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Comment by jefftk | Hacker News Reader

The article explains it well:

Here’s a thought experiment: Take a 1000 cubic feet room and a purifier that processes 100 cubic feet of air per minute. (I follow Wirecutter in using vulgar imperial units.) Assume pessimistically that all particles are the worst-case size. If you run that purifier with an E12 filter, the fraction of particles that will remain after one minute is

     .1 × (1-.995) + .9 = 0.9005.

That’s because 10% of the air goes through the purifier and has 99.5% of particles removed, while 90% of the air doesn’t go through the purifier at all.
Meanwhile, if you run that purifier with an H13 filter instead then the fraction of particles that remain will be
.1 × (1-.9995) + .9 = 0.90005.
If you noticed that 0.9005 and 0.90005 are almost identical then congratulations—you understand air filters better than the Wirecutter. Both 99.5% and 99.95% are close enough to 100% that performance is almost entirely determined by the volume of air they process.

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Another application of Amdahl's law.

"the overall performance improvement gained by optimizing a single part of a system is limited by the fraction of time that the improved part is actually used"

Thanks for teaching me the name for this principle!

This is how I feel every time HN suggests rewriting every website in C while ignoring the fact the database takes most of the time for average web apps.

Or the articles about how Python is causing climate disaster while the author continues to drive an oversized SUV.

I really don't like this math, no one actually stops at that point, you take the output, la, "0.9005" and re-run it

     (0.9005 × .1) × (1-.995) + (0.9005 × .9)

and again, and again, etc, point being that over time it does cycle the entire room, due to entropy, and then suddenly the differences start to stack up a bit, not a lot, but when one filter is letting 10x the particles through vs the other filter, it'll show

> suddenly the differences start to stack up a bit, not a lot, but when one filter is letting 10x the particles through vs the other filter, it'll show

It doesn't show: the difference is too small to notice at every stage: https://www.jefftk.com/remaining-particles-by-minute-995-vs-...

Even comparing 95% vs 99.95% it's barely noticeable: https://www.jefftk.com/remaining-particles-by-minute-95-vs-9...

Sheet: https://docs.google.com/spreadsheets/d/1wVIdWR0lWgZRt4gbnVaH...

fair

The idea that the difference between 0.9005 and 0.90005 is "small" is … weird.

The moment I read that I checked out on the rest of the authors opinions.

Why do you feel it is weird? They are both 90%, because 90% * 100% + 10% * "effectively 0%" is completely dominated by the first term.

> The idea that the difference between 0.9005 and 0.90005 is "small" is … weird.

it is. that's one minute of filtration and the difference is minuscule. over time, this would trend to zero. in 10 minutes you'd expect to be near the steady state of the room. (obviously not completely steady state since you are filtering some already filtered air and probably introducing more particulates but close enough for an approximation)

It's a 10x difference. It's not small.

In a sealed environment, you're right, you'd eventually end up with all particles filtered.

But homes are not sealed environments.

(0.9005 - 0.90005) / 0.90005 = 0.00049997222

It's a 0.049997% difference, not a 10x difference.

In an unsealed environment, the steady state will be related to amount filtered * % filtered / amount exchanged for any given time period. The difference in % filtered is not a significant factor in the above ratio.

What? Where are you getting the 10x from? Both numbers are about 0.9 and the difference is about 0, not 10. If you are refering to the sticker number, yeah the whole point of that calculation is that a 10x sticker number does absolutely not translate to a 10x difference.

> but homes are not sealed.

Correct, but neither are they ultra high throughput (at which point any filter sitting in the room would be useless anyway, since you never get the filtered air). So "not sealed" is too vague to make any conclusion.

No, a 10x difference would be between 0.9 and 0.09. What was given was about a 1.0005x difference. If you had a child that was .9005 meters tall and one that was .90005 meters tall, you couldn't tell which was taller without a precision ruler.

A E12 filter filters out 99.5% of particles above 0.3 microns.

An H13 filter filters out 99.95% of particles above 0.3 microns.

Assuming a volume of 10000 particles above 0.3 microns:

An E12 filter will leave 50 particles.

An H13 filter will leave 5 particles.

That's a 10x difference.

The only point the author was making with the 0.9005 vs 0.90005 example was that if you're only processing 10% of the air, then the efficiency of your filter doesn't matter. The entire section honestly would have been better without numbers, because they cause some amount of confusion and they don't really help make the point since it's obvious. If your room's air is recycled with outside air fully over the course of one day, and your filters take ten days to work through the volume of air in your room, then the efficiency of your filter doesn't matter.

That's it. Yes, one filter is 10x as efficient. It doesn't matter because in this example they aren't moving enough air relative to the room size/leakiness for it to matter.

Saying, "OK, but as a total percent of the air in your home at any given time, the difference is tiny" is … weird. It's still a 10x difference.

Using small numbers to make the difference look small doesn't hide that.

Sorry, you’re still missing the point.

If you are taking air, running it once through a filter, and using the air that comes out for an application that needs very few particles, then a 99.99% filter is “10x” as efficient as a 99.9% filter in the sense that the air coming out will have 1/10 as many particles. For example, a 99% efficient face mask is “10x” as efficient as a 90% efficient mask (assuming both fit perfectly, which they don’t, although a PAPR approximates a perfect fit).

But an air purifier doesn’t do this at all. It continuously sucks in air, removes particles from it, and sends the filtered air right back into the room to mix with all the other air. The performance of a 95% filter in this context is barely distinguishable from that of a hypothetical 100% filter. Your characterization would have the 100% air purifier being “infinitely” more efficient.

I understand the point. The point doesn't matter.

Air purifiers operate on a fraction of available air. That air supply is continually being cycled, refreshed and mixed. Particulate matter within that air is not evenly dispersed.

That, for a single minute, as a percentage of total air, a 99.5% and a 99.95% purifier produce a minor difference in total air quality is deeply irrelevant to the overall performance of the purifier over any length of time. The 10x difference, however, will matter over time.

This is why the tests, which the author dismissed without any reasoning beyond "looks wrong!", in the original WireCutter article showed such stark differences between the performance of the Förnuftig and the Levoit Core 300, over a 30 minute span.

If you were correct, over those 30 minutes, the amount of particulate in the test room would have been roughly equal for both purifiers. It wasn't. The Förnuftig removed only 64.5% of the particulate while the Levoit removed 97.4%.

Can you point to a test which shows dramatically different results than the ones the WireCutter reported?

You started this subthread with the comment:

> The idea that the difference between 0.9005 and 0.90005 is "small" is … weird.

When you use the author's numbers, 0.9005 and 0.90005, the implication is that you're taking the parameters of they hypothetical as given. You then go on to say that the difference between those numbers is significant. Remember that in this abstract, idealized scenario, the air filters are only able to process 1/10th of the air in the room (hence the shared 0.9, the dominant portion of the magnitude). Perhaps the room reciculates with its environment at the rate of one room volume per day, and the filters can only process 1/10th of the room per day. Given that, do you still think the difference is significant? Or are you just outright refusing to participate in the thought experiment at all? Because that's what it seems like now that you're trying to broaden the scope of your contention to the other sections of the article.

I started the subthread with the comment because saying the difference between those numbers is small is weird, … because it is.

Those numbers represent percentages (90.005% and 90.0005%) and those two inputs, especially when applied to a chaotic system, will produce outsized differences over time.

And the data shows that the two filters produced outsized differences over time.

I'm not broadening the scope of my contention. I'm pointing out that my contention (there is a large difference in those numbers that is hidden by the way the author presents them) is confirmed by the data.

The data have nothing to do with the hypothetical where the air filters process 10% of the air in the room. They also have nothing to do with the air filters in the thought experiment, which are simplified, ideal filters that have the exact characteristics we say they do. Nobody is applying the numbers in question to any "chaotic system", because this is just a simple framing designed to illustrate exactly one, utterly banal point: If you don't process most of the air, it doesn't matter how efficient you are. In fact, that's all the section should have been. That one sentence. No numbers (it doesn't need them), and very little detail. Just, "if you don't process most of the air, the efficiency doesn't matter". You can't disagree with that conditional statment. You can argue that the premise is flawed, or irrelevant, or unrealistic to the point of uselessness, but you can't argue with the totally boringly obvious statement that if you aren't processing 90% of the air at all, then your efficiency doesn't matter. It's not more controversial than saying that "if your air filters are turned off, their efficiency doesn't matter."

> That, for a single minute, as a percentage of total air, a 99.5% and a 99.95% purifier produce a minor difference in total air quality is deeply irrelevant to the overall performance of the purifier over any length of time. The 10x difference, however, will matter over time.

Can you explain, with actual math, what you’re trying to say?

There are plenty of plausible explanations for Wirecutter’s unexpected results. They could have messed up (quite likely). The difference in the behavior of the fans could be circulating the air differently (also seems reasonably likely). The conditions of the test could be such that the difference in CADR was relevant (possible but doesn’t seem likely). They could have failed to set up the IKEA filter correctly (I once failed to set up a Conway filter correctly — it was somewhat embarrassing). Or, by pure magic, the fact that the extremely clean outgoing air from the IKEA filter was less extremely clean than the extremely clean outgoing air from the other filter made a difference (seems very unlikely).

> the original WireCutter article showed such stark differences between the performance of the Förnuftig and the Levoit Core 300, over a 30 minute span. If you were correct, over those 30 minutes, the amount of particulate in the test room would have been roughly equal for both purifiers. It wasn't. The Förnuftig removed only 64.5% of the particulate while the Levoit removed 97.4%.

Note that you are talking about the 0.3 micron measurements: if we look at larger particles the difference is smaller. But that's fine!

There are two big ways that that comparison is different from what we're talking about here:

* Those two purifiers have very different capacities: 135 CFM (CADR) for the Levoit, 82 for the Förnuftig

* The filter on the Förnuftig is much less effective against very small particles. The math above is comparing filters that are 99.5% vs 99.95% effective, while in this case it's more like 70% vs 99.97%.

We're talking about 0.3 micron measurements because the input value for his numbers is the efficiency of the filters in removing 0.3 micron particles (99.5 vs 99.95).

The author claimed the difference between the purified air, as a percentage of total air volume, was small. He used percentages expressed as a decimal to make that difference look small (0.9005 vs 0.90005). But a clever observer would translate those numbers back into their percentages (90.05 vs 90.005), start applying some math (i.e. 100000 x 0.9005 vs 0.90005), see the 10x difference, understand how that 10x different is going to multiply over time in a chaotic system, check the data to see if that's true, and then throw away the author's point.

Multiplying .9005 and .90005 by 10000 does not actually cause a 10x difference to appear. No, really, try it!

If your goal is to play with numbers, you could raise them both to a large power. You would discover that the ratio between them increases exponentially, but this would pale in comparison to the fact that both results would exponentially approach zero much faster than the ratio would increase.

10000 x .9005 is 9,005. 10000 x .90005 is 9,000.5.

Meaning that the first filter left 5 particles vs the second filter leaving .5 particles.

A 10x difference.

The goal isn't to "play with numbers" but to understand why/if the relative effectiveness of a filter results in a substantive difference in air quality.

The data shows it does.

> The data shows it does

As I described above [1] the data show that the difference between 70% and 99.95% matters, not that the difference between 99.5% and 99.95% does. (And that's ignoring difference in flow rates, which is also very large.)

[1] https://news.ycombinator.com/item?id=31823047

The 70% and 99.95% difference is because of the 10x filtration efficiency difference, combined with the CADR difference.

This is very straight-forward.

The "10x" you've been referring to is about the difference in how many particles make it through filters of 99.5% vs 99.95% efficacy [1], not 70% vs 99.95%, which would be 600x [2].

[1] (1−.995)÷(1−.9995)

[2] (1−.7)÷(1−.9995)

"CADR" is a word I included for a reason.

Extraordinary claims require extraordinary evidence. The data shows a huge difference which cannot be explained by the difference between 5 particles and 0.5 particles.

As noted in the article, the Wirecutter does not explain its methodology or give particularly complete data, and what explanations they do give about filtration make no sense.

It's a 10x difference in filtration efficiency, combined with a CADR difference between the two units.

I don't understand why you think more of an explanation is required?

There is a 10x difference between (1-efficiency) for the two filter media choices. Explanation needed as to why this is at all relevant.

Your comment is like observing that car A burns 87 octane gasoline and another burns 89 octane gasoline and claiming, without explanation, that one of them accelerates faster because (90-octane) is 3x lower.

hint: the bigger purifier wins because it has a more powerful, more power hungry fan pushing air through it. Its performance might be further improved (depending on the fan and motor characteristics) by putting a less efficient, lower pressure drop filter in because more air would go through it per unit time.

Meanwhile, two IKEA filters will outperform it in every measure, including cost, noise, and power consumption. But their efficiency will still be lower.

So? We're talking about practical effectiveness here. The difference really only matters if you only have one chance to filter the air, like the filter in a ventilation system bringing air into a cleanroom (the article goes into this).

Since the air purifier intakes and exhausts in the same space (meaning filtered air gets re-filtered), all the slightly worse filter means it that you'd need to run it for a couple more minutes to get the room down to a similar concentration of particulate per unit volume... So the difference in particulate concentration would likely not be anywhere near 10x at steady state, it would be much smaller (but depends how much air leaks into the room from outside, the particulate content of the outside air, the volume of air you're getting through the purifier per unit time, etc.)

in practice the filtration in a room goes down exponentially and quite quickly even with budget filters that only filter out 90%. even in shops where you are sanding.

https://woodgears.ca/dust/dylos.html

The author explicitly states that it's small in the home use context. If you're talking about medical or cleanroom manufacturing contexts, yes it's a huge difference.

The difference between 0.9005 and 0.90005 is not huge in a medical context, or in a chip fab context, or in any other practical context. We're not talking about the difference between 0.0005 and 0.00005. The numbers in question are 0.9005 and 0.90005, and the point being made is that the 0.9 problem dwarfs the 10x efficiency difference way over in the thousandths place.

That difference is from his comments on the toy model of 1000 cubic feet room and 100 cubic feet per minute recirculating air.

In an operating room or chip fab, the room would be over pressure and the new air coming into the room would be filtered. The cleanliness of that air would be determined by the quality of the filter.

Also, if you need air that clean, you need to have strategies for all sorts of things besides filtering.

The point is, you need to be very careful when you put numbers on the internet, and when you read numbers on the internet. Numbers make things feel more real than they are.

For me to actually trust the numbers here, I would need to see the graphs for multiple runs of each filter.

Yes, but in that case you wouldn't be comparing 0.9005 and 0.90005, but rather 0.0005 and 0.00005. No one is arguing that the difference in filters wouldn't matter in a cleanroom context, just that recirculating air in a home the difference in filtration is more like 0.9005 vs 0.90005, and the difference between those numbers is small (in any context to which they apply).

> In an operating room or chip fab, the room would be over pressure and the new air coming into the room would be filtered.

You're describing a situation where the filter is on the intake, but this thread and article are about purifiers within rooms. I agree that the math is really different in your situation.

Yes, the numbers are from a toy example, one that the author used to make one uncontroversial point in one section of the post. Those are the numbers we are discussing in this subthread, which began with:

> The idea that the difference between 0.9005 and 0.90005 is "small" is … weird.

We aren't talking about a situation where both filters are processing all the air in the room. We're talking about a situation where the filters are only processing 10% of the air in the room. That's the defining characteristic of the hypothetical.

I did a poor job of being explicit in my first reply to etchalon.

Even in a medical context, the difference, when operated like an air purifier, is negligible.

The genuine HEPA filter in a cleanroom [0] is not sitting in front of a fan in the middle of the room. It’s very carefully installed such that all the air coming into the clean area goes through it once. The calculation is entirely different. (A medical or industrial HEPA filter may well be in the exhaust, in which case the considerations are again different.)

[0] There’s none of this “true HEPA” stuff in a cleanroom. There is a filter that meets a specific standard, and that filter will have a gasket that seals with considerable force against the air handling equipment. The “true HEPA” filter in a Wirecutter-approved air purifier achieves nowhere near 99.97% due to the lack of the aforementioned gasket regardless of how amazing the filter media may be.

Small home or not, homes are not sealed environments. A 10x difference is a 10x difference.

Using one small number or produce another small number, so the difference looks small, doesn't hide the 10x change.

0.90005 times 10 is 9.0005, not 0.9005 (I.e. the two fractions presented are 90.005% and 90.05%). Even if you look at the complement you get 9.995% vs 9.95% which is small. One could imagine that these differences could also arise from eg obstructions to airflow or positioning in the room or the direction of the wind outside. The point is that the difference is dominated by air flow in a typical environment rather than filtering differences.

How is there a 10x change? I see a 0.1% change.

Is it a 10x difference? If you used the better filter, you would still have 99.95% of the particles you would have had if you used the worse filter

This is how it works because the room is not sealed, nor is the filter being used to filter outside air into a positive pressure area.

It is (hopefully) easy to see that e.g. a filter that removes 99.5% of particles, but moves twice as much air per minute will remove almost twice as many particles per minute as a filter that removes 99.95% of particles.

Using the numbers from TFA (20% of the room for the 99.5 rather than 10%):

  .2 × (1-.995) + .8 = 0.801

  .1 × (1-.9995) + .9 = 0.90005

Thus proving the point in TFA that the airflow matters more than E12 vs H13. The fact that the steady state (given that "dirty" air is being introduced somehow) is lower for the filter that moves more air follows from the fact that it removes particles at a faster rate.

Why? It's a 0.05% difference, seems pretty small to me.

The difference between 99.5% and 99.95 is the difference between an event happening 1 in 200 times and happening and 1 in 2000 times.

It's a 10x difference.

The author's "I'll just times .1 by the percent of flow, and produce very small numbers that look fine! See! The numbers are so small!" trick is just … wrong.

The author implies that the difference can be made up by the volume of air being processed, but that would only be true of a sealed environment, where no new pollutants are added to the air.

Setting aside the basic misunderstanding of probability, and ignoring that home purifiers don't operate in sealed environments, the IKEA unit does not process 10x the amount of air as the other units, so the point is mute.

Probabilities and amounts are not comparable even though they both use % notation.

In this case they are measuring the % of particles captured (an amount), not the likelihood a particle is captured (a probability). The parent is right, it’s a tiny difference.

Consider a purifier that purifies 99.995%. According to your "probabilities", that's a 100x improvement. Now consider this purifier purifies 1 cubic millimeter of air per hour. That is to say, each hour 1 cubic millimeter of air is 99.995% purified (no probability). Would you say that this purifier is 100x better than the IKEA one with 99.5% purification at 1 cubic feet of air per minute? Considering air flow is not a trick.

A E12 filter filters out 99.5% of particles above 0.3 microns.

An H13 filter filters out 99.95% of particles above 0.3 microns.

Assuming a volume of 10000 particles above 0.3 microns:

An E12 filter will leave 50 particles.

An H13 filter will leave 5 particles.

The "rootlocus" filter would leave 0.5 particles.

So yes, I would say your filter is 100x better because it literally is.

> Assuming a volume of 10000 particles above 0.3 microns:

That volume is not the same volume processed by all filters in the same amount of time.

In the first minute:

    E12  filters 10000 particles   @ 99.5% performance   -> removes 9950,   leaves 50
    H13  filters 10000 particles   @ 99.95% performance  -> removes 9995,   leaves 5
    RLv1 filters 10 particles      @ 99.995% performance -> removes 10,     leaves 0
    RLv2 filters 1000000 particles @ 99% performance     -> removes 990000, leaves 10000

RLv1 only filters a tiny amount of air each minute, while RLv2 filters a lot of air each minute (I've improved the flow, but drastically botched the performance)

By your method, RLv2 is 2000x slower than H13, but in the same ammount of time filtered 99x more particles. RLv1 needs to run 99000 minutes to filter the same amount of particles RLv2 does in one minute.

The example is meant to show air flow totaly dominates performance, and it's not "a trick" to multiply by it. I also want to point out that comparing the amount of particles "left" (50 vs 5 vs 0 vs 10000) is nonsense and absolutely no indication of performance in any way.

In other words: you miss 100% of the particles you don't circulate

(With apologies to Wayne Gretzky)

This entire subthread is the math version of "most programmers can't even do fizz buzz".

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