Total instances where the control procedure failed.
5% (High Assurance)
10% (Moderate Assurance)
Acceptable risk that the sample supports the conclusion that the control is effective when it is not.
Sample Deviation Rate:–
Upper Deviation Limit (UDL):–
This is the maximum projected population deviation rate.
Allowance for Sampling Risk:–
Understanding Tolerable Deviation Rate in Auditing
In the field of auditing and internal control testing, the Tolerable Deviation Rate (TDR) is a critical parameter defined during the planning phase of attribute sampling. It represents the maximum rate of deviation from a prescribed control procedure that the auditor is willing to accept without altering the planned assessed level of control risk.
While the TDR is usually a fixed benchmark set by the auditor based on professional judgment (often between 3% and 10%), the calculation that occurs after testing involves determining the Upper Deviation Limit (UDL). This calculator helps auditors perform that evaluation by computing the statistical upper limit based on the sample results.
How to Calculate Sample and Upper Deviation Rates
To evaluate the results of an audit sample, three key metrics are derived:
Sample Deviation Rate: The simple percentage of errors found in the sample. Formula: Number of Deviations / Sample Size
Upper Deviation Limit (UDL): A statistical projection that adjusts the sample rate to account for sampling risk. This figure represents the "worst-case scenario" for the population at a given confidence level.
Allowance for Sampling Risk: The difference between the UDL and the Sample Deviation Rate.
Using the Calculator
This tool uses the standard AICPA reliability factors (based on the Poisson distribution approximation) to calculate the Upper Deviation Limit. Here is how to interpret the inputs:
Sample Size (n): Enter the total number of invoices, checks, or items examined.
Deviations Found (k): Enter the count of items that failed the specific control test (e.g., missing signature).
Risk of Overreliance: Select your confidence requirement. A 5% risk corresponds to a 95% confidence level, while a 10% risk corresponds to a 90% confidence level.
Interpreting the Results
Once you calculate the Upper Deviation Limit (UDL), compare it to your pre-defined Tolerable Deviation Rate (TDR):
If UDL ≤ TDR: The sample results support the planned assessed level of control risk. You can rely on the control.
If UDL > TDR: The sample results do not support the planned reliance. The control may be ineffective, and substantive testing may need to be increased.
Why Simple Division Isn't Enough
Simply dividing the deviations by the sample size (Sample Deviation Rate) is insufficient for auditing purposes because it does not account for sampling risk. Even if 0 deviations are found in a sample of 25, the true population deviation rate could still be higher. The Upper Deviation Limit calculation provides the statistical buffer necessary to make a conclusion with the desired level of confidence.
function calculateDeviationRates() {
// 1. Get Inputs
var sampleSizeInput = document.getElementById('sampleSize');
var deviationsInput = document.getElementById('deviationsFound');
var riskInput = document.getElementById('riskOfOverreliance');
var n = parseFloat(sampleSizeInput.value);
var k = parseFloat(deviationsInput.value);
var risk = parseInt(riskInput.value);
// 2. Validation
if (isNaN(n) || n <= 0) {
alert("Please enter a valid Sample Size greater than 0.");
return;
}
if (isNaN(k) || k n) {
alert("Number of deviations cannot exceed the sample size.");
return;
}
// 3. Define Reliability Factors (AICPA Table approximations for Poisson/Attribute Sampling)
// Index represents number of deviations (0 to 10)
var factors5Percent = [3.0, 4.8, 6.3, 7.8, 9.1, 10.5, 11.8, 13.2, 14.6, 15.9, 17.3];
var factors10Percent = [2.3, 3.9, 5.3, 6.7, 8.0, 9.3, 10.5, 11.8, 13.0, 14.3, 15.5];
var rFactor = 0;
// Logic to select factor
var selectedFactors = (risk === 5) ? factors5Percent : factors10Percent;
if (k 10 if not in table
// Approximation: Factor ~= Deviations + small buffer based on risk, or alert user.
// For robust calc, we alert user or use simple linear extrapolation for demo purposes
// Let's use a basic linear extrapolation based on the last gap
var lastFactor = selectedFactors[selectedFactors.length – 1];
var secondLast = selectedFactors[selectedFactors.length – 2];
var gap = lastFactor – secondLast;
var extraDeviations = k – (selectedFactors.length – 1);
rFactor = lastFactor + (extraDeviations * gap);
}
// 4. Calculations
// Sample Deviation Rate %
var sampleRate = (k / n) * 100;
// Upper Deviation Limit (UDL) % = (Reliability Factor / Sample Size) * 100
var udl = (rFactor / n) * 100;
// Allowance for Sampling Risk %
var samplingRisk = udl – sampleRate;
// Cap results at 100% just in case
if (udl > 100) udl = 100;
if (samplingRisk < 0) samplingRisk = 0;
// 5. Display Results
document.getElementById('sampleRateResult').innerHTML = sampleRate.toFixed(2) + "%";
document.getElementById('udlResult').innerHTML = udl.toFixed(2) + "%";
document.getElementById('samplingRiskResult').innerHTML = samplingRisk.toFixed(2) + "%";
// Logic for interpretation message
var msgDiv = document.getElementById('evaluation-msg');
if (k === 0) {
msgDiv.innerHTML = "With 0 deviations found, your UDL is driven entirely by sampling risk.";
msgDiv.style.color = "#0056b3";
} else {
msgDiv.innerHTML = "Sampling risk accounts for " + ((samplingRisk/udl)*100).toFixed(0) + "% of your Upper Limit.";
msgDiv.style.color = "#495057";
}
document.getElementById('results-area').style.display = 'block';
}