WEBVTT 00:00:00.000 --> 00:00:02.960 If your stock portfolio loses 50 percent on Tuesday 00:00:02.960 --> 00:00:06.219 and then gains 50 percent on Wednesday, how much 00:00:06.219 --> 00:00:08.759 money did you lose? Most people would say zero, 00:00:08.759 --> 00:00:12.400 right? Exactly. If your brain immediately answered, 00:00:12.519 --> 00:00:14.580 oh, I broke even, well, your brain is lying to 00:00:14.580 --> 00:00:17.679 you. And today we're going to show you exactly 00:00:17.679 --> 00:00:20.500 how. Yeah, it's a massive blind spot for human 00:00:20.500 --> 00:00:23.440 intuition. Because, you know, you are constantly 00:00:23.440 --> 00:00:26.199 bombarded with numbers and headlines, corporate 00:00:26.199 --> 00:00:29.179 reports, political polls, and they feel concrete. 00:00:29.289 --> 00:00:31.570 They feel objective. Right. Numbers feel like 00:00:31.570 --> 00:00:34.409 facts. But the way data is mathematically framed, 00:00:34.469 --> 00:00:36.630 like before it ever reaches your screen, can 00:00:36.630 --> 00:00:38.969 completely alter your perception of reality. 00:00:39.750 --> 00:00:42.009 So our mission today is to act as your shortcut. 00:00:42.509 --> 00:00:44.549 We want you to be truly well informed about the 00:00:44.549 --> 00:00:46.570 hidden mechanics of data presentation. Which 00:00:46.570 --> 00:00:49.049 is so crucial right now. It really is. We're 00:00:49.049 --> 00:00:50.829 doing a deep dive into some fascinating source 00:00:50.829 --> 00:00:53.369 material on the underlying math of relative change. 00:00:53.829 --> 00:00:55.950 We're exploring why absolute numbers deceive 00:00:55.950 --> 00:00:58.590 you. the specific vocabulary used to manipulate 00:00:58.590 --> 00:01:01.710 percentages, and the mathematical workarounds 00:01:01.710 --> 00:01:04.069 professionals use to find the actual truth. It's 00:01:04.069 --> 00:01:07.069 a profound shift in perspective, really. Once 00:01:07.069 --> 00:01:08.989 you learn to read the mechanical structure of 00:01:08.989 --> 00:01:11.849 the math rather than just, you know, the surface 00:01:11.849 --> 00:01:13.890 level numbers, you start seeing these subtle 00:01:13.890 --> 00:01:16.769 framing choices everywhere. Everywhere. And the 00:01:16.769 --> 00:01:18.890 illusion usually starts simply because our human 00:01:18.890 --> 00:01:21.489 brains naturally default to absolute numbers. 00:01:21.769 --> 00:01:24.769 We just prefer raw counts. OK, let's unpack this. 00:01:25.030 --> 00:01:27.879 Because To really understand how we get tricked, 00:01:28.599 --> 00:01:31.060 we need a functional way to think about it. I 00:01:31.060 --> 00:01:34.159 agree. Visuals help a lot here. Right. So think 00:01:34.159 --> 00:01:37.079 of an absolute number like saying a car costs 00:01:37.079 --> 00:01:40.879 $50 ,000 as a photograph of a mountain. It shows 00:01:40.879 --> 00:01:43.719 you the object, sure. But relative change is 00:01:43.719 --> 00:01:46.340 the scale on a map of that same mountain. Oh, 00:01:46.400 --> 00:01:48.640 that's a great way to frame it. Without the scale, 00:01:48.859 --> 00:01:50.299 you have absolutely no idea if you're looking 00:01:50.299 --> 00:01:52.849 at a tiny hill or Mount Everest. I mean, the 00:01:52.849 --> 00:01:54.409 photograph gives you the what, but the scale 00:01:54.409 --> 00:01:56.989 gives you the context. Exactly. And the problem 00:01:56.989 --> 00:01:58.909 with relying solely on the photograph, you know, 00:01:59.010 --> 00:02:01.349 the absolute change, is that it leaves you highly 00:02:01.349 --> 00:02:04.329 susceptible to the unit illusion. The unit illusion. 00:02:04.590 --> 00:02:07.450 Yeah. Absolute change is just taking a new value 00:02:07.450 --> 00:02:10.270 and subtracting the reference value. But the 00:02:10.270 --> 00:02:13.729 unit you choose to measure that change in completely 00:02:13.729 --> 00:02:15.889 dictates how your brain reacts. Right, because 00:02:15.889 --> 00:02:18.729 units feel arbitrary sometimes. They do. If you 00:02:18.729 --> 00:02:20.569 measure the difference between a two meter piece 00:02:20.569 --> 00:02:23.550 of wood and a one meter piece of wood, the absolute 00:02:23.550 --> 00:02:25.669 difference is just one. A difference of one meter. 00:02:25.870 --> 00:02:29.810 It sounds completely manageable. But if we measure 00:02:29.810 --> 00:02:32.270 that exact same physical distance in centimeters, 00:02:32.569 --> 00:02:34.909 you are now subtracting 100 centimeters from 00:02:34.909 --> 00:02:37.650 200 centimeters. Oh, wow. Right. The absolute 00:02:37.650 --> 00:02:40.550 difference is 100. Mechanically, the physical 00:02:40.550 --> 00:02:43.229 reality in space is identical. But the number 00:02:43.229 --> 00:02:46.610 100 triggers a significantly larger cognitive 00:02:46.610 --> 00:02:49.090 reaction than the number 1. Exactly. It feels 00:02:49.090 --> 00:02:51.270 like a bigger gap simply because of the unit 00:02:51.270 --> 00:02:54.129 chosen. And relative change neutralizes that 00:02:54.129 --> 00:02:57.110 illusion by mathematically stripping away the 00:02:57.110 --> 00:02:59.729 units entirely. You take that actual change and 00:02:59.729 --> 00:03:01.669 divide it by your reference value. Precisely. 00:03:01.750 --> 00:03:04.370 It levels the playing field. So a change of one 00:03:04.370 --> 00:03:06.770 meter divided by the starting one meter is one. 00:03:07.180 --> 00:03:09.759 And a change of 100 centimeters divided by the 00:03:09.759 --> 00:03:11.639 starting 100 centimeters is also one. Right. 00:03:11.740 --> 00:03:15.300 It yields a 100 % increase either way. You get 00:03:15.300 --> 00:03:18.199 a pure unit -less number that represents the 00:03:18.199 --> 00:03:20.919 actual proportion of the shift. If we connect 00:03:20.919 --> 00:03:23.800 this to the bigger picture, relative change is 00:03:23.800 --> 00:03:26.759 the great equalizer. It judges the underlying 00:03:26.759 --> 00:03:29.560 importance of a change rather than just the raw 00:03:29.560 --> 00:03:31.919 amount. Give me a real -world example of that. 00:03:32.009 --> 00:03:35.009 Well, if an item costs $50 and the price goes 00:03:35.009 --> 00:03:38.449 up by an absolute value of $100 to a total of 00:03:38.449 --> 00:03:41.669 $150, that item just tripled in cost. Yeah, the 00:03:41.669 --> 00:03:43.569 relative change is massive. That hurts your wallet. 00:03:43.819 --> 00:03:47.699 But if you are buying a $10 ,000 piece of industrial 00:03:47.699 --> 00:03:50.479 equipment, and the price goes up by that same 00:03:50.479 --> 00:03:53.719 absolute $100... The relative change is miniscule. 00:03:53.900 --> 00:03:56.099 Exactly. The absolute $100 bill is physically 00:03:56.099 --> 00:03:58.080 identical, but the mathematical importance of 00:03:58.080 --> 00:04:00.360 that bill changes depending on the reference 00:04:00.360 --> 00:04:03.259 value. I follow the logic, but I'm going to push 00:04:03.259 --> 00:04:06.080 back slightly here. Is absolute change strictly 00:04:06.080 --> 00:04:08.599 a trap, or is there a functional utility to it? 00:04:08.740 --> 00:04:11.379 Oh, there's definitely utility. Because if I 00:04:11.379 --> 00:04:13.719 think about my own paycheck... You know, if my 00:04:13.719 --> 00:04:16.060 boss tells me I'm getting $100 bonus this week, 00:04:16.379 --> 00:04:19.319 I care deeply about that absolute $100. Of course 00:04:19.319 --> 00:04:22.060 you do. That is real utility. I can buy groceries 00:04:22.060 --> 00:04:25.360 with that. Even if I calculate the relative change 00:04:25.360 --> 00:04:28.860 to my total annual salary and realize it's a 00:04:28.860 --> 00:04:31.920 fraction of a percent, the relative math doesn't 00:04:31.920 --> 00:04:34.660 diminish the absolute utility of the cash in 00:04:34.660 --> 00:04:37.170 my pocket. And that's a vital distinction, really. 00:04:37.649 --> 00:04:40.209 Absolute numbers are completely necessary when 00:04:40.209 --> 00:04:42.389 the real world utility of the unit matters to 00:04:42.389 --> 00:04:45.129 you directly. A dollar has a fixed purchasing 00:04:45.129 --> 00:04:47.649 power regardless of your total salary. Right. 00:04:47.910 --> 00:04:50.449 The mathematical trap only springs when you are 00:04:50.449 --> 00:04:52.410 attempting to compare two different scenarios 00:04:52.410 --> 00:04:55.449 or, and this is key, when a narrator is attempting 00:04:55.449 --> 00:04:58.870 to frame a change as inherently massive or insignificant. 00:04:59.230 --> 00:05:01.589 Ah, so when someone is trying to sell you a narrative. 00:05:01.689 --> 00:05:04.629 Yes, that is when relying on the absolute number 00:05:04.720 --> 00:05:07.819 leaves your perception highly vulnerable to manipulation. 00:05:08.240 --> 00:05:11.519 And if relative change is how we judge the importance 00:05:11.519 --> 00:05:14.379 of a comparison, percentages are the vocabulary 00:05:14.379 --> 00:05:17.060 we use to communicate it. But that vocabulary 00:05:17.060 --> 00:05:19.639 is incredibly fragile, isn't it? Fragile is the 00:05:19.639 --> 00:05:22.100 perfect word for it. It breaks so easily. Take 00:05:22.100 --> 00:05:24.379 the example of two vehicles from our source material. 00:05:24.560 --> 00:05:28.899 Let's say car M cost $50 ,000 and car L cost 00:05:28.899 --> 00:05:32.800 $40 ,000. Depending entirely on which car you 00:05:32.800 --> 00:05:35.740 anchor as your reference value, the grammatical 00:05:35.740 --> 00:05:38.000 phrasing of the comparison dictates a completely 00:05:38.000 --> 00:05:41.160 different mathematical reality. Walk me through 00:05:41.160 --> 00:05:44.800 that. Well, if car L is the reference, the absolute 00:05:44.800 --> 00:05:48.180 difference is $10 ,000. You divide 10 ,000 by 00:05:48.180 --> 00:05:52.240 car L's price of 40 ,000 and you get 0 .25. So 00:05:52.240 --> 00:05:54.800 you would say car M is 25 % more than car L. 00:05:54.920 --> 00:05:57.560 Right. Or if you express it as a direct ratio 00:05:57.560 --> 00:06:00.939 rather than a difference, car M is 125 % of the 00:06:00.939 --> 00:06:03.620 cost of car L. Oh, I see. The specific words 00:06:03.620 --> 00:06:06.920 more than versus of signal whether you are citing 00:06:06.920 --> 00:06:09.360 the relative difference or the total ratio. Exactly. 00:06:09.480 --> 00:06:11.899 The grammar literally dictates the math. But 00:06:11.899 --> 00:06:15.699 if you flip the anchor, the math changes completely. 00:06:16.040 --> 00:06:19.019 Like, if we start with CAR -M at $50 ,000 as 00:06:19.019 --> 00:06:21.339 the reference, the difference is now negative 00:06:21.339 --> 00:06:24.379 $10 ,000. Which changes everything. Right. Divide 00:06:24.379 --> 00:06:27.759 that by $50 ,000, and you get negative 0 .20. 00:06:28.079 --> 00:06:32.779 So suddenly, CAR -L is 20 % less than CAR -M, 00:06:32.939 --> 00:06:36.339 or 80 % of what CAR -M costs. Yes. The physical 00:06:36.339 --> 00:06:38.899 prices of the car is never moved. Yet you can 00:06:38.899 --> 00:06:42.779 truthfully report the gap as either 25 % or 20%, 00:06:42.779 --> 00:06:44.879 purely based on which bumper you decided to point 00:06:44.879 --> 00:06:47.819 at first. It really requires hypervigilance from 00:06:47.819 --> 00:06:49.720 the listener. I mean, if you aren't paying attention 00:06:49.720 --> 00:06:51.959 to the specific noun acting as the baseline, 00:06:52.459 --> 00:06:54.639 your brain will misinterpret the scale of the 00:06:54.639 --> 00:06:56.300 math. Here's where it gets really interesting, 00:06:56.459 --> 00:06:58.959 though. The ultimate manipulation of this vocabulary 00:06:58.959 --> 00:07:01.639 is the percentages of percentages trap. Oh, this 00:07:01.639 --> 00:07:04.180 one drives mathematicians crazy. We see this 00:07:04.180 --> 00:07:06.800 constantly with interest rates. Let's say a bank 00:07:06.800 --> 00:07:09.000 raises the interest rate on a mortgage from 3 00:07:09.000 --> 00:07:11.939 % to 4%. The bank sends out a notice saying, 00:07:12.360 --> 00:07:15.240 we have implemented a 1 % increase in your rate. 00:07:15.439 --> 00:07:18.579 And that phrasing is fundamentally incorrect. 00:07:19.040 --> 00:07:21.379 Completely wrong. Yeah. It is a conflation of 00:07:21.379 --> 00:07:25.079 absolute and relative terms. The absolute change 00:07:25.079 --> 00:07:28.339 between 3 % and 4 % is one percentage point. 00:07:28.500 --> 00:07:31.660 Right. A point, not a percent. Exactly. But to 00:07:31.660 --> 00:07:34.279 find the relative change, you must take that 00:07:34.279 --> 00:07:37.199 one point absolute difference and divide it by 00:07:37.199 --> 00:07:40.040 the original reference value of 3%. Which means 00:07:40.040 --> 00:07:42.500 the actual relative change to your interest rate 00:07:42.560 --> 00:07:46.560 is 33 and one -third percent. Yes. Your interest 00:07:46.560 --> 00:07:48.420 burden didn't go up by one percent, it went up 00:07:48.420 --> 00:07:52.339 by a third. It's difficult not to assume that 00:07:52.339 --> 00:07:55.500 marketers and financial institutions intentionally 00:07:55.500 --> 00:07:58.180 exploit the public's misunderstanding of a percentage 00:07:58.180 --> 00:08:00.610 point versus a percent increase. It's a very 00:08:00.610 --> 00:08:03.189 convenient confusion for them. Calling a massive 00:08:03.189 --> 00:08:06.889 33 % spike in borrowing costs a 1 % increase. 00:08:07.430 --> 00:08:09.750 Just brilliantly downplays the bad news. When 00:08:09.750 --> 00:08:11.730 the underlying variable you are measuring is 00:08:11.730 --> 00:08:13.889 already formatted as a percentage, the linguistic 00:08:13.889 --> 00:08:16.550 waters become very muddy. Muddy and expensive. 00:08:16.750 --> 00:08:20.069 Very true. It requires strict discipline to separate 00:08:20.069 --> 00:08:22.930 absolute percentage points from relative percentage 00:08:22.930 --> 00:08:25.550 change. Otherwise, the magnitude of the shift 00:08:25.550 --> 00:08:28.170 is completely obscured. So the language of relative 00:08:28.170 --> 00:08:31.420 change is easily weaponized? But the actual formula 00:08:31.420 --> 00:08:34.039 for relative change has its own mechanical failures, 00:08:34.039 --> 00:08:37.120 doesn't it? There are situations where taking 00:08:37.120 --> 00:08:39.940 the new value, subtracting the old value, and 00:08:39.940 --> 00:08:42.080 dividing by the old value just mathematically 00:08:42.080 --> 00:08:44.840 breaks down. It does. The most obvious failure 00:08:44.840 --> 00:08:47.340 is when your reference value is zero. You cannot 00:08:47.340 --> 00:08:49.759 divide by zero. The function is undefined. Just 00:08:49.759 --> 00:08:52.990 a classic math error. Right. But the far more 00:08:52.990 --> 00:08:55.509 counterintuitive failure occurs when your reference 00:08:55.509 --> 00:08:58.710 value is negative. The denominator sign completely 00:08:58.710 --> 00:09:00.850 corrupts the logic of the result. Let's use temperature 00:09:00.850 --> 00:09:03.250 to visualize this. Imagine the weather outside 00:09:03.250 --> 00:09:06.049 goes from negative 10 degrees to negative 6 degrees. 00:09:06.570 --> 00:09:08.750 The temperature got warmer. it increased by a 00:09:08.750 --> 00:09:11.289 positive 4 degrees. But if we plug those numbers 00:09:11.289 --> 00:09:14.049 into the standard relative change formula, we 00:09:14.049 --> 00:09:16.470 take the actual change, which is positive 4, 00:09:16.669 --> 00:09:18.669 and we divide it by the reference value, which 00:09:18.669 --> 00:09:21.230 is negative 10. OK, so positive 4 divided by 00:09:21.230 --> 00:09:25.070 negative 10. Which yields negative 0 .4. So the 00:09:25.070 --> 00:09:27.870 formula outputs a relative change of negative 00:09:27.870 --> 00:09:30.230 40%. It's telling me the temperature decreased 00:09:30.230 --> 00:09:32.769 by 40%, even though I know definitively that 00:09:32.769 --> 00:09:35.600 the absolute temperature increased. Yes. The 00:09:35.600 --> 00:09:38.580 math is outputting a direct contradiction to 00:09:38.580 --> 00:09:41.960 physical reality. That is wild. The classic formula 00:09:41.960 --> 00:09:44.259 simply doesn't know how to handle the crossing 00:09:44.259 --> 00:09:47.139 of the negative threshold. It yields a negative 00:09:47.139 --> 00:09:50.019 relative change for a positive absolute increase. 00:09:50.200 --> 00:09:53.159 So how do people fix that? Well... To bypass 00:09:53.159 --> 00:09:55.419 this mechanical flaw, professionals use a special 00:09:55.419 --> 00:09:58.600 variation called percent error. It's heavily 00:09:58.600 --> 00:10:00.799 utilized in scientific fields when comparing 00:10:00.799 --> 00:10:03.159 an experimental value, like a measurement taken 00:10:03.159 --> 00:10:06.360 in a lab, to a theoretical or widely accepted 00:10:06.360 --> 00:10:08.919 value. I like to visualize this using a dartboard. 00:10:09.419 --> 00:10:11.799 Treat the theoretical accepted value like the 00:10:11.799 --> 00:10:14.590 bullseye. The standard formula for percent error 00:10:14.590 --> 00:10:17.250 puts absolute value brackets around the top part 00:10:17.250 --> 00:10:19.250 of the fraction. Right. It forces the numerator 00:10:19.250 --> 00:10:22.289 to be positive. So if you throw a dart, that 00:10:22.289 --> 00:10:25.409 standard formula just measures the pure magnitude 00:10:25.409 --> 00:10:27.769 of your miss. It tells you that you missed the 00:10:27.769 --> 00:10:30.690 bullseye by two inches, but it strips away the 00:10:30.690 --> 00:10:33.629 context of where the dart landed. What's fascinating 00:10:33.629 --> 00:10:36.230 here is that scientists will frequently and intentionally 00:10:36.230 --> 00:10:40.210 remove those absolute value brackets. Why? Because 00:10:40.210 --> 00:10:43.370 the negative or positive sign of the error holds 00:10:43.370 --> 00:10:46.929 vital physical context. The directionality matters 00:10:46.929 --> 00:10:49.330 just as much as the magnitude. Right, because 00:10:49.330 --> 00:10:51.809 if you take the brackets away, the math tells 00:10:51.809 --> 00:10:54.490 you whether you overthrew or underthrew the dart. 00:10:54.950 --> 00:10:56.830 And there's a brilliant example in the sources 00:10:56.830 --> 00:10:59.370 regarding the speed of light. Oh yes, that's 00:10:59.370 --> 00:11:01.970 a perfect illustration. If a physicist measures 00:11:01.970 --> 00:11:05.289 the speed of light in a vacuum, and they calculate 00:11:05.289 --> 00:11:07.649 a negative percent error without the brackets, 00:11:08.250 --> 00:11:10.450 it simply means their experimental velocity was 00:11:10.450 --> 00:11:13.299 slower than the accepted speed of light. The 00:11:13.299 --> 00:11:15.799 dart landed short. They just need to recalibrate 00:11:15.799 --> 00:11:18.120 their lasers. But if they calculate a positive 00:11:18.120 --> 00:11:20.240 percent error without those brackets... Then 00:11:20.240 --> 00:11:21.740 the dart went straight through the backboard. 00:11:21.840 --> 00:11:24.039 A positive percent error means they recorded 00:11:24.039 --> 00:11:26.580 a speed faster than light. They just mathematically 00:11:26.580 --> 00:11:29.480 violated the theory of relativity. Which is huge. 00:11:29.639 --> 00:11:31.440 That isn't a measurement error you just file 00:11:31.440 --> 00:11:34.480 away in a drawer. That is data that rewrites 00:11:34.480 --> 00:11:37.779 the laws of physics. The presence or absence 00:11:37.779 --> 00:11:40.539 of that single negative sign completely alters 00:11:40.539 --> 00:11:43.120 the interpretation of the data. And this raises 00:11:43.120 --> 00:11:45.340 an important question about the rigidity of these 00:11:45.340 --> 00:11:48.399 formulas. In percent error, you must adhere to 00:11:48.399 --> 00:11:51.440 a strict operational order. It has to be experimental 00:11:51.440 --> 00:11:55.360 value minus theoretical value. The order matters. 00:11:55.700 --> 00:11:59.220 Yes, the values do not commute. If you flip them, 00:11:59.360 --> 00:12:01.860 the sign flips, and a scientist might mistakenly 00:12:01.860 --> 00:12:03.940 believe they broke relativity when in reality 00:12:03.940 --> 00:12:05.980 they simply subtracted in the wrong direction. 00:12:06.139 --> 00:12:08.279 Which introduces a massive mechanical headache 00:12:08.279 --> 00:12:10.779 for systems that rely on automated math. Oh, 00:12:11.000 --> 00:12:14.320 a nightmare for programmers. Right. If the basic 00:12:14.320 --> 00:12:17.080 formula explodes when it hits zero and outputs 00:12:17.080 --> 00:12:20.039 contradictions on negative numbers, how do software 00:12:20.039 --> 00:12:22.519 programmers or economists stabilize their models? 00:12:22.879 --> 00:12:25.779 They can't have a global financial model crash 00:12:25.779 --> 00:12:28.059 just because a localized metric dropped below 00:12:28.059 --> 00:12:31.840 zero. They stabilize it by fundamentally reinventing 00:12:31.840 --> 00:12:35.120 the denominator. You see, the core mechanical 00:12:35.120 --> 00:12:37.840 flaw with the classic relative change formula 00:12:37.840 --> 00:12:40.899 is that it lacks symmetry. Standard division 00:12:40.899 --> 00:12:44.120 inherently anchors to the past. The starting 00:12:44.120 --> 00:12:46.700 reference value. Let's walk through why that 00:12:46.700 --> 00:12:49.879 asymmetry happens. If a stock goes from $100 00:12:50.320 --> 00:12:54.519 up to 125, the absolute change is $25. Yeah. 00:12:54.799 --> 00:12:56.759 We divide that by the starting anchor of 100, 00:12:57.120 --> 00:12:59.899 which gives us a 25 % increase. But if that same 00:12:59.899 --> 00:13:04.080 stock drops from 125 back down to 100, the absolute 00:13:04.080 --> 00:13:07.360 change is negative $25. OK, sure. However, your 00:13:07.360 --> 00:13:10.049 anchor has now shifted. You are now dividing 00:13:10.049 --> 00:13:14.669 that negative 25 by 125, which yields a 20 % 00:13:14.669 --> 00:13:17.070 decrease. The journey is physically identical 00:13:17.070 --> 00:13:19.730 in distance. But because the denominator changed, 00:13:19.909 --> 00:13:22.610 the relative math is totally skewed. Going up 00:13:22.610 --> 00:13:26.210 is 25%, coming down is 20%. It feels mathematically 00:13:26.210 --> 00:13:29.269 unbalanced. It is. And in data science, that 00:13:29.269 --> 00:13:31.769 lack of balance creates compounding errors over 00:13:31.769 --> 00:13:34.580 time. A relative change from state A to state 00:13:34.580 --> 00:13:38.100 B inherently must have the exact same magnitude 00:13:38.100 --> 00:13:41.000 as moving from state B back to state A. So how 00:13:41.000 --> 00:13:44.179 do they force this symmetry? Experts use indicators 00:13:44.179 --> 00:13:46.840 of relative change. Instead of dividing by the 00:13:46.840 --> 00:13:49.179 starting reference value, they replace the denominator 00:13:49.179 --> 00:13:52.019 with a mean, an average of the two numbers. Wait, 00:13:52.019 --> 00:13:53.620 so if I use the average of the two numbers as 00:13:53.620 --> 00:13:56.440 the denominator, the math stabilizes. Let's test 00:13:56.440 --> 00:13:59.039 it on the stock. Okay, the mean of 100 to 125 00:13:59.039 --> 00:14:03.870 is 112 .5. So if the stock goes up, we divide 00:14:03.870 --> 00:14:08.409 the positive $25 change by 1 .12 .5, which is 00:14:08.409 --> 00:14:11.750 about 22 .2%. And if the stock goes down, you 00:14:11.750 --> 00:14:14.590 divide the negative $25 change by that exact 00:14:14.590 --> 00:14:19.190 same mean of 1 .12 .5, yielding negative 22 .2%. 00:14:19.190 --> 00:14:21.610 The symmetry is perfectly restored. Whether you're 00:14:21.610 --> 00:14:24.149 moving forward or backward, the denominator acts 00:14:24.149 --> 00:14:26.769 as a fixed anchor midway between the two states. 00:14:27.100 --> 00:14:30.080 Exactly. But mathematicians don't just use a 00:14:30.080 --> 00:14:33.080 simple average. They tailor the type of mean 00:14:33.080 --> 00:14:35.580 to the specific vulnerabilities of their field. 00:14:36.059 --> 00:14:38.500 Like what? For instance, programmers checking 00:14:38.500 --> 00:14:40.940 floating point values in computer code are heavily 00:14:40.940 --> 00:14:44.299 advised to use maximum mean change. Floating 00:14:44.299 --> 00:14:47.139 point values are essentially decimal calculations 00:14:47.139 --> 00:14:49.740 in software, right? Why do they specifically 00:14:49.740 --> 00:14:52.889 need the maximum mean? Because computers struggle 00:14:52.889 --> 00:14:56.210 with dividing by incredibly tiny numbers. If 00:14:56.210 --> 00:14:58.629 a programmer uses a standard average and the 00:14:58.629 --> 00:15:01.330 numbers are close to zero, the resulting fraction 00:15:01.330 --> 00:15:03.929 could become massively inflated. Which causes 00:15:03.929 --> 00:15:07.110 a crash. Right, a floating point overflow. By 00:15:07.110 --> 00:15:09.529 forcing the denominator to be the maximum, the 00:15:09.529 --> 00:15:11.490 larger of the two numbers being compared, they 00:15:11.490 --> 00:15:14.250 guarantee the denominator is as robust as possible, 00:15:14.769 --> 00:15:16.669 mathematically minimizing the resulting fraction 00:15:16.669 --> 00:15:18.950 and keeping the software stable. That is incredibly 00:15:18.950 --> 00:15:21.370 clever. They are choosing a mathematical flavor 00:15:21.370 --> 00:15:23.509 that actively defends against the hardware's 00:15:23.509 --> 00:15:25.950 weakness. And economists do the exact opposite 00:15:25.950 --> 00:15:29.850 right. They use minimum mean change. Yes. Economists 00:15:29.850 --> 00:15:32.049 modeling massive financial systems will often 00:15:32.049 --> 00:15:34.690 use the smaller of the two numbers as their denominator. 00:15:35.450 --> 00:15:37.509 Dividing by the minimum value yields the largest 00:15:37.509 --> 00:15:40.190 possible relative change. So it exaggerates the 00:15:40.190 --> 00:15:43.230 shift. In econometrics, this acts as a built 00:15:43.230 --> 00:15:46.409 -in stress test. It forces the economic model 00:15:46.409 --> 00:15:49.090 to react to the most volatile possible interpretation 00:15:49.090 --> 00:15:51.590 of the data, ensuring the system can survive 00:15:51.590 --> 00:15:54.990 extreme variance. It's fascinating how the specific 00:15:54.990 --> 00:15:57.529 denominator is chosen not just for accuracy but 00:15:57.529 --> 00:16:00.330 for survival. But, you know, averaging the denominator 00:16:00.330 --> 00:16:02.529 is still essentially just patching a broken system. 00:16:03.029 --> 00:16:05.950 If we want a mathematical framework that is fundamentally 00:16:05.950 --> 00:16:08.809 unbroken from the ground up, we have to abandon 00:16:08.809 --> 00:16:11.149 standard percentages entirely. We have to change 00:16:11.149 --> 00:16:13.779 the scale. Which brings us to logarithmic change. 00:16:14.139 --> 00:16:16.700 It is arguably the most elegant solution in mathematics 00:16:16.700 --> 00:16:19.100 for tracking relative proportion over time. So 00:16:19.100 --> 00:16:20.580 what does this all mean? Let's break down the 00:16:20.580 --> 00:16:23.299 mechanics. Logarithmic change uses the natural 00:16:23.299 --> 00:16:26.279 logarithm of the ratio of the two numbers. Scientists 00:16:26.279 --> 00:16:29.139 scale this result by 100 to create units called 00:16:29.139 --> 00:16:32.210 log points, or centinovers. The brilliance of 00:16:32.210 --> 00:16:34.669 log points is that for very small fluctuations, 00:16:35.129 --> 00:16:37.830 they are virtually indistinguishable from standard 00:16:37.830 --> 00:16:41.669 percentages. A 1 % increase in a standard formula 00:16:41.669 --> 00:16:46.669 translates to roughly 0 .995 centintopers. So 00:16:46.669 --> 00:16:49.309 to the layperson, the day -to -day numbers look 00:16:49.309 --> 00:16:52.629 identical. Exactly. But log points possess a 00:16:52.629 --> 00:16:55.070 mathematical superpower that standard percentages 00:16:55.070 --> 00:16:58.639 lack, true additivity. And to understand why 00:16:58.639 --> 00:17:01.379 additivity matters, we have to resolve the trap 00:17:01.379 --> 00:17:03.320 I mentioned at the very beginning of this deep 00:17:03.320 --> 00:17:07.740 dive, the 50 % trap. Ah, yes. Standard percentages 00:17:07.740 --> 00:17:10.299 are inherently multiplicative, but they masquerade 00:17:10.299 --> 00:17:12.700 as additive. This is why the human brain gets 00:17:12.700 --> 00:17:15.140 wiped out by volatility. Let's run the math on 00:17:15.140 --> 00:17:18.140 the casual investor. You put $100 into a stock, 00:17:18.279 --> 00:17:21.839 it goes up 50%. You now have $150. Pretty straightforward. 00:17:22.140 --> 00:17:24.440 Then the next day, the market corrects and your 00:17:24.440 --> 00:17:27.279 stock drops 50%. Your brain naturally adds plus 00:17:27.279 --> 00:17:30.119 50 and minus 50 and assumes you're back at zero. 00:17:30.299 --> 00:17:32.180 You assume you still have your original $100. 00:17:32.759 --> 00:17:37.839 But a 50 % drop on $150 is $75. You are left 00:17:37.839 --> 00:17:41.460 with $75. You just lost a quarter of your entire 00:17:41.460 --> 00:17:43.720 portfolio, even though the headline percentages 00:17:43.720 --> 00:17:45.900 were equal and opposite. Standard percentages 00:17:45.900 --> 00:17:48.440 simply cannot be added together over time. A 00:17:48.440 --> 00:17:51.759 50 % loss actually requires a 100 % gain just 00:17:51.759 --> 00:17:55.319 to break even. But log points mechanically translate 00:17:55.480 --> 00:17:58.539 multiplicative relationships like scaling a portfolio 00:17:58.539 --> 00:18:01.759 up and down onto a genuinely additive scale, 00:18:02.680 --> 00:18:05.200 they calculate the continuous area of change. 00:18:05.319 --> 00:18:08.279 Which means if we map that exact same stock market 00:18:08.279 --> 00:18:10.920 journey using logarithmic change? If your stock 00:18:10.920 --> 00:18:13.779 gains 50 cent in tulpers and then loses 50 cent 00:18:13.779 --> 00:18:16.049 in tulpers, Well, you can literally just sum 00:18:16.049 --> 00:18:19.069 them up. Plus 50 minus 50 equals zero. You are 00:18:19.069 --> 00:18:21.869 left with exactly $100. Log points allow you 00:18:21.869 --> 00:18:24.250 to perfectly sum up a series of volatile changes 00:18:24.250 --> 00:18:26.809 over time without the math wildly skewing as 00:18:26.809 --> 00:18:28.730 you move up and down the scale. It completely 00:18:28.730 --> 00:18:31.450 neutralizes the 50 % trap. It's so clean. It 00:18:31.450 --> 00:18:35.279 really is. Log change is the unique two variable 00:18:35.279 --> 00:18:37.920 function that is both perfectly additive and 00:18:37.920 --> 00:18:40.960 matches relative change linearly. It removes 00:18:40.960 --> 00:18:43.880 the illusion of volatility and provides a stable 00:18:43.880 --> 00:18:46.980 baseline for complex systems. So you now possess 00:18:46.980 --> 00:18:49.440 the ultimate decoder ring for the data you encounter 00:18:49.440 --> 00:18:52.259 every day. You understand the mechanics of the 00:18:52.259 --> 00:18:55.259 unit illusion and why 100 centimeters feels scarier 00:18:55.259 --> 00:18:57.420 than one meter. And you know how to parse the 00:18:57.420 --> 00:19:00.180 vocabulary of deception, separating absolute 00:19:00.180 --> 00:19:02.700 percentage points from relative percent increases. 00:19:02.940 --> 00:19:05.440 You've seen how mean denominators force symmetry 00:19:05.440 --> 00:19:07.839 in computer models and how the additivity of 00:19:07.839 --> 00:19:10.259 log points saves you from the stock market's 00:19:10.259 --> 00:19:12.839 50 % trap. And this raises an important question 00:19:12.839 --> 00:19:15.279 for you to take forward. The next time you see 00:19:15.279 --> 00:19:17.500 a news headline declaring a massive percentage 00:19:17.500 --> 00:19:20.240 drop in a market or a huge swing in a political 00:19:20.240 --> 00:19:22.400 poll stop and check the math. Always check the 00:19:22.400 --> 00:19:25.079 math. What was the reference value? Is the narrator 00:19:25.079 --> 00:19:27.559 using a specific unit or a shifted denominator 00:19:27.559 --> 00:19:29.960 to artificially inflate the scale of the change? 00:19:30.640 --> 00:19:33.279 Look past the math they want you to see and calculate 00:19:33.279 --> 00:19:35.579 the math that actually matters. Remember the 00:19:35.579 --> 00:19:37.359 mountain. Don't just stare at the photograph 00:19:37.359 --> 00:19:40.240 and let the raw numbers induce panic. Find the 00:19:40.240 --> 00:19:42.160 scale on the map. Find your reference value. 00:19:42.670 --> 00:19:44.890 Thank you for joining us on this custom Taylor 00:19:44.890 --> 00:19:47.410 deep dive. Keep questioning the numbers around 00:19:47.410 --> 00:19:48.490 you and we'll catch you next time.