Confessions of a Recovering Auditor OR Ye Shall Know the Formula, and the Formula Shall Set Ye Free!

By Bruce Truitt

The Formula

Welcome back, Dear Reader.

I’m here to chat about “The Formula.” Our first installment in these tales of woe and intrigue (q.v., “Is Statistics a Criminal Act?“) ended with your humble scribe positing that “the seed, the primum mobile, the Alpha-Omega, the thread, the DNA, the glue that ties it all together” was the following, AKA “The Formula”:

Sample Size = Confidence x Variation
Precision

In Part One, we also established that a whole passel of folks sprinted from their final “Sadistics” exam to the nearest trading post to turn their textbook into cash (and immediately thereafter into either anti-inflammatories or an adult beverage), assuming that the dumpster wasn’t too seductive en route, and noting that “passel” really should be officially recognized by the National Bureau of Standards.

Yet, those of you who avoided this rush to rubles and somehow found the Herculean stamina required to take Statistics in multiple collegiate departments (“Inconceivable!” quoth Wallace Shawn) found yourselves in Dante’s eighth level of academic Hell. You likely discovered that statistics terminology varies by department and—will the fun never stop?—by textbook or author. Next time the Ambien fails, compare statistical revelations from engineering, physics, psychology, sociology, math, education, and business textbooks. Vertigo by variation is assured! Even limiting ourselves to the parlance of auditing texts is daunting:

• “Risk of over-reliance” AKA “Confidence level” AKA “Alpha”
• “Estimated deviation rate” AKA “Expected population attribute error percentage” AKA “Standard deviation” AKA “Standard error, nee Relative error”
• “Tolerable error rate” AKA “Upper error limit” AKA “Desired precision” AKA “Margin of error” AKA “Confidence interval”

With this many aliases, crimes must be afoot, or, at least, quests for immortality in a footnote.

(This part of the page deliberately blank for deep, cleansing breaths.)

Anyhoo, fact is all these terminologies do collapse into one of the three words in “The Formula”—confidence, variation, or precision. Repeat with me: Confidence. Variation. Precision.

Or, even more logically:

Sample Size = How Confident Must I Be In My Results x
How Much Variation Is In The Population
How Precise Do I Want To Be

Or, to totally eliminate the lingering stench of math:

The Amount Of Work I Gotta Do = How Often I Wanna Be Right x
How Screwy The World Is
How Close I Wanna Be To The Bullseye

So, let’s play with The Formula a bit. For example, what happens to sample size if confidence goes up, i.e., if you want to be right more often?

Sample Size ↕ ? = Confidence ↑ x Variation
Precision

Right you are! Sample size goes up if you want to be more confident in your results. That’s reasonable with or without math. You gotta do more work to be right more often.

Okus dokus. What happens to sample size if variation in the population increases, that is, if the world gets more screwy?

Sample Size ↕ ? = Confidence x Variation ↑
Precision

Right again—you’re good! The sample size must again go up. This also makes sense. If the stuff you are sampling turns out to be more messed up than you thought it would be, you have to sample more to figure out how screwy it really is.

Then, what happens to sample size if you want to be more precise, i.e., the numeric value of precision goes down. Does sample size go up or down if you want to get closer to the bull’s eye?

Sample Size ↕ ? = Confidence x Variation
Precision ↓

No math needed here either. Sample size has to rise. If you wanna get closer to the bulls eye, you gotta fire more arrows (sample more). Take it from one who rarely hit the blasted bull’s eye and often missed the whole target.

All this makes perfect logical sense and is easily understood without those pesky Greek and Latin letters and {formulas [formulas (formulas) formulas] formulas}.

More confidence—more work. More variation—more work. More precise—more work. Period.

It does not matter whether you think about The Formula logically, linguistically, or mathematically, as long as you think about, speak it, share it, dream it, mantra it … without ceasing until next we speak …

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision

Confidence – Variation – Precision …

About Bruce Truitt

Bruce Truitt has 25+ years’ experience in applied statistics and government auditing, with particular focus on quantitative methods and reporting in health and human services fraud, waste, and abuse. His tools and methods are used by public and private sector entities in all 50 states and 33 foreign countries and have been recognized by the National State Auditors Association for Excellence in Accountability.

He also teaches the US Government Auditor’s Training Institute’s “Practical Statistical Sampling for Auditors” course, is on the National Medicaid Integrity Institute’s faculty, and taught Quantitative Methods in Saint Edward’s University’s Graduate School of Business.

Bruce holds a Master of Public Affairs from the LBJ School of Public Affairs, as well as Masters’ Degrees in Foreign Language Education and Russian and East European Studies from The University of Texas at Austin.