 864.36, rounded up to 865.

Results: the 865 items have an average audited value of $795 and a sample standard deviation of $150.

Point estimate: 4,000 x $795 = $3,180,000. Achieved precision: 1.96 x 4,000 x 150 / square root of 865 = 1,176,000 / 29.41 = $39,985. Interval: $3,180,000 plus or minus $39,985 = $3,140,015 to $3,219,985.

Book value of $3,200,000 falls inside the interval, and the farther limit, $3,140,015, is $59,985 from book value, within tolerable misstatement of $90,000. Projected misstatement is $3,200,000 - $3,180,000 = $20,000 overstatement. The team accepts the balance, records the $20,000 projected misstatement on the summary schedule, and notes that the finite population correction, which would narrow the interval, was not applied.

Self-check

Answer each question before you open the answer.

Give two situations where classical variables sampling fits better than MUS.
Show answer
Any two of: testing understatement, populations with many zero or negative balances, or populations where many misstatements are expected.
What drives the size of a mean-per-unit sample most?
Show answer
The population’s standard deviation relative to the allowance for sampling risk. Sample size grows with the square of SD divided by the allowance, so stratification that reduces SD can shrink the sample sharply.
Book value falls outside the computed interval. What does that indicate?
Show answer
At the chosen confidence, the population is likely misstated. The difference between book value and the point estimate is the projected misstatement to evaluate against tolerable misstatement.

Sources

Back to the syllabus

Standards are paraphrased from the primary sources listed above; read the source before relying on a requirement. This lesson is training material, not professional advice. Report an error