For the battle of Gettysburg, the southern reports were systematically low by about 50%. My interpretation is that Lee falsified the reports so the war would continue.
http://gburginfo.brinkster.net/Casualties.htm
For the Confederate Army of Northern Virginia, the return of casualties is far more problematic. The official casualty return compiled by Lee’s medical director in late July 1863 was manifestly incomplete: a total of 20,451, comprising 2,592 killed in action, 12,709 wounded and 5,150 missing. It was apparent at the time that this was not a complete count: the trauma of defeat and retreat and the loss of many officers prevented a proper assessment: many units grossly under reported their losses, and in some cases the returns included none of the “missing”, and when the missing were counted, it was only too often the case that they were in reality either dead or wounded . In the absence of complete official returns, historians have attempted to estimate Confederate casualties at Gettysburg, using different methods of compilation. Prominent among these was Livermore, who, in his widely used essay NUMBERS AND LOSSES, proffered a total of 28,063 rebel casualties at Gettysburg, including 3,903 killed, 18,735 wounded and 5,425 unwounded prisoners. His method was simple but controversial…….the report of the US Provost Marshall gave a count of 5,425 unwounded and 6,802 wounded Confederate prisoners in Federal hands, these men being captured between July 1st and July 5th. Livermore then cites the report of the medical director of the Army of Northern Virginia, which admitted that only 776 wounded men were left behind. Extrapolating from this, Livermore concluded that this indicates that 6,026 Confederate wounded were not reported, and should be added to the 12,709 that were counted, thereby increasing the number of wounded to 18,735. Using the ratio of killed to wounded that pertained to the Federal army in the battle, which amounted to 4.6 wounded to every one killed in action, Livermore applies this to the extra 6,026 Confederate wounded, and comes up with an additional 1,311 killed, increasing the officially reported number of 2,592 killed in action to 3,903, a rise of over fifty per cent.
Lee didn't have the luxury of time, resources or hindsight that Krick had when he compiled his numbers "using newspaper reports and other sources".
However, from the link you provided:
"The summation provided by Krick raises the total of Confederate dead to about four and a half thousand, and the wounded to roughly twelve and a half thousand, but, of course, this is an incomplete total.Unfortunately, some accounts adopt the total of approximately four and a half thousand rebel deaths and present them as killed in action, thereby suggesting that Confederate losses in killed exceeded Union losses in this category by roughly fifty per cent….the fact that this figure allows for mortally wounded is apparently not understood, or ignored .This is a prime example of the most excellent statistical research being used improperly." (emphasis added)