Welcome to the deep dive. Whether you are prepping
for a massive meeting or cashing up on data analysis,
or you're just insanely curious about how the
world works, we are thrilled to have you here,
the learner. Yeah, it's great to be here. So
today, our mission is to permanently upgrade
how you read statistics, news headlines, and
just everyday data. We're looking at some incredibly
detailed source material today, mainly technical
articles and mathematical definitions that break
down this concept of relative change. Which sounds
simple but really isn't. Exactly. We are going
to expose the invisible mechanics and honestly
the hidden traps of relative versus absolute
change. It is a perfect topic because There's
this inherent assumption we all make about basic
math, right? We expect it to represent absolute
rock -solid reality, like, you know, counting
stones. Right. Two stones are obviously more
than one. Exactly. You add, you subtract, and
the math just objectively works. We like our
numbers to tell us exactly what is happening
in the physical world without any ambiguity.
But the sources reveal that, well, the moment
you step into the world of comparing two numbers,
that solid ground just completely crumbles beneath
your feet. I mean... We're looking at a mathematical
landscape that is honestly riddled with illusions.
It really is the definition of numerical muddy
waters. The standard formula we use every single
day to compare numbers. I mean, percentages.
It's remarkably fragile. Oh, yeah. It breaks
down entirely under certain conditions to the
point where scientists and mathematicians have
had to engineer completely different systems.
like logarithmic points just to make reality
make sense. Wow. But before we explore how the
math shatters, we kind of have to establish why
we even use relative change in the first place
rather than just subtracting one number from
another. Right. Let's ground this for a second.
So if I have five apples and you have three,
I have two more apples than you. That is absolute
change, just simple subtraction. My value minus
the reference value. Yes. That is the actual
difference. or the absolute difference. But absolute
difference has a fatal flaw when we try to use
it to understand the world. It is entirely dependent
on the arbitrary units of measurement you happen
to choose. OK, what do you mean by that? Well,
that unit choice heavily distorts human perception.
Think about measuring physical distance, right?
If you compare a two meter length of rope to
a one meter length of rope, the absolute difference
is exactly one. Makes sense. One meter. Right.
But if you measure that exact same rope in centimeters,
you are now comparing 200 centimeters to 100
centimeters. The absolute difference is now 100.
Oh, I see. And to the human brain, that latter
comparison, it feels subjectively bigger. Like,
it feels like a much more significant difference,
even though it describes the exact same physical
reality. It gives the impression of a larger
gap simply because we change the word at the
end of the number. Exactly. We see this in daily
life all the time, actually. Imagine you're buying
a simple everyday item, like a $5 cup of coffee.
If the cafe suddenly raises the price by an absolute
difference of $100, so now you're paying $105
for a standard latte. There would be riots in
the streets. Right. complete chaos. But if you're
buying a $10 ,000 used car and the dealer tax
on an absolute difference of $100 for, I don't
know, some premium floor mats, raising the total
to $10 ,100... you probably wouldn't even blink.
You wouldn't even notice it. Exactly. The absolute
change is mathematically identical in both scenarios.
It's $100. But it doesn't tell us anything about
the severity of the event. That is a brilliant
illustration. And this is exactly why mathematicians
introduced relative change. To find the relative
change, you take that actual change, the $100,
and you divide it by your reference value. Okay,
so going back to the coffee. Right, so going
from a $5 coffee to $105 coffee, the change is
100 divided by the reference of 5 giving you
20. That is a 2000 % increase. Which explains
the riots. Exactly. But with the car, the change
of 100 divided by the reference of 10 ,000 is
0 .01, a mere 1 % increase. And the beauty of
that calculation is that relative change is perfectly
unit -less. It strips away whether we are talking
about dollars or centimeters or meters, and just
gives us the pure proportional impact. It does.
But... Relative change introduces its own psychological
quirks. Well, of course it does. Yeah, specifically
based on what you choose to put at the bottom
of that fraction as your reference value. The
source material highlights this classic scenario
we can call the valuable assets example. Imagine
two cars on a lot. Car M costs $50 ,000. Car
L costs $40 ,000. Okay, so let's compare them.
If we use the cheaper car, Car L, as our baseline
reference value, the absolute difference is $10
,000. We divide that by the $40 ,000 reference,
which gives us 0 .25. Got it. Therefore, Car
M costs 25 % more than Car L. Or, you know, phrase
differently. It is 125 % of the cost of Car L.
But the narrative completely flips if we change
our anchor, doesn't it? Like, what if we choose
the more expensive car, car M, as our baseline
reference? Then the absolute difference is still
$10 ,000. But now we divide it by the new reference
value of $50 ,000. 10 ,000 divided by 50 ,000
is 0 .20. Wow. So now, car L costs 20 % less
than car M, or it is 80 % of the cost. Exactly.
The math is completely accurate in both statements,
but the story it tells changes drastically depending
on where you plant your flag. And marketers absolutely
exploit this on purpose. I mean, if a salesperson
is trying to upsell you to the luxury model,
they are going to say for just 20 percent less
than the premium model, you can get the standard
one. Yeah, they definitely won't say the premium
model is 25 percent more expensive. Right. They
deliberately choose the reference value that
outputs the smaller percentage because 20 percent
feels like a smaller gap to bridge. They're basically
reverse engineering the math to make their sales
pitch sound more persuasive to the human brain.
The narrative is entirely shaped by whatever
number we subtly place in the denominator, but
the confusion gets infinitely worse when the
thing we are measuring is already a percentage.
Oh, this leads to one of the most common statistical
errors people make in everyday life, especially
in finance and the news. The sources call this
the percentages of percentages trap. It happens
constantly. Let's say you have a savings account,
right? Yeah. and the bank raises your interest
rate from 3 % up to 4%. If a journalist reporting
on this says the interest rate increased by 1%,
they are making a highly misleading, factually
incorrect statement. Because 4 minus 3 is 1.
That's the absolute change. But we're dealing
with a rate here. Right. It's like comparing
the speed of a car with its acceleration. Like,
they sound like they belong to the same category,
but they measure entirely different dimensions
of reality. I like that. Yeah. Speed is your
current state, say. 60 miles per hour. Acceleration
is the change in that state. If you confuse your
current speed with your rate of acceleration,
you're going to completely misunderstand the
physics of your road trip. That is exactly the
dynamic at play. The crucial distinction you
have to make is between percent and percentage
points. Percentage points. Yes. The raw subtraction
of 4 minus 3 is one percentage point. But if
we want the true relative change, we have to
use the formula. OK, walk me through it. The
actual change is 1. The reference value where
we started is 3. 1 divided by 3 is 0 .333. So
the relative change is actually a 33 .3 % increase.
Wow. Saying your rate went up by 1 % implies
a tiny, almost negligible bump. But saying your
rate increased by 33 .3 % sounds massive. Exactly.
And yet the 33 .3 % figure is the mathematically
honest description of the relative change. Precision
with these terms really dictates how we interpret
massive systemic changes in our economy, like
inflation or mortgage rates. If you just subtracted
two percentages, you must say percentage points.
If you divided them to find the proportion, use
the percent sign. OK, so we see how this math
can be manipulated by a car salesman or misunderstood
in a bank lobby. But surely the hard sciences,
you know, where precision is a matter of life
and death, they must have figured out a way around
this ambiguity. Well, in the scientific community,
this concept is formalized as percent error.
This happens when you are testing a physical
measurement against a theoretically accepted
actual value. The formula is the absolute change
between your experimental measured value and
the theoretical value divided by that theoretical
value. But there is a fascinating debate in the
literature about how to handle the output of
that formula, specifically regarding absolute
value brackets. Yes. Usually when calculating
percent error, engineers put absolute value brackets
around the top part of the fraction. This forces
the resulting percentage to always be positive.
You just say, I had a 5 % error. regardless of
whether your measurement was too high or too
low. So we're basically intentionally putting
blinders on. We're saying, I don't care if I
overshot or undershot the target, I just want
to know how far the arrow landed from the bullseye.
And that makes sense for a factory floor, you
know. In basic quality control context, the direction
of the error often doesn't dictate the next step.
Any error beyond a certain threshold just means
you throw the part away. Right. But the source
text discusses the strategic choice in advanced
fields to purposely drop those absolute value
brackets. If you drop them, the sign of the number
positive or negative suddenly provides critical
directional context. I was completely blown away
by the speed of light example in the research.
Because it shows how dropping those brackets
changes everything. Like if you are in a physics
lab calculating the speed of light and you just
let the formula run with the negative sign. If
your calculation yields a negative percent error,
it simply means your measured velocity was less
than the accepted speed of light. You're a little
slow, no big deal, just a normal day in the lab.
Right, but a positive percent error means your
experimental number was bigger than the theoretical
number. A positive percent error means you have
just measured something moving faster than the
speed of light. You have just violated Albert
Einstein's theory of relativity. That is not
just a rounding error on a spreadsheet. That
is a massively newsworthy Nobel Prize -winning
result. In theoretical physics, retaining that
negative sign is paramount. The formula clearly
works well when we use it carefully and understand
the context, but we teased at the very beginning
of this deep dive that this elegant formula harbors
a dark side. It really does. There are places
on the number line where the standard formula
for relative change completely self -destructs.
The flaw is built right into the architecture
of the formula. To find relative change, you
must divide by the reference value. So we immediately
run into a fundamental mathematical domain restriction.
What happens if your reference value is exactly
zero? You can't divide by zero. It's a mathematical
black hole. If a startup company made zero dollars
last year and one million dollars this year,
you literally cannot calculate a percentage increase.
It is mathematically undefined. But it actually
gets stranger than zero. What happens when your
reference value is a negative number? The classical
formula literally breaks. Let's walk through
a temperature reading, or perhaps a financial
deficit. Let's say your reading starts at minus
10. That is your reference value. Negative 10.
Then the reading goes up to minus 6. OK, so from
negative 10 to negative 6, the absolute change
is an increase of positive 4. Like, I am getting
warmer, or I am climbing out of debt. Exactly.
Now, plug that into the relative change formula.
Your actual change, which is positive 4, divided
by the reference value, which is negative 10.
OK, 4 divided by negative 10. That gives us negative
0 .4. which translates to a negative 40%. Wait,
the temperature physically went up by four units?
Yep. But the relative change formula spits out
a negative percentage, actively telling us we're
getting colder when we are physically getting
warmer. The math is straight up lying to us.
It is entirely broken for negative reference
values. And because the classical formula lies
to us with zeros and negatives, mathematicians
have had to engineer specialized patches. They've
created alternative formulas called indicators
of relative change. But how do you even patch
a broken formula? You fix the denominator. Instead
of just dividing blindly by the initial reference
value, which we'll call x, Mathematicians divide
by a function of both values involved, x and
y. They replace the denominator with a mean,
an average of sorts, between the starting and
ending points. Like the arithmetic mean, just
adding the two numbers and dividing by two. Yes,
the arithmetic mean is one option, but there
are others, like the geometric mean, which multiplies
the numbers and takes the root, or the harmonic
mean. The text actually highlights some highly
specific use cases, where fields choose a different
denominator to protect their work. Like what?
Take computer science. When a programmer is writing
code that compares two floating point numbers,
essentially numbers, with decimals that can get
infinitely tiny, they have to be very careful.
Because if one of those numbers is incredibly
close to zero, dividing by it could cause a division
by zero error or an overflow, which literally
crashes the software. Exactly. So the recommended
patch is the maximum mean. You always divide
by whichever of the two numbers is larger, creating
a safe, stable anchor that prevents the program
from imploding. Okay, and what about econometrics?
The sources mention they do the exact opposite.
Econometrics often recommends the minimum mean
change, using the smaller of the two numbers
as the denominator. Economists often deal with
growth from a small base. By intentionally dividing
by the smaller number, it yields a larger percentage
change, which establishes the maximum possible
bounds of a change, setting a strict upper limit
on volatility. Honestly, I have to push back
on this entire concept because arguing that we
should just change the denominator based on the
field you're in, like econometrics versus programming,
it feels a little bit like cheating. It does
sound like that, yeah. It feels like we are just
engineering a number that behaves nicely for
our specific spreadsheet. Is there a true relative
change or is it all just human invention? That
is a fair critique. But mathematicians aren't
just doing this to cheat. They're trying to introduce
a vital property called symmetry. Symmetry. Yes.
Let's look at classical relative change again.
Say you start at 40 and go to 50. The increase
is 10. 10 divided by 40 is a 25 % increase. Now
go backward. Start at 50 and drop to 40. The
decrease is 10. 10 divided by 50 is a 20 % decrease.
OK. The physical distance is identical. 10 units.
Yeah. But the percentages don't match. 25 versus
20. They are completely asymmetrical. But look
what happens if you use a mean in the denominator.
Let's use the arithmetic mean. The average of
40 and 50 is 45. Now going up 10 divided by 45
is roughly 22 .2 percent. Okay. And going down
negative 10 divided by 45 is roughly negative
22 .2 percent. Oh wow! The relative change from
X to Y has the exact same magnitude as the change
from Y to X. The mirror is perfect. It creates
a perfectly symmetrical mathematical reflection.
But to answer your question about whether it's
cheating, many mathematicians actually agree
with you. These mean -based indicators are just
situational patches. They're workarounds. They
aren't the ultimate cure. So what is the actual
cure? The source material points to a mathematically
superior general purpose replacement that solves
almost all of these problems. It's called logarithmic
change. Logarithmic change. So instead of using
simple division, you take the natural logarithm
of the ratio of the two numbers. But what does
that actually look like for someone trying to
picture this mentally? Think of a standard ruler.
Classical percentages treat every jump like a
fixed inch on that ruler. But because the starting
point keeps changing, the inches warp and stretch,
which is why 25 % up doesn't equal 20 % down.
Right. That makes sense. A logarithmic function
stops looking at the flat distance. Instead,
it measures the proportional steps. It asks,
what is the fundamental ratio of change? And
translates that into a smooth, continuous curve.
So instead of taking a clunky percentage of whatever
arbitrary number you happen to be standing on,
log points measure the actual continuous momentum
of the change. And when you scale this mathematical
output by 100, the resulting units are called
log points. You might also see them called centi
-napers in the literature. Centi meaning 100th
and never named after John Napier, the mathematician
who invented logarithms. It's a fantastic unit
because for small changes, log points mirror
our classical percentages perfectly. An increase
of 1 % in classical math is roughly an increase
of 0 .95 log points. They look practically identical
to the naked eye. But they possess a superpower
that classical percentages completely lack. which
is additivity. Let's walk through the ultimate
trap of classical percentages. Imagine you invest
$100. The market goes up and you see a 50 % increase.
Okay, 50 % of $100 is $50. I now have $150. I'm
feeling great. But the next day the market crashes
and you suffer a 50 % decrease. Well... 50 %
of my current $150 is $75. So I subtract $75.
I am left with $75. Wait. See the problem? Yeah,
I went up 50 % and down 50%. I should be back
exactly where I started at $100. But I'm down
$25. Plus 50 % and minus 50 % leaves you at negative
25 % of your starting point. Classical percentages
do not add up to zero. They compound their errors.
That's crazy. But here is the magic of logarithmic
change. A 50 log point increase followed by a
50 log point decrease leaves you exactly at zero.
It perfectly sums up. It's like classical percentages
are playing a bad game of telephone, like every
time you make a change up or down, the errors
compound and you drift further and further from
reality. But log points are like a perfect, incorruptible
financial ledger. Every single transaction balances
out flawlessly no matter how many times the market
bounces up and down. Mathematically speaking,
log change is the unique two variable function
that perfectly balances a ledger every single
time, meaning it's perfectly additive. while
still mapping smoothly onto our intuitive human
understanding of what a percentage should look
like for small, everyday numbers. It is mathematically
flawless in ways that percentages simply aren't.
Okay, let's recap this journey we've just been
on. We started by realizing how simply changing
a reference value can manipulate the perceived
price of a car to make a sales pitch sound better.
We discovered the vocabulary trap of percentage
points, and while your bank rate didn't go up
1%, it went up 33 .3%. Right. We saw how classical
math actively breaks down and lies to us when
we dip into negative numbers. We looked at how
computer scientists and economists patch the
formula. And finally, we discovered the flawless
balancing ledger of log points. I'd say you are
now officially equipped to see right through
the mathematical illusions hiding in everyday
statistics. But I want to leave you with a final
lingering thought, directly inspired by the conclusions
in our source text. The authors explicitly note
that despite log points being mathematically
superior, perfectly additive, and highly recommended
by statisticians, they have not seen widespread
use. Almost no one outside of highly specific
technical fields uses them. Which is wild considering
they fix the exact math we all use every single
day. Why is that? What is it about human psychology
or perhaps the ingrained structures of our financial
and media institutions that makes us cling to
the flawed asymmetrical classical percentage,
even when a perfect mathematical alternative
is sitting right in front of us. Well, think
about the stock market example we just did. Because
classical percentages are asymmetrical, they
allow financial institutions to mask volatility.
If a stock drops 50 percent, it needs a massive
100 percent gain just to recover to where it
started. But a broker can simply tell their clients,
we're up 75 percent this year. after a massive
crash and it sounds fantastic even though the
client is technically still underwater. We choose
the broken math because the broken math is incredibly
effective at selling us a better narrative. It
is definitely something to ponder the next time
you read a dramatic headline about a 50 % drop
in the market. Keep your eyes open, remember
the broken foundation beneath the numbers and
we'll see you on the next deep dive.