In auditing and statistical sampling, the Upper Deviation Rate (UDR) is a measure used to determine the maximum rate of deviation from a prescribed control that an auditor is willing to accept. It represents the "worst-case scenario" for the population error rate based on the sample results and a specific confidence level.
The UDR Formula
The calculation of UDR is based on the Poisson distribution or Binomial distribution. The general formula is:
UDR = Reliability Factor (R) / Sample Size (n)
Where:
Reliability Factor (R): A factor derived from statistical tables (like the AICPA tables) based on the number of deviations found and the desired confidence level.
Sample Size (n): The total number of items inspected in the audit test.
How to Calculate UDR: Step-by-Step
Identify the Sample Size: Determine how many items were tested (e.g., 100 invoices).
Count Deviations: Record the number of errors or control failures found in that sample.
Select Confidence Level: Usually 90% or 95% in financial auditing.
Find the Reliability Factor: Use a statistical table or our calculator to find the factor associated with your deviations and confidence level.
Divide: Divide the factor by the sample size and multiply by 100 to get the percentage.
Practical Example
Imagine an auditor tests a sample of 50 purchase orders for proper authorization. They find 1 deviation (an unauthorized order). Using a 95% confidence level:
Sample Size (n) = 50
Deviations (d) = 1
Reliability Factor (R) for 1 deviation at 95% = 4.75 (approx)
UDR = 4.75 / 50 = 0.095 or 9.5%
If the auditor's Tolerable Deviation Rate was 7%, they would conclude the control is not operating effectively because the UDR (9.5%) exceeds the tolerable rate.
function calculateUDR() {
var n = parseFloat(document.getElementById('sampleSize').value);
var d = parseInt(document.getElementById('deviationsCount').value);
var cl = document.getElementById('confidenceLevel').value;
if (isNaN(n) || n <= 0) {
alert("Please enter a valid sample size greater than 0.");
return;
}
if (isNaN(d) || d n) {
alert("Number of deviations cannot exceed the sample size.");
return;
}
// Reliability Factors (R) based on Poisson Distribution Table for common audit values
// Row index = number of deviations (0 to 10)
var rFactors95 = [3.00, 4.75, 6.30, 7.76, 9.16, 10.52, 11.85, 13.15, 14.44, 15.71, 16.97];
var rFactors90 = [2.31, 3.89, 5.33, 6.69, 8.00, 9.28, 10.54, 11.78, 13.00, 14.21, 15.41];
var rFactors99 = [4.61, 6.64, 8.41, 10.05, 11.61, 13.11, 14.58, 16.00, 17.41, 18.79, 20.15];
var rValue = 0;
var tableToUse = [];
if (cl === "95") {
tableToUse = rFactors95;
} else if (cl === "90") {
tableToUse = rFactors90;
} else {
tableToUse = rFactors99;
}
// If d is within our pre-defined table, use it. Otherwise, use Chi-square approximation
if (d 10 using Chi-square inverse logic
// For 95% CL: R approx = (d + 1) + 1.645 * sqrt(d + 1)
// This is a simplified proxy for standard Poisson upper limits
if (cl === "95") rValue = d + 1 + (1.96 * Math.sqrt(d + 1));
else if (cl === "90") rValue = d + 1 + (1.645 * Math.sqrt(d + 1));
else rValue = d + 1 + (2.576 * Math.sqrt(d + 1));
}
var udr = (rValue / n) * 100;
// Display Result
document.getElementById('resultArea').style.display = 'block';
document.getElementById('udrValue').innerHTML = udr.toFixed(2) + "%";
var interpretationText = "Based on " + d + " deviation(s) in a sample of " + n + ", we are " + cl + "% confident that the actual population deviation rate does not exceed " + udr.toFixed(2) + "%.";
document.getElementById('interpretation').innerHTML = interpretationText;
// Scroll to result
document.getElementById('resultArea').scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}