Breakpoints, Categories and Margins of Error

There are infinitely many quantitative differences between analog factors. But there are only a small number of relevant qualitative differences for a particular context with a few goals. Qualitative, notable, important, meaningful differences – differences that connect to some kinda relevant intellectual concept – are sparse and rare.

There are also digital factors that already involve only a few categories or just two categories. The analog factors are harder to deal with so I give them more attention. If a factor uses 2-5 discrete categories then it’s often already like the result of finding 1-4 breakpoints for an analog factor. That’s like skipping the step of finding breakpoints.

If a digital factor involves millions of categories (or even just twenty), then it should be treated like an analog factor and breakpointed. It approximates analog.

You can breakpoint anything into just two categories with a pass/fail (good enough?) goal criterion. However, getting things down to a small number of categories is good and often matches a conceptual understanding of the issue. Then you can define and consider multiple goals with different requirements for success. If there are three categories, A, B, C, then you could have the following six goals: A, B, C, A or B, A or C, B or C. Usually only some goals make sense. E.g. A is the best and B is second best, so there would be two goals to consider: either just A or else A or B. Note that if all the categories were good enough then it wouldn’t make sense to have a goal since all outcomes are fine. And if none of the categories were good enough, a goal wouldn’t work either.

Categories are normally groupings with a qualitative/conceptual difference between them. You can also define categories fully arbitrarily (don’t do that) or because you need categories but don’t know of qualitative differences. You can’t work with infinitely many real numbers. You need to organize them. So, at worst, you can use a generic, default categorization scheme like 0-20, 20-40, 40-60, up to 100. (You would also specify which category a result of exactly 20 goes in, e.g. if it’s right on the border use the lower category. You can have a standard policy like round ties down. Not that it matters much because it’s within the margin of error of the category boundary. When something is within the margin of error, we either examine more closely or proceed in a way that works with either of the two categories it could be. BTW, if the margin of error around your category boundary is larger than a category, so there are 3+ categories a data point could be in, then you need to fix that. Either look at the issues more precisely to reduce the margin of error or change the categories by e.g. making them bigger.)

Typical categories involve finding reasons that certain quantities matter, e.g. enough food to not be hungry. Or a piece of furniture small enough to fit through your door or too big to get into your home. And you could have more than one breakpoint, like small enough to fit into your truck and small enough to fit through your door are two separate breakpoints, though often one breakpoint matters most. In this case, even if it fits in your truck, if it won’t fit through the door you’ve got a major problem. And if it does fit through your door, then it will generally also fit in your truck, so you don’t really have to worry about both issues, just the smaller size constraint. Though you could imagine something long and skinny that fits through the door but won’t fit in the truck. You could just make the breakpoint be “fits in truck and through door” – just categorize on shapes that fit both or not. A separate breakpoint related to shape could be whether it’s comfortable to you. Or whether your feet can touch the ground while sitting on it. That’s a different kind of conceptual issue than whether it fits in the space. I’d just treat that as a separate factor – comfort – rather than combining it with the stuff about how bulky it is. It’s related to a different goal. One goal is to transport it. Another is to enjoy it. Enjoying involves other factors too like color, feel, smell, style/fashion, etc.

Breakpoints have margins of error – safety, buffer – because we’re not perfectly precise. How much food is enough not to be hungry? You don’t know exactly. But you do know approximately. Even with fitting furniture through a door, measurements are never perfectly exact. If you measure to an accuracy of a hundredth of an inch – meaning you’ve correctly identified which hundredth of an inch it’s closest to (e.g. 1.52 inches), then your margin of error is around 5 thousands of an inch. It could be from 1.515 to 1.525 inches. (I won’t address the minor detail about what if it’s exactly in the middle between two hundredths of an inch.)

Most data points are outside the margins of error of all relevant breakpoints. Why? Imagine the real number line or, as an approximation, some large stretch of the integer number line like from zero to a billion. Now add ten breakpoints at random spots. Ten is on the high side – one breakpoint is the most common, two is the second most common, three is the third most common, and so on. Ten is rare. Now give the breakpoints significant margins of error, e.g. plus or minus one hundred. What does this look like? Ten little dots (for the breakpoints plus the whole range of their margin of error), barely visible, on a huge line that’s almost entirely empty. The breakpoints are very likely to be far apart from each other.

The reason we deal with stuff within the margin of error reasonably often, and have memories of doing that, is due to selective focus. We pay more attention to cases that are near breakpoints and pay less attention to the many, many other cases. That’s fine but it’s misleading about what the typical case looks like. We spend a lot of our time dealing with atypical cases. Most possible things to do are either very easy or very hard, and either way we don’t pay much attention to them.

In general, margins of errors are much smaller than categories. So even with safe, conservative margins of error, most data points can be categorized easily and accurately. We don’t even have to try very hard to measure precisely. We can have a large margin of error around a breakpoint, and also a large margin of error around a measurement, and those two margins usually won’t overlap.

When margins of error do matter, we can look for a different solution. Why bother with a close call solution, that might barely work, when a lot of other solutions are not close calls? In the atypical case that we’re having trouble finding any solutions that are further from a success/failure breakpoint, our first thought should be to abandon the project. It’s hard, unreliable and inefficient. Maybe we should do something else and perhaps revisit this when we’re better at stuff. In general, it’s a bad idea to put a bunch of effort into doing things we can barely do. We could do them later for less effort. They’re an inefficient way to make progress.

If we do want to proceed with a goal using solutions near a breakpoint, we can investigate the issue more precisely. We can put in extra effort to shrink the margin of error. We can measure things more precisely. We can think about the breakpoint, and how and why it works, more. If we understand the qualitative difference more, we can better understand what it takes to be on the right side of it.

And the other option is just try it and hope. We can do this if we’re in a bad situation with little choice (we’re about to die if we don’t succeed, so we might as well make a risky attempt), or if the stakes are low and there’s little downside to failure (e.g. in video games like Mario Bros., failure often means playing for 10 seconds to get back where you were to try again, so it generally wouldn’t make sense to spend minutes of effort to try to avoid having any failures).

Category groupings like splitting a numeric range into fifths work OK because they correspond to concepts like “tiny, small, medium, large, huge”. They are conceptual breakpoints, just not ideal breakpoints because they use generic concepts instead of figuring out what’s important in this particular case. This approach doesn’t work as well – and the categories are less useful – if you make 50 categories instead of 5. A lot of the point of doing this is to get a small number of things to think about. It’s OK to have the wrong 5 categories and be able to think them over and figure out what’s wrong with them and make changes. It’s worse to be overwhelmed with 50 categories which is too many to think about and requires further simplification (and they’re wrong anyway, because using 50 categories is ~never right). If you have the wrong 5 categories, at least you can get started by thinking clearly about them, which will help you learn more about the issues and the flaws in that categorization, so you can recategorize once you understand more.

Generic size-based categories (or amount-based) aren’t great because they are disconnected from your goal(s). Often you know that either more is better or less is better, or that there’s a right amount and you don’t want too much or too little. You can then take some arbitrary categories and try to figure out if each one is too much or too little, or good. This is good to do if you’re getting stuck. If you could just directly find the breakpoints, do that. But if you can’t then you can divide into a few categories, pick a random data point in each category, and then try it out. You can try to get a feel for these categories. Working with concrete examples can help. You could also just use random examples without categorizing first. I’m not really a fan of categorizing by counting by 20s, or similar, I just want to say it’s an option so if you’re really stuck on categories you’ve got that. (Another generic option for categorizing is “most” and “not the most”. You can categorize values as either the best available or not.)

Say you’re shy and want to go to a party but don’t want it to be a disaster. Don’t try to add up pros and cons to find out whether to go. Listing pros and cons is good. But then you should try to look for solutions to the cons. Look for failure modes and think about what can be done about them. Lots are just discrete or binary issues in the first place, but some aren’t and they can be breakpointed. Like you might worry your heart will race. Number of heartbeats per minute is an analog factor. It could be 100, 100.0003, 100.00031, etc. Any real number in some range like 0-300. There are multiple meaningful breakpoints that relate to goals you have. Like how fast is going to do medical harm? That’s some number, e.g. 200 (with some margin of error around it – it’s not a perfectly accurate single number, but risk starts going up significantly somewhere around there). How fast will get you to notice your pulse and feel like your heart is racing, which could distract you or affect your demeanor? That’s some number, e.g. 100, though it depends how much attention you’re paying to worrying about stuff like that – the more you’re focused on a conversation and forget your anxiety, the less you’ll notice a mild heart rate increase.

Similarly, the amount you sweat is a quantity. It can be breakpointed on whether you sweat enough for you to notice, or for other people to notice. We could be more precise and look at multiple breakpoints. Is it enough sweat for them to notice by seeing it from across the room? By seeing from a typical distance for chatting? By seeing from up close right before kissing? Is it noticeable by sight when in certain poses and not others? E.g. a handshake means you reveal your armpit more than if you’re standing with your hands at your sides. Or a dance pose with your hands over your head reveals your armpits more.

I don’t recommend trying to put numbers on these things. I’m arguing against numbers. I’m just saying how you can use breakpoints instead of score systems. They make more sense here. But what’s even better is to look at higher level conceptual issues, like what sort of social interactions you’re worried about and don’t know how to deal with, and then making some plans for those (and perhaps not going if you’re dissatisfied with your plans). Like what if someone teases you, how will you handle that? And you might split that category into several types of teasing with qualitative differences and plan a response for each.