## Statistical Forensics

#### One Digit Too Many!

When the original handwritten data contain only *three* decimal places, and the same data entered into a Microsoft Excel7 spreadsheet appear with *four* decimal places, there is cause for suspicion. Where did that extra digit come from? And when that fourth digit is either a *zero* or a *five*, (and no other digit, e.g., *one* or *nine*) the suspicion is that division by *two* produced the fourth decimal digit. Thus, division by *two* of a decimal number whose last digit is *odd* leaves a remainder of *one-half* which produces an extra decimal digit, *five*. In contrast, when the last digit is even, there is no remainder and the extra digit is *zero*. Division by two yields no other extra digits.

In one case, these clues led investigators of ORI's Division of Investigative Oversight (DIO) to determine that the data for a third rat were fabricated by averaging the data for two others. This case concerned a study of the effect of rhythmic contractions of skeletal muscle in the hind limbs of rats where blood flow was measured at rest and during nerve stimulation. Measurements of blood flow and muscle weight were recorded.

The respondent presented results for six rats in a lab seminar. Sometime after that, a co-worker discovered that the original data sheets for two rats were blank. The respondent furnished the university with copies of tracings of continuous measurements, handwritten recorded data for muscles for six rats, and printouts of a Microsoft Excel7 spreadsheet that contained blood pressure measurements and muscle weights for six rats.

DIO investigators concentrated on the measurements of muscle weight. The investigators observed that the entries in the spreadsheet for Rat-3 and Rat-6, purportedly copied from the handwritten data sheets, had an extra decimal digit. Therefore, the spreadsheet numbers had not been copied from the handwritten sheets. They further observed that the extra digit was either *five* or *zero.* This observation led to the hypothesis that the Rat-3 and Rat-6 measurements were the average of two measurements. Investigators then verified that the purported measurements for "Rat-3" and "Rat-6" were the averages of the corresponding measurements, respectively, for Rat-1 with Rat-2, and Rat-4 with Rat-5. This irrefutable demonstration that the weights for both Rat-3 and Rat-6 were fabricated by calculation facilitated the voluntary exclusion of the respondent from receiving Federal funds for 3 years.