Internal Revenue Bulletin:  2004-20 

May 17, 2004 

TECHNICAL FORMULAS

The formulas below are included to clarify the statistical sampling terms used and to ensure consistent application of the procedures described in the revenue procedure.

UNSTRATIFIED (SIMPLE RANDOM SAMPLE) MEAN ESTIMATOR STRATIFIED MEAN ESTIMATOR
Sample Mean of Audited Amounts
 
Estimate of Total Audited Amount
Estimated Standard Deviation of the Audited Amount
 
Estimated Standard Error of the Total Audited Amount
Achieved Precision of the Total Audited Amount
UNSTRATIFIED (SIMPLE RANDOM SAMPLE) DIFFERENCE ESTIMATOR STRATIFIED DIFFERENCE ESTIMATOR
Estimate of Total Difference
Estimate of Total Audited Amount
Estimated Standard Deviation of the Difference Amount
 
Estimated Standard Error of the Difference Amount
Achieved Precision of the Difference Amount
UNSTRATIFIED (SIMPLE RANDOM SAMPLE) RATIO ESTIMATOR STRATIFIED COMBINED RATIO ESTIMATOR
Estimated Ratio of Audited Amount to Recorded Amount
Estimate of Total Audited Amount
Estimated Standard Deviation of the Ratio
 
Estimated Standard Deviation of the Ratio in i th Stratum
Estimated Standard Error of the Ratio Amounts
Achieved Precision of the Ratio Amounts
UNSTRATIFIED (SIMPLE RANDOM SAMPLE) REGRESSION ESTIMATOR STRATIFIED COMBINED REGRESSION ESTIMATOR
Estimated Regression Coefficient
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Estimate of Total Audited Amount
Estimated Standard Deviation of the Regression Amounts
 
Estimated Covariance between the Audited and Recorded Amounts in i th Stratum
 
Estimated Standard Deviation between the Audited and Recorded Amounts in i th Stratum
 
Estimated Standard Error of the Audited and Recorded Amounts
Achieved Precision of the Audited and Recorded Amounts
Definition of Symbols  
TERM DEFINITION
n Sample Size
N Population Size
x The value of the sampling unit that is being used as the primary variable of interest. In audit sampling, this would be the audited (or revised) value of the transaction.
y The value of the sampling unit that is being used as the “paired” variable that is related to the variable of interest. In audit sampling, this would be the reported (or original) value of the transaction.
d The value of the sampling unit that is the difference between “paired” variable (y) and the variable of interest (x). That is, d = x - y. In audit sampling, this would be the difference (or the change) of each transaction’s value.
X The total value of the primary variable of interest. In audit sampling, this would be the estimated total audited value of the population. Typically, this value is not known for the entire population and is estimated based on the probability sample selected.
Y The total value of the variable that is paired with variable of interest. In audit sampling, this would be the total reported value of the population. Typically, this value is known for the entire population and may be estimated based on the probability sample selected.
D The total value of the difference between the “paired” variable and the variable of interest. In audit sampling, this would be the estimated total difference of the population. Typically, this value is not known for the entire population and is estimated based on the probability sample selected.
UR The confidence coefficient which is based on either the Student’s t-distribution or the normal distribution. For example, a 95% one-sided confidence coefficient based on the normal distribution is 1.645. This term is often referred to as the t-value and the z-value.

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