Internal Revenue Bulletin: 200420 
May 17, 2004 
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. 
U_{R}  The confidence coefficient which is based on either the Student’s tdistribution or the normal distribution. For example, a 95% onesided confidence coefficient based on the normal distribution is 1.645. This term is often referred to as the tvalue and the zvalue. 
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