Casualty numbers seem to be a popular thing to taunt around in pop-history and often the first number people memorize about a battle or war, next to the size of armies. But, from a historical perspective, they are arguably one of the lesser interesting statistics. Or put more accurately: "They are among the most superficial and least useful of statistics for purposes of analyzing a conflict in detail. The reasons for this are numerous; first off:

They are by no means reliable. Casualties are frequently recorded as being far larger or smaller by the sources than they really were, as are army sizes. This can be because the source has purposefully inflated/deflated the numbers of either side due to personal motivations, or it could just be a general case of imperfect information, where the source of the statistics really did try his hardest to get accurate numbers, but was fed wrong information himself, or made some other mistake (or both!). The latter -imperfect information- is by no means to be underestimated or seen as limited to old conflicts. Conflicts are a messy affair and large conflicts doubly so. Casualty numbers in WW2 operations for example, are disputed or rounded to (tens of) thousands almost without exception. You mention the battle of Stalingrad and that makes me so happy, because it's an excellent example of a time period where the administrative capabilities were, on paper, certainly there to manage numbers in the millions, yet if we take a look at what several sources have to say about it we see all kinds of numbers!^1; as low as 150,000 and as high as 800,000 (interestingly, the 900,000 you mention, isn't mentioned in these at all). The Soviets have even more outrageous variance, with numbers as low as 140,000 and up to 2 million. So what's going wrong here? Are these number just wrong? No, they aren't necessarily wrong. A great amount of variance stems from the lack of unitary definitions. If we really want numbers to line up correctly we need to have some questions answered consistently:

1. What measure is a casualty?

2. What measure is a soldier?

3. What measure is a battle/operation/war?

These questions are not answered consistently across the sources on the page I linked to. Some record only kills, some captured OR deserted OR wounded OR killed. Some simply record people as 'lost'. Some are estimates based on the actual number of physical corpses counted after a battle. Most rely on the statistics that the army itself keeps, but some others rely on things like the national record-keeping to track losses! This can obviously lead up to enormous errors all on its own if handled improperly and there are countless of instances of baaad history on the internet and other media where an enthusiastic journalist or whatever tried to get his sources right but did not differentiate between one that mentions confirmed combat kills and one that includes frostbite and hunger and desertion....

... and civilians. Because civilians can also screw things up a lot. Some conflicts are more clear than others and in theory the second question could simply be answered with something like "only enlisted soldiers" or "anything in a uniform". But for something like the Battle of Stalingrad that breaks down so hard it's not even funny. You might think that that old grandma limping around in a dress with her aching back is a civilian, but she secretly blew up several tanks, so should we count her as a Soviet casualty too? In a total war like the one on the East front every man and woman and regrettably even child was potentially a combatant, but discriminating this effectively afterwards is of course very hard and made harder by the fact that many combatants might be (deliberately) left off any paper records. As a side note, "What measure is a soldier" also extends to the rather morbid question of "Is this arm a casualty?". Which is a concern when bodies get mangled beyond all recognition.

Skipping the third question makes things very blurry very quickly. A single sortie or short skirmish might be reasonably clearly defined, but for something like the Battle of Stalingrad which wasn't a single battle of course but a whole series of offensive and defensive operations, this becomes nigh impossible. Casualties aren't exactly timestamped either, so it's hard to tell whether some corpse they found beneath the snow on the outskirts of Stalingrad was a soldier who got there in the midst of the defensive or happened to get lost and frozen to death on a drunken stroll a year earlier. And yet another common mistake is simply the various nationalities and sides and such. How many times is German (nationality) confused with with German (ethnicity) or Axis (the faction, including Hungarians, Italians et cetera)? Same for partisan activity. So you found a Polish corpse with a rifle next to it in Poland in WW2. Was he fighting for the Polish government during the invasion? Or the government in exile? Or was he a partisan who helped the Soviets? Should we count him as a Soviet casualty then? I deliberately use singular examples, but the same applies on a large scale just as well. What is included?

Lastly, there's something to consider, which is that the big statistics of a complete war are pieced together from smaller reports/sources (not always. They can also be estimates based on a larger census or some other large record), which are in turn based on smaller sources/reports. Each of these can and will make mistakes and each of these can and most will forget to define one of the aforementioned three questions, leaving it up to the analyst or historian to consider if he should include the data for something and how to include that data.^2

Now I don't want to get stuck in the problems involved with getting a casualty number, only to completely skip over your actual question, which was about using whatever best estimate we got to assess army effectiveness. So on to that! Battle casualties alone are not a great way to assess an army's effectiveness in a war except on the most superficial level. This is assuming the number is clearly defined as for example by the NATO^3, as

"Any casualty incurred as the direct result of hostile action, sustained in combat or relating thereto or sustained going to or returning from a combat mission"

and for the sake of simplicity, the battle itself was clearly defined as well, without any civilian involvement. Why not? Because armies are more than a number of soldiers. Take Operation Barbarossa for example. At first glance, just looking at the battle casualties, the invasion was a staggering succes, with several millions of battle casualties for the Soviets and much fewer for the Axis forces, and that'd raise the question why the Germans didn't steamroll all that remained. But taking a closer look, it's less of a surprise it didn't turn out that way! Because at the start of the invasion in June, the Germans had around 400,000 reservists, and in October that same year, they had lost all of their reserves, and they lacked another 340,000. Taking a closer look at what constitutes these 'battle casualties', we see that the Soviet casualties were mostly caused by encirclements where everyone was killed or captured. This includes many, many soldiers that normally don't do any actual fighting. These are all soldiers and all considered battle casualties! Many of them may have carried a weapon, but were by no means the real fighting force. The Germans on the other hand suffered no big encirclements early on, hence their battle casualties mostly consist of the actual, hard core of soldiers that were actively fighting^4. Turns out that of those 'only' 740,000 the Germans lost, most were the backbone of the experienced infantry. The replacements, if they came in at all, were not at nearly the same effectiveness of the soldiers they replaced. In addition, they lost staggering numbers of trucks and other critical material and as such the effective fighting strength of the German army was reduced much further in Barbarossa than mere battle casualties would indicate. So the effects of the invasion are radically different from what is superficially implied by the kill-death-ratio.

part 2 below