WEBVTT 00:00:00.000 --> 00:00:03.620 Welcome to the deep dive. Whether you are prepping 00:00:03.620 --> 00:00:06.660 for a massive meeting or cashing up on data analysis, 00:00:07.059 --> 00:00:09.400 or you're just insanely curious about how the 00:00:09.400 --> 00:00:11.519 world works, we are thrilled to have you here, 00:00:11.539 --> 00:00:13.900 the learner. Yeah, it's great to be here. So 00:00:13.900 --> 00:00:16.519 today, our mission is to permanently upgrade 00:00:16.519 --> 00:00:20.460 how you read statistics, news headlines, and 00:00:20.460 --> 00:00:23.260 just everyday data. We're looking at some incredibly 00:00:23.260 --> 00:00:26.760 detailed source material today, mainly technical 00:00:26.760 --> 00:00:29.300 articles and mathematical definitions that break 00:00:29.300 --> 00:00:32.039 down this concept of relative change. Which sounds 00:00:32.039 --> 00:00:34.960 simple but really isn't. Exactly. We are going 00:00:34.960 --> 00:00:37.560 to expose the invisible mechanics and honestly 00:00:37.560 --> 00:00:40.000 the hidden traps of relative versus absolute 00:00:40.000 --> 00:00:43.490 change. It is a perfect topic because There's 00:00:43.490 --> 00:00:45.450 this inherent assumption we all make about basic 00:00:45.450 --> 00:00:48.009 math, right? We expect it to represent absolute 00:00:48.009 --> 00:00:50.369 rock -solid reality, like, you know, counting 00:00:50.369 --> 00:00:52.229 stones. Right. Two stones are obviously more 00:00:52.229 --> 00:00:54.450 than one. Exactly. You add, you subtract, and 00:00:54.450 --> 00:00:56.710 the math just objectively works. We like our 00:00:56.710 --> 00:00:58.490 numbers to tell us exactly what is happening 00:00:58.490 --> 00:01:01.109 in the physical world without any ambiguity. 00:01:01.530 --> 00:01:04.430 But the sources reveal that, well, the moment 00:01:04.430 --> 00:01:06.569 you step into the world of comparing two numbers, 00:01:06.969 --> 00:01:09.670 that solid ground just completely crumbles beneath 00:01:09.670 --> 00:01:12.829 your feet. I mean... We're looking at a mathematical 00:01:12.829 --> 00:01:16.250 landscape that is honestly riddled with illusions. 00:01:16.609 --> 00:01:18.750 It really is the definition of numerical muddy 00:01:18.750 --> 00:01:21.870 waters. The standard formula we use every single 00:01:21.870 --> 00:01:25.069 day to compare numbers. I mean, percentages. 00:01:25.629 --> 00:01:27.810 It's remarkably fragile. Oh, yeah. It breaks 00:01:27.810 --> 00:01:30.549 down entirely under certain conditions to the 00:01:30.549 --> 00:01:32.549 point where scientists and mathematicians have 00:01:32.549 --> 00:01:34.930 had to engineer completely different systems. 00:01:36.219 --> 00:01:38.280 like logarithmic points just to make reality 00:01:38.280 --> 00:01:41.260 make sense. Wow. But before we explore how the 00:01:41.260 --> 00:01:43.939 math shatters, we kind of have to establish why 00:01:43.939 --> 00:01:45.920 we even use relative change in the first place 00:01:45.920 --> 00:01:48.459 rather than just subtracting one number from 00:01:48.459 --> 00:01:49.939 another. Right. Let's ground this for a second. 00:01:50.000 --> 00:01:51.980 So if I have five apples and you have three, 00:01:52.060 --> 00:01:54.079 I have two more apples than you. That is absolute 00:01:54.079 --> 00:01:56.319 change, just simple subtraction. My value minus 00:01:56.319 --> 00:01:59.739 the reference value. Yes. That is the actual 00:01:59.739 --> 00:02:03.329 difference. or the absolute difference. But absolute 00:02:03.329 --> 00:02:06.209 difference has a fatal flaw when we try to use 00:02:06.209 --> 00:02:09.590 it to understand the world. It is entirely dependent 00:02:09.590 --> 00:02:12.610 on the arbitrary units of measurement you happen 00:02:12.610 --> 00:02:15.830 to choose. OK, what do you mean by that? Well, 00:02:16.009 --> 00:02:19.509 that unit choice heavily distorts human perception. 00:02:20.169 --> 00:02:22.030 Think about measuring physical distance, right? 00:02:22.310 --> 00:02:25.340 If you compare a two meter length of rope to 00:02:25.340 --> 00:02:28.219 a one meter length of rope, the absolute difference 00:02:28.219 --> 00:02:31.259 is exactly one. Makes sense. One meter. Right. 00:02:31.620 --> 00:02:33.900 But if you measure that exact same rope in centimeters, 00:02:34.400 --> 00:02:36.659 you are now comparing 200 centimeters to 100 00:02:36.659 --> 00:02:39.020 centimeters. The absolute difference is now 100. 00:02:39.159 --> 00:02:42.860 Oh, I see. And to the human brain, that latter 00:02:42.860 --> 00:02:46.870 comparison, it feels subjectively bigger. Like, 00:02:46.889 --> 00:02:48.689 it feels like a much more significant difference, 00:02:49.090 --> 00:02:51.349 even though it describes the exact same physical 00:02:51.349 --> 00:02:53.389 reality. It gives the impression of a larger 00:02:53.389 --> 00:02:55.370 gap simply because we change the word at the 00:02:55.370 --> 00:02:57.590 end of the number. Exactly. We see this in daily 00:02:57.590 --> 00:03:00.110 life all the time, actually. Imagine you're buying 00:03:00.110 --> 00:03:04.509 a simple everyday item, like a $5 cup of coffee. 00:03:04.849 --> 00:03:07.889 If the cafe suddenly raises the price by an absolute 00:03:07.889 --> 00:03:11.430 difference of $100, so now you're paying $105 00:03:11.430 --> 00:03:14.150 for a standard latte. There would be riots in 00:03:14.150 --> 00:03:18.080 the streets. Right. complete chaos. But if you're 00:03:18.080 --> 00:03:22.460 buying a $10 ,000 used car and the dealer tax 00:03:22.460 --> 00:03:25.500 on an absolute difference of $100 for, I don't 00:03:25.500 --> 00:03:28.180 know, some premium floor mats, raising the total 00:03:28.180 --> 00:03:31.930 to $10 ,100... you probably wouldn't even blink. 00:03:32.030 --> 00:03:34.449 You wouldn't even notice it. Exactly. The absolute 00:03:34.449 --> 00:03:37.469 change is mathematically identical in both scenarios. 00:03:37.909 --> 00:03:41.389 It's $100. But it doesn't tell us anything about 00:03:41.389 --> 00:03:43.949 the severity of the event. That is a brilliant 00:03:43.949 --> 00:03:47.129 illustration. And this is exactly why mathematicians 00:03:47.129 --> 00:03:49.990 introduced relative change. To find the relative 00:03:49.990 --> 00:03:53.349 change, you take that actual change, the $100, 00:03:53.990 --> 00:03:56.379 and you divide it by your reference value. Okay, 00:03:56.400 --> 00:03:58.259 so going back to the coffee. Right, so going 00:03:58.259 --> 00:04:01.620 from a $5 coffee to $105 coffee, the change is 00:04:01.620 --> 00:04:04.080 100 divided by the reference of 5 giving you 00:04:04.080 --> 00:04:07.319 20. That is a 2000 % increase. Which explains 00:04:07.319 --> 00:04:10.990 the riots. Exactly. But with the car, the change 00:04:10.990 --> 00:04:14.189 of 100 divided by the reference of 10 ,000 is 00:04:14.189 --> 00:04:18.370 0 .01, a mere 1 % increase. And the beauty of 00:04:18.370 --> 00:04:20.750 that calculation is that relative change is perfectly 00:04:20.750 --> 00:04:24.769 unit -less. It strips away whether we are talking 00:04:24.769 --> 00:04:27.470 about dollars or centimeters or meters, and just 00:04:27.470 --> 00:04:30.269 gives us the pure proportional impact. It does. 00:04:30.649 --> 00:04:33.829 But... Relative change introduces its own psychological 00:04:33.829 --> 00:04:36.230 quirks. Well, of course it does. Yeah, specifically 00:04:36.230 --> 00:04:38.610 based on what you choose to put at the bottom 00:04:38.610 --> 00:04:41.410 of that fraction as your reference value. The 00:04:41.410 --> 00:04:43.449 source material highlights this classic scenario 00:04:43.449 --> 00:04:46.089 we can call the valuable assets example. Imagine 00:04:46.089 --> 00:04:49.829 two cars on a lot. Car M costs $50 ,000. Car 00:04:49.829 --> 00:04:52.529 L costs $40 ,000. Okay, so let's compare them. 00:04:52.670 --> 00:04:55.709 If we use the cheaper car, Car L, as our baseline 00:04:55.709 --> 00:04:58.290 reference value, the absolute difference is $10 00:04:58.290 --> 00:05:00.730 ,000. We divide that by the $40 ,000 reference, 00:05:00.810 --> 00:05:04.009 which gives us 0 .25. Got it. Therefore, Car 00:05:04.009 --> 00:05:08.889 M costs 25 % more than Car L. Or, you know, phrase 00:05:08.889 --> 00:05:12.310 differently. It is 125 % of the cost of Car L. 00:05:12.410 --> 00:05:15.050 But the narrative completely flips if we change 00:05:15.050 --> 00:05:17.389 our anchor, doesn't it? Like, what if we choose 00:05:17.389 --> 00:05:20.370 the more expensive car, car M, as our baseline 00:05:20.370 --> 00:05:22.850 reference? Then the absolute difference is still 00:05:22.850 --> 00:05:26.569 $10 ,000. But now we divide it by the new reference 00:05:26.569 --> 00:05:30.389 value of $50 ,000. 10 ,000 divided by 50 ,000 00:05:30.389 --> 00:05:35.850 is 0 .20. Wow. So now, car L costs 20 % less 00:05:35.850 --> 00:05:39.389 than car M, or it is 80 % of the cost. Exactly. 00:05:39.850 --> 00:05:42.129 The math is completely accurate in both statements, 00:05:42.610 --> 00:05:45.550 but the story it tells changes drastically depending 00:05:45.550 --> 00:05:48.470 on where you plant your flag. And marketers absolutely 00:05:48.470 --> 00:05:51.430 exploit this on purpose. I mean, if a salesperson 00:05:51.430 --> 00:05:53.629 is trying to upsell you to the luxury model, 00:05:54.009 --> 00:05:56.930 they are going to say for just 20 percent less 00:05:56.930 --> 00:05:58.750 than the premium model, you can get the standard 00:05:58.750 --> 00:06:00.529 one. Yeah, they definitely won't say the premium 00:06:00.529 --> 00:06:03.569 model is 25 percent more expensive. Right. They 00:06:03.569 --> 00:06:05.730 deliberately choose the reference value that 00:06:05.730 --> 00:06:08.290 outputs the smaller percentage because 20 percent 00:06:08.290 --> 00:06:11.430 feels like a smaller gap to bridge. They're basically 00:06:11.430 --> 00:06:13.569 reverse engineering the math to make their sales 00:06:13.569 --> 00:06:16.009 pitch sound more persuasive to the human brain. 00:06:16.160 --> 00:06:18.860 The narrative is entirely shaped by whatever 00:06:18.860 --> 00:06:22.000 number we subtly place in the denominator, but 00:06:22.000 --> 00:06:26.279 the confusion gets infinitely worse when the 00:06:26.279 --> 00:06:28.579 thing we are measuring is already a percentage. 00:06:29.000 --> 00:06:31.860 Oh, this leads to one of the most common statistical 00:06:31.860 --> 00:06:34.860 errors people make in everyday life, especially 00:06:34.860 --> 00:06:37.839 in finance and the news. The sources call this 00:06:37.839 --> 00:06:40.139 the percentages of percentages trap. It happens 00:06:40.139 --> 00:06:42.120 constantly. Let's say you have a savings account, 00:06:42.220 --> 00:06:44.920 right? Yeah. and the bank raises your interest 00:06:44.920 --> 00:06:48.399 rate from 3 % up to 4%. If a journalist reporting 00:06:48.399 --> 00:06:51.379 on this says the interest rate increased by 1%, 00:06:51.379 --> 00:06:54.339 they are making a highly misleading, factually 00:06:54.339 --> 00:06:57.019 incorrect statement. Because 4 minus 3 is 1. 00:06:57.079 --> 00:06:59.800 That's the absolute change. But we're dealing 00:06:59.800 --> 00:07:01.779 with a rate here. Right. It's like comparing 00:07:01.779 --> 00:07:04.019 the speed of a car with its acceleration. Like, 00:07:04.139 --> 00:07:06.220 they sound like they belong to the same category, 00:07:06.220 --> 00:07:08.300 but they measure entirely different dimensions 00:07:08.300 --> 00:07:11.180 of reality. I like that. Yeah. Speed is your 00:07:11.180 --> 00:07:14.350 current state, say. 60 miles per hour. Acceleration 00:07:14.350 --> 00:07:16.850 is the change in that state. If you confuse your 00:07:16.850 --> 00:07:18.750 current speed with your rate of acceleration, 00:07:19.329 --> 00:07:21.569 you're going to completely misunderstand the 00:07:21.569 --> 00:07:24.129 physics of your road trip. That is exactly the 00:07:24.129 --> 00:07:26.709 dynamic at play. The crucial distinction you 00:07:26.709 --> 00:07:28.790 have to make is between percent and percentage 00:07:28.790 --> 00:07:33.889 points. Percentage points. Yes. The raw subtraction 00:07:33.889 --> 00:07:37.589 of 4 minus 3 is one percentage point. But if 00:07:37.589 --> 00:07:41.199 we want the true relative change, we have to 00:07:41.199 --> 00:07:43.379 use the formula. OK, walk me through it. The 00:07:43.379 --> 00:07:45.420 actual change is 1. The reference value where 00:07:45.420 --> 00:07:49.939 we started is 3. 1 divided by 3 is 0 .333. So 00:07:49.939 --> 00:07:53.060 the relative change is actually a 33 .3 % increase. 00:07:53.259 --> 00:07:56.720 Wow. Saying your rate went up by 1 % implies 00:07:56.720 --> 00:08:00.230 a tiny, almost negligible bump. But saying your 00:08:00.230 --> 00:08:03.730 rate increased by 33 .3 % sounds massive. Exactly. 00:08:03.889 --> 00:08:06.790 And yet the 33 .3 % figure is the mathematically 00:08:06.790 --> 00:08:09.649 honest description of the relative change. Precision 00:08:09.649 --> 00:08:12.430 with these terms really dictates how we interpret 00:08:12.430 --> 00:08:15.230 massive systemic changes in our economy, like 00:08:15.230 --> 00:08:18.009 inflation or mortgage rates. If you just subtracted 00:08:18.009 --> 00:08:19.829 two percentages, you must say percentage points. 00:08:20.110 --> 00:08:22.490 If you divided them to find the proportion, use 00:08:22.490 --> 00:08:25.050 the percent sign. OK, so we see how this math 00:08:25.050 --> 00:08:27.750 can be manipulated by a car salesman or misunderstood 00:08:27.750 --> 00:08:31.420 in a bank lobby. But surely the hard sciences, 00:08:31.639 --> 00:08:33.659 you know, where precision is a matter of life 00:08:33.659 --> 00:08:36.240 and death, they must have figured out a way around 00:08:36.240 --> 00:08:38.740 this ambiguity. Well, in the scientific community, 00:08:38.940 --> 00:08:41.240 this concept is formalized as percent error. 00:08:41.700 --> 00:08:43.960 This happens when you are testing a physical 00:08:43.960 --> 00:08:46.720 measurement against a theoretically accepted 00:08:46.720 --> 00:08:50.460 actual value. The formula is the absolute change 00:08:50.460 --> 00:08:53.690 between your experimental measured value and 00:08:53.690 --> 00:08:56.389 the theoretical value divided by that theoretical 00:08:56.389 --> 00:08:59.009 value. But there is a fascinating debate in the 00:08:59.009 --> 00:09:01.450 literature about how to handle the output of 00:09:01.450 --> 00:09:04.090 that formula, specifically regarding absolute 00:09:04.090 --> 00:09:07.090 value brackets. Yes. Usually when calculating 00:09:07.090 --> 00:09:10.110 percent error, engineers put absolute value brackets 00:09:10.110 --> 00:09:12.970 around the top part of the fraction. This forces 00:09:12.970 --> 00:09:15.149 the resulting percentage to always be positive. 00:09:15.570 --> 00:09:18.840 You just say, I had a 5 % error. regardless of 00:09:18.840 --> 00:09:20.700 whether your measurement was too high or too 00:09:20.700 --> 00:09:22.720 low. So we're basically intentionally putting 00:09:22.720 --> 00:09:24.659 blinders on. We're saying, I don't care if I 00:09:24.659 --> 00:09:26.759 overshot or undershot the target, I just want 00:09:26.759 --> 00:09:28.960 to know how far the arrow landed from the bullseye. 00:09:29.259 --> 00:09:31.639 And that makes sense for a factory floor, you 00:09:31.639 --> 00:09:34.860 know. In basic quality control context, the direction 00:09:34.860 --> 00:09:37.120 of the error often doesn't dictate the next step. 00:09:37.740 --> 00:09:39.960 Any error beyond a certain threshold just means 00:09:39.960 --> 00:09:42.940 you throw the part away. Right. But the source 00:09:42.940 --> 00:09:45.740 text discusses the strategic choice in advanced 00:09:45.740 --> 00:09:49.019 fields to purposely drop those absolute value 00:09:49.019 --> 00:09:52.600 brackets. If you drop them, the sign of the number 00:09:52.600 --> 00:09:55.639 positive or negative suddenly provides critical 00:09:55.639 --> 00:09:58.779 directional context. I was completely blown away 00:09:58.779 --> 00:10:00.799 by the speed of light example in the research. 00:10:01.039 --> 00:10:03.440 Because it shows how dropping those brackets 00:10:03.440 --> 00:10:06.000 changes everything. Like if you are in a physics 00:10:06.000 --> 00:10:08.779 lab calculating the speed of light and you just 00:10:08.779 --> 00:10:11.000 let the formula run with the negative sign. If 00:10:11.000 --> 00:10:13.539 your calculation yields a negative percent error, 00:10:13.799 --> 00:10:16.320 it simply means your measured velocity was less 00:10:16.320 --> 00:10:18.100 than the accepted speed of light. You're a little 00:10:18.100 --> 00:10:20.120 slow, no big deal, just a normal day in the lab. 00:10:20.279 --> 00:10:23.159 Right, but a positive percent error means your 00:10:23.159 --> 00:10:25.360 experimental number was bigger than the theoretical 00:10:25.360 --> 00:10:27.500 number. A positive percent error means you have 00:10:27.500 --> 00:10:29.639 just measured something moving faster than the 00:10:29.639 --> 00:10:32.159 speed of light. You have just violated Albert 00:10:32.159 --> 00:10:34.960 Einstein's theory of relativity. That is not 00:10:34.960 --> 00:10:37.179 just a rounding error on a spreadsheet. That 00:10:37.179 --> 00:10:40.559 is a massively newsworthy Nobel Prize -winning 00:10:40.559 --> 00:10:45.289 result. In theoretical physics, retaining that 00:10:45.289 --> 00:10:48.730 negative sign is paramount. The formula clearly 00:10:48.730 --> 00:10:50.909 works well when we use it carefully and understand 00:10:50.909 --> 00:10:54.269 the context, but we teased at the very beginning 00:10:54.269 --> 00:10:57.029 of this deep dive that this elegant formula harbors 00:10:57.029 --> 00:10:59.210 a dark side. It really does. There are places 00:10:59.210 --> 00:11:01.230 on the number line where the standard formula 00:11:01.230 --> 00:11:03.809 for relative change completely self -destructs. 00:11:03.870 --> 00:11:05.690 The flaw is built right into the architecture 00:11:05.690 --> 00:11:08.389 of the formula. To find relative change, you 00:11:08.389 --> 00:11:11.769 must divide by the reference value. So we immediately 00:11:11.769 --> 00:11:14.169 run into a fundamental mathematical domain restriction. 00:11:14.769 --> 00:11:16.990 What happens if your reference value is exactly 00:11:16.990 --> 00:11:19.710 zero? You can't divide by zero. It's a mathematical 00:11:19.710 --> 00:11:22.230 black hole. If a startup company made zero dollars 00:11:22.230 --> 00:11:24.149 last year and one million dollars this year, 00:11:24.370 --> 00:11:26.929 you literally cannot calculate a percentage increase. 00:11:27.230 --> 00:11:29.669 It is mathematically undefined. But it actually 00:11:29.669 --> 00:11:31.830 gets stranger than zero. What happens when your 00:11:31.830 --> 00:11:34.049 reference value is a negative number? The classical 00:11:34.049 --> 00:11:36.549 formula literally breaks. Let's walk through 00:11:36.549 --> 00:11:38.570 a temperature reading, or perhaps a financial 00:11:38.570 --> 00:11:41.509 deficit. Let's say your reading starts at minus 00:11:41.509 --> 00:11:43.730 10. That is your reference value. Negative 10. 00:11:44.330 --> 00:11:46.809 Then the reading goes up to minus 6. OK, so from 00:11:46.809 --> 00:11:50.230 negative 10 to negative 6, the absolute change 00:11:50.230 --> 00:11:54.289 is an increase of positive 4. Like, I am getting 00:11:54.289 --> 00:11:57.309 warmer, or I am climbing out of debt. Exactly. 00:11:57.470 --> 00:11:59.509 Now, plug that into the relative change formula. 00:11:59.669 --> 00:12:02.350 Your actual change, which is positive 4, divided 00:12:02.350 --> 00:12:04.490 by the reference value, which is negative 10. 00:12:04.690 --> 00:12:07.409 OK, 4 divided by negative 10. That gives us negative 00:12:07.409 --> 00:12:11.669 0 .4. which translates to a negative 40%. Wait, 00:12:12.090 --> 00:12:14.450 the temperature physically went up by four units? 00:12:14.690 --> 00:12:17.330 Yep. But the relative change formula spits out 00:12:17.330 --> 00:12:19.830 a negative percentage, actively telling us we're 00:12:19.830 --> 00:12:21.590 getting colder when we are physically getting 00:12:21.590 --> 00:12:24.840 warmer. The math is straight up lying to us. 00:12:25.100 --> 00:12:27.120 It is entirely broken for negative reference 00:12:27.120 --> 00:12:29.700 values. And because the classical formula lies 00:12:29.700 --> 00:12:32.240 to us with zeros and negatives, mathematicians 00:12:32.240 --> 00:12:35.139 have had to engineer specialized patches. They've 00:12:35.139 --> 00:12:37.500 created alternative formulas called indicators 00:12:37.500 --> 00:12:39.720 of relative change. But how do you even patch 00:12:39.720 --> 00:12:43.100 a broken formula? You fix the denominator. Instead 00:12:43.100 --> 00:12:45.320 of just dividing blindly by the initial reference 00:12:45.320 --> 00:12:48.490 value, which we'll call x, Mathematicians divide 00:12:48.490 --> 00:12:50.789 by a function of both values involved, x and 00:12:50.789 --> 00:12:53.889 y. They replace the denominator with a mean, 00:12:54.269 --> 00:12:56.429 an average of sorts, between the starting and 00:12:56.429 --> 00:12:58.789 ending points. Like the arithmetic mean, just 00:12:58.789 --> 00:13:01.049 adding the two numbers and dividing by two. Yes, 00:13:01.110 --> 00:13:03.090 the arithmetic mean is one option, but there 00:13:03.090 --> 00:13:05.730 are others, like the geometric mean, which multiplies 00:13:05.730 --> 00:13:08.110 the numbers and takes the root, or the harmonic 00:13:08.110 --> 00:13:11.049 mean. The text actually highlights some highly 00:13:11.049 --> 00:13:14.009 specific use cases, where fields choose a different 00:13:14.009 --> 00:13:16.210 denominator to protect their work. Like what? 00:13:16.299 --> 00:13:19.279 Take computer science. When a programmer is writing 00:13:19.279 --> 00:13:22.399 code that compares two floating point numbers, 00:13:22.659 --> 00:13:25.059 essentially numbers, with decimals that can get 00:13:25.059 --> 00:13:27.299 infinitely tiny, they have to be very careful. 00:13:27.519 --> 00:13:29.379 Because if one of those numbers is incredibly 00:13:29.379 --> 00:13:32.519 close to zero, dividing by it could cause a division 00:13:32.519 --> 00:13:35.440 by zero error or an overflow, which literally 00:13:35.440 --> 00:13:37.960 crashes the software. Exactly. So the recommended 00:13:37.960 --> 00:13:40.159 patch is the maximum mean. You always divide 00:13:40.159 --> 00:13:42.580 by whichever of the two numbers is larger, creating 00:13:42.580 --> 00:13:44.659 a safe, stable anchor that prevents the program 00:13:44.659 --> 00:13:47.340 from imploding. Okay, and what about econometrics? 00:13:47.840 --> 00:13:50.100 The sources mention they do the exact opposite. 00:13:50.799 --> 00:13:53.019 Econometrics often recommends the minimum mean 00:13:53.019 --> 00:13:55.879 change, using the smaller of the two numbers 00:13:55.879 --> 00:13:59.000 as the denominator. Economists often deal with 00:13:59.000 --> 00:14:02.799 growth from a small base. By intentionally dividing 00:14:02.799 --> 00:14:05.860 by the smaller number, it yields a larger percentage 00:14:05.860 --> 00:14:09.000 change, which establishes the maximum possible 00:14:09.000 --> 00:14:11.539 bounds of a change, setting a strict upper limit 00:14:11.539 --> 00:14:13.929 on volatility. Honestly, I have to push back 00:14:13.929 --> 00:14:16.330 on this entire concept because arguing that we 00:14:16.330 --> 00:14:18.289 should just change the denominator based on the 00:14:18.289 --> 00:14:20.990 field you're in, like econometrics versus programming, 00:14:21.710 --> 00:14:23.929 it feels a little bit like cheating. It does 00:14:23.929 --> 00:14:25.830 sound like that, yeah. It feels like we are just 00:14:25.830 --> 00:14:28.029 engineering a number that behaves nicely for 00:14:28.029 --> 00:14:31.669 our specific spreadsheet. Is there a true relative 00:14:31.669 --> 00:14:34.330 change or is it all just human invention? That 00:14:34.330 --> 00:14:36.960 is a fair critique. But mathematicians aren't 00:14:36.960 --> 00:14:39.500 just doing this to cheat. They're trying to introduce 00:14:39.500 --> 00:14:42.200 a vital property called symmetry. Symmetry. Yes. 00:14:42.539 --> 00:14:44.340 Let's look at classical relative change again. 00:14:44.440 --> 00:14:47.000 Say you start at 40 and go to 50. The increase 00:14:47.000 --> 00:14:50.399 is 10. 10 divided by 40 is a 25 % increase. Now 00:14:50.399 --> 00:14:54.379 go backward. Start at 50 and drop to 40. The 00:14:54.379 --> 00:14:58.220 decrease is 10. 10 divided by 50 is a 20 % decrease. 00:14:58.259 --> 00:15:01.120 OK. The physical distance is identical. 10 units. 00:15:01.320 --> 00:15:04.440 Yeah. But the percentages don't match. 25 versus 00:15:04.440 --> 00:15:07.070 20. They are completely asymmetrical. But look 00:15:07.070 --> 00:15:09.269 what happens if you use a mean in the denominator. 00:15:10.149 --> 00:15:12.409 Let's use the arithmetic mean. The average of 00:15:12.409 --> 00:15:16.440 40 and 50 is 45. Now going up 10 divided by 45 00:15:16.440 --> 00:15:20.080 is roughly 22 .2 percent. Okay. And going down 00:15:20.080 --> 00:15:22.679 negative 10 divided by 45 is roughly negative 00:15:22.679 --> 00:15:26.139 22 .2 percent. Oh wow! The relative change from 00:15:26.139 --> 00:15:29.419 X to Y has the exact same magnitude as the change 00:15:29.419 --> 00:15:32.559 from Y to X. The mirror is perfect. It creates 00:15:32.559 --> 00:15:35.019 a perfectly symmetrical mathematical reflection. 00:15:35.480 --> 00:15:37.980 But to answer your question about whether it's 00:15:37.980 --> 00:15:39.720 cheating, many mathematicians actually agree 00:15:39.720 --> 00:15:42.340 with you. These mean -based indicators are just 00:15:42.340 --> 00:15:44.740 situational patches. They're workarounds. They 00:15:44.740 --> 00:15:47.240 aren't the ultimate cure. So what is the actual 00:15:47.240 --> 00:15:49.860 cure? The source material points to a mathematically 00:15:49.860 --> 00:15:52.740 superior general purpose replacement that solves 00:15:52.740 --> 00:15:55.639 almost all of these problems. It's called logarithmic 00:15:55.639 --> 00:15:58.620 change. Logarithmic change. So instead of using 00:15:58.620 --> 00:16:01.200 simple division, you take the natural logarithm 00:16:01.200 --> 00:16:04.120 of the ratio of the two numbers. But what does 00:16:04.120 --> 00:16:06.100 that actually look like for someone trying to 00:16:06.100 --> 00:16:08.679 picture this mentally? Think of a standard ruler. 00:16:09.720 --> 00:16:12.440 Classical percentages treat every jump like a 00:16:12.440 --> 00:16:15.320 fixed inch on that ruler. But because the starting 00:16:15.320 --> 00:16:18.539 point keeps changing, the inches warp and stretch, 00:16:18.879 --> 00:16:22.320 which is why 25 % up doesn't equal 20 % down. 00:16:22.480 --> 00:16:24.799 Right. That makes sense. A logarithmic function 00:16:24.799 --> 00:16:27.799 stops looking at the flat distance. Instead, 00:16:28.200 --> 00:16:30.980 it measures the proportional steps. It asks, 00:16:31.419 --> 00:16:34.460 what is the fundamental ratio of change? And 00:16:34.460 --> 00:16:36.679 translates that into a smooth, continuous curve. 00:16:37.139 --> 00:16:39.639 So instead of taking a clunky percentage of whatever 00:16:39.639 --> 00:16:41.879 arbitrary number you happen to be standing on, 00:16:42.240 --> 00:16:44.480 log points measure the actual continuous momentum 00:16:44.480 --> 00:16:46.340 of the change. And when you scale this mathematical 00:16:46.340 --> 00:16:49.019 output by 100, the resulting units are called 00:16:49.019 --> 00:16:51.600 log points. You might also see them called centi 00:16:51.600 --> 00:16:54.519 -napers in the literature. Centi meaning 100th 00:16:54.519 --> 00:16:57.159 and never named after John Napier, the mathematician 00:16:57.159 --> 00:16:59.779 who invented logarithms. It's a fantastic unit 00:16:59.779 --> 00:17:03.299 because for small changes, log points mirror 00:17:03.299 --> 00:17:06.319 our classical percentages perfectly. An increase 00:17:06.319 --> 00:17:09.380 of 1 % in classical math is roughly an increase 00:17:09.380 --> 00:17:14.119 of 0 .95 log points. They look practically identical 00:17:14.119 --> 00:17:16.359 to the naked eye. But they possess a superpower 00:17:16.359 --> 00:17:19.150 that classical percentages completely lack. which 00:17:19.150 --> 00:17:22.589 is additivity. Let's walk through the ultimate 00:17:22.589 --> 00:17:25.950 trap of classical percentages. Imagine you invest 00:17:25.950 --> 00:17:29.990 $100. The market goes up and you see a 50 % increase. 00:17:30.369 --> 00:17:34.430 Okay, 50 % of $100 is $50. I now have $150. I'm 00:17:34.430 --> 00:17:36.990 feeling great. But the next day the market crashes 00:17:36.990 --> 00:17:40.400 and you suffer a 50 % decrease. Well... 50 % 00:17:40.400 --> 00:17:44.240 of my current $150 is $75. So I subtract $75. 00:17:44.279 --> 00:17:47.180 I am left with $75. Wait. See the problem? Yeah, 00:17:47.240 --> 00:17:50.039 I went up 50 % and down 50%. I should be back 00:17:50.039 --> 00:17:52.640 exactly where I started at $100. But I'm down 00:17:52.640 --> 00:17:57.380 $25. Plus 50 % and minus 50 % leaves you at negative 00:17:57.380 --> 00:18:00.519 25 % of your starting point. Classical percentages 00:18:00.519 --> 00:18:02.720 do not add up to zero. They compound their errors. 00:18:02.880 --> 00:18:05.619 That's crazy. But here is the magic of logarithmic 00:18:05.619 --> 00:18:09.079 change. A 50 log point increase followed by a 00:18:09.079 --> 00:18:12.279 50 log point decrease leaves you exactly at zero. 00:18:12.420 --> 00:18:15.410 It perfectly sums up. It's like classical percentages 00:18:15.410 --> 00:18:18.109 are playing a bad game of telephone, like every 00:18:18.109 --> 00:18:20.250 time you make a change up or down, the errors 00:18:20.250 --> 00:18:22.309 compound and you drift further and further from 00:18:22.309 --> 00:18:26.730 reality. But log points are like a perfect, incorruptible 00:18:26.730 --> 00:18:30.390 financial ledger. Every single transaction balances 00:18:30.390 --> 00:18:32.730 out flawlessly no matter how many times the market 00:18:32.730 --> 00:18:35.470 bounces up and down. Mathematically speaking, 00:18:35.890 --> 00:18:38.170 log change is the unique two variable function 00:18:38.170 --> 00:18:40.289 that perfectly balances a ledger every single 00:18:40.289 --> 00:18:43.099 time, meaning it's perfectly additive. while 00:18:43.099 --> 00:18:45.380 still mapping smoothly onto our intuitive human 00:18:45.380 --> 00:18:47.039 understanding of what a percentage should look 00:18:47.039 --> 00:18:50.039 like for small, everyday numbers. It is mathematically 00:18:50.039 --> 00:18:52.519 flawless in ways that percentages simply aren't. 00:18:52.680 --> 00:18:55.200 Okay, let's recap this journey we've just been 00:18:55.200 --> 00:18:57.920 on. We started by realizing how simply changing 00:18:57.920 --> 00:19:00.660 a reference value can manipulate the perceived 00:19:00.660 --> 00:19:03.119 price of a car to make a sales pitch sound better. 00:19:03.369 --> 00:19:05.970 We discovered the vocabulary trap of percentage 00:19:05.970 --> 00:19:07.869 points, and while your bank rate didn't go up 00:19:07.869 --> 00:19:12.289 1%, it went up 33 .3%. Right. We saw how classical 00:19:12.289 --> 00:19:14.990 math actively breaks down and lies to us when 00:19:14.990 --> 00:19:17.430 we dip into negative numbers. We looked at how 00:19:17.430 --> 00:19:19.769 computer scientists and economists patch the 00:19:19.769 --> 00:19:23.390 formula. And finally, we discovered the flawless 00:19:23.390 --> 00:19:27.509 balancing ledger of log points. I'd say you are 00:19:27.509 --> 00:19:29.309 now officially equipped to see right through 00:19:29.309 --> 00:19:31.569 the mathematical illusions hiding in everyday 00:19:31.569 --> 00:19:34.099 statistics. But I want to leave you with a final 00:19:34.099 --> 00:19:36.980 lingering thought, directly inspired by the conclusions 00:19:36.980 --> 00:19:39.660 in our source text. The authors explicitly note 00:19:39.660 --> 00:19:41.579 that despite log points being mathematically 00:19:41.579 --> 00:19:44.460 superior, perfectly additive, and highly recommended 00:19:44.460 --> 00:19:47.160 by statisticians, they have not seen widespread 00:19:47.160 --> 00:19:49.980 use. Almost no one outside of highly specific 00:19:49.980 --> 00:19:52.680 technical fields uses them. Which is wild considering 00:19:52.680 --> 00:19:54.920 they fix the exact math we all use every single 00:19:54.920 --> 00:19:59.089 day. Why is that? What is it about human psychology 00:19:59.089 --> 00:20:02.230 or perhaps the ingrained structures of our financial 00:20:02.230 --> 00:20:05.930 and media institutions that makes us cling to 00:20:05.930 --> 00:20:09.779 the flawed asymmetrical classical percentage, 00:20:10.119 --> 00:20:12.480 even when a perfect mathematical alternative 00:20:12.480 --> 00:20:14.200 is sitting right in front of us. Well, think 00:20:14.200 --> 00:20:16.920 about the stock market example we just did. Because 00:20:16.920 --> 00:20:19.380 classical percentages are asymmetrical, they 00:20:19.380 --> 00:20:22.339 allow financial institutions to mask volatility. 00:20:22.859 --> 00:20:25.759 If a stock drops 50 percent, it needs a massive 00:20:25.759 --> 00:20:28.680 100 percent gain just to recover to where it 00:20:28.680 --> 00:20:31.240 started. But a broker can simply tell their clients, 00:20:31.619 --> 00:20:34.220 we're up 75 percent this year. after a massive 00:20:34.220 --> 00:20:36.319 crash and it sounds fantastic even though the 00:20:36.319 --> 00:20:39.180 client is technically still underwater. We choose 00:20:39.180 --> 00:20:41.619 the broken math because the broken math is incredibly 00:20:41.619 --> 00:20:43.960 effective at selling us a better narrative. It 00:20:43.960 --> 00:20:45.660 is definitely something to ponder the next time 00:20:45.660 --> 00:20:48.140 you read a dramatic headline about a 50 % drop 00:20:48.140 --> 00:20:50.660 in the market. Keep your eyes open, remember 00:20:50.660 --> 00:20:52.559 the broken foundation beneath the numbers and 00:20:52.559 --> 00:20:54.059 we'll see you on the next deep dive.