# The Deceptiveness of Percent Effectiveness

## Microbiology

Bacteria grow exponentially. One becomes two, two becomes four, four becomes eight, etc. This is also sometimes referred to logarithmic growth or bacterial doubling (in reality there are multiple phases to bacterial growth lag, log, stationary, and death, but I am ignoring these for sake of simplicity). The upshot of this is that bacterial populations can reach very large numbers very quickly. See link below for more on this.

## Large Numbers

Five million is a large number. It is a 5 followed by six zeroes or 5,000,000. Large numbers like this are often written in a different way, 5 x 10⁶ (five times ten to the sixth), which is 5 times 1 million, which is 5 million (5 x 10⁶). In microbiology bacterial populations are almost always written out in this format because they span a very large range of numbers from as low as 1 (1 x 10⁰) to 1000 (1 x 10³) to 100,000 (1 x 10⁵) to one billion (1 x 10⁹) or even higher in some rare cases.

(note: You may also sometimes see these numbers written as for example 5e6. eX means the same as 10 to the X power. I can’t remember actually why this is the case, but I am sure it makes total sense and is not in the least bit confusing or stupid).

In case you were wondering the units we use in microbiology to express bacterial counts are called colony forming units (CFU) and these are most often shown per milliliter or (CFU/ml). Basically the CFU/ml number tells you how many bacteria can be grown and counted on an agar plate out of one milliliter of liquid. No need to delve into the details here and all you need to know for purposes of this discussion is that bacteria counts/populations are typically expressed as CFU/ml and can be very, very large numbers.

## Percentages

A percent of something is 1/100th of that thing. One per cent is one per hundred. Stupid right?

How is a cent a hundred (I guess because a penny is one one hundredth of a dollar) and why does 1/100th of a thing get its own special designation that everyone talks about when 1/10th of a thing or 1/1000th of a thing does not? Great question but it is what it is.

The one and the one hundred can be anything, and I mean anything, solid, liquid, gel, made up fictional entity, anything). For example, 1g in 100ml is 1%, 1 apple in 100 apples is 1%, 1 bacteria in 100 bacteria is 1%, etc. 1Q from Star Trek TNG in 100 Q is 1%. By the same token 50g in 100ml is 50%, as is 50 apples in 100 apples, and 50 bacteria in 100 bacteria. The two things don’t have to be the same either. 50 apples in a mix of 100 apples and oranges is still 50% of the total fruit present. Importantly, it also does not have to be out of a hundred though it is always broken down and expressed as such, 250 apples out of 500 apples is still 50%, 50 out of every 100 is another way you could say the same thing. A percent is essentially just a way of expressing how big a proportion out of one hundred something is. It is also confusing as all get out and really stupid as I already mentioned above. However, for whatever reason, people are “used to” seeing and working with percentages. At least they think they are and for many things like 25% off discounts they work just fine and are fairly straightforward and not deceptive in the least. When it comes to microbiology and large numbers however they can be very misleading.

Get out your calculator and actually plug in the numbers. Type a five than type six zeroes then hit the times button then hit zero then hit the little point button that looks just like a period, then nine three times, 5,000,000 x 0.999 (remember 99.9% is 99.9/100 or 0.999). After hitting the equal button the result will be 4,995,000 (4.995 x 10⁶).

Now what is the difference between 5 x 10⁶ (the number of bacteria we started with) and 4.995 x 10⁶ (the number of bacteria we killed)? That difference represents the number of bacteria still remaining. At first blush you would think that killing that huge a number of bacteria would surely eliminate any potential risk of infection or illness and there must be almost no bacteria remaining alive. However, if you actually do the math

(get out your calculator again and do it. Seriously, it’s way easier that way)

, that number is 5,000 (5 x 10³). There are still 5,000 bacteria remaining even with a 99.9% effective antimicrobial. Remember what I said above in the microbiology section about the infectious dose for many common human pathogens, Five thousand is more than enough to get you sick. What if we do the same math for a 99.99% effective antimicrobial? The situation improves, but not by as much as you might think. In this case there would be 500 bacteria remaining, and even at 99.999% you still would have 50 bacteria remaining (in both cases for some pathogens that is still enough to make you sick).

## Logarithms

Good old logarithms. The Eurythmics of math.

I never thought it would be possible to hate a log until I was introduced to logarithms in math class in 7th or 8th grade (I can’t remember which likely because the memories are so traumatic or possibly because I blacked out from boredom and did not wake up again until a grade later.) Does the period go inside or outside of the parentheses? I can never freakin remember that for some reason. Probably because of all those damn logarithms. Stupid logarithms.

In any event, in addition to inducing comas, logarithms can be helpful in understanding percent effectiveness claims and microbiological populations. This is because every log10 (note: this applies to log base 10 only and the little 10 is supposed to be subscript but I cant figure out how to do a subscript in Medium’s text editor so fuck it. There are other log base values possible but for the most part they are useless and stupid and so I ignore them.) change represents a ten fold change in something. A change from 1 to 10 is a one log increase, 1 to 100 a two log increase, 1 to 1000 three logs, etc. and so on. The same applies in the reverse, a change from 1000 to 1 is a three log decrease.

Time to see what you have learned. Nothing? Crap. Oh well I will just go ahead and tell you what you should have learned then by discussing the example above of the 99.9% effectiveness claim.

A change from five million (5,000,000) to five thousand (5,000) is a three log decrease. Were you able to figure that out on your own? No? Congratulations you are a normal person that sucks at logarithms. A 99.9% reduction translates to a three log reduction.

Can you guess how man logs a 99.99% reduction translates to? Maybe you should try calculating or thinking instead of guessing. That usually works better. The answer is four you big dummy.

99.999% is a five log reduction, etc. and so on. At this point, if you are still awake, you might be asking yourself where is all of this heading. It’s heading straight to hell at warp factor seven. Nah. It is actually heading to the following point. Remember what I said way up in the microbiology section, or maybe the large numbers part, I can’t actually remember as it was so long ago. In any event I said something somewhere about bacterial population numbers being large, very large, ranging from 1 to 1 billion or higher. Populations of a million are commonplace. How many logs are there in a million smart guy or gal? Huh? It should be obvious unless you are a dullard that there are six logs in a million. You didn’t get that? Because you are a dullard probably.

Taking three logs out of six logs still leaves three logs (1000) remaining and take my word for it three logs is a lot of logs, plenty of logs to make you sick. It’s so obvious even a PhD in micro and molecular biology like myself can understand it. What’s your problem? Dummy.

And there you have it. I guess. What was I talking about again? Oh yeah, percent effectiveness claims are very misleading. Get it?

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