If your stock portfolio loses 50 percent on Tuesday
and then gains 50 percent on Wednesday, how much
money did you lose? Most people would say zero,
right? Exactly. If your brain immediately answered,
oh, I broke even, well, your brain is lying to
you. And today we're going to show you exactly
how. Yeah, it's a massive blind spot for human
intuition. Because, you know, you are constantly
bombarded with numbers and headlines, corporate
reports, political polls, and they feel concrete.
They feel objective. Right. Numbers feel like
facts. But the way data is mathematically framed,
like before it ever reaches your screen, can
completely alter your perception of reality.
So our mission today is to act as your shortcut.
We want you to be truly well informed about the
hidden mechanics of data presentation. Which
is so crucial right now. It really is. We're
doing a deep dive into some fascinating source
material on the underlying math of relative change.
We're exploring why absolute numbers deceive
you. the specific vocabulary used to manipulate
percentages, and the mathematical workarounds
professionals use to find the actual truth. It's
a profound shift in perspective, really. Once
you learn to read the mechanical structure of
the math rather than just, you know, the surface
level numbers, you start seeing these subtle
framing choices everywhere. Everywhere. And the
illusion usually starts simply because our human
brains naturally default to absolute numbers.
We just prefer raw counts. OK, let's unpack this.
Because To really understand how we get tricked,
we need a functional way to think about it. I
agree. Visuals help a lot here. Right. So think
of an absolute number like saying a car costs
$50 ,000 as a photograph of a mountain. It shows
you the object, sure. But relative change is
the scale on a map of that same mountain. Oh,
that's a great way to frame it. Without the scale,
you have absolutely no idea if you're looking
at a tiny hill or Mount Everest. I mean, the
photograph gives you the what, but the scale
gives you the context. Exactly. And the problem
with relying solely on the photograph, you know,
the absolute change, is that it leaves you highly
susceptible to the unit illusion. The unit illusion.
Yeah. Absolute change is just taking a new value
and subtracting the reference value. But the
unit you choose to measure that change in completely
dictates how your brain reacts. Right, because
units feel arbitrary sometimes. They do. If you
measure the difference between a two meter piece
of wood and a one meter piece of wood, the absolute
difference is just one. A difference of one meter.
It sounds completely manageable. But if we measure
that exact same physical distance in centimeters,
you are now subtracting 100 centimeters from
200 centimeters. Oh, wow. Right. The absolute
difference is 100. Mechanically, the physical
reality in space is identical. But the number
100 triggers a significantly larger cognitive
reaction than the number 1. Exactly. It feels
like a bigger gap simply because of the unit
chosen. And relative change neutralizes that
illusion by mathematically stripping away the
units entirely. You take that actual change and
divide it by your reference value. Precisely.
It levels the playing field. So a change of one
meter divided by the starting one meter is one.
And a change of 100 centimeters divided by the
starting 100 centimeters is also one. Right.
It yields a 100 % increase either way. You get
a pure unit -less number that represents the
actual proportion of the shift. If we connect
this to the bigger picture, relative change is
the great equalizer. It judges the underlying
importance of a change rather than just the raw
amount. Give me a real -world example of that.
Well, if an item costs $50 and the price goes
up by an absolute value of $100 to a total of
$150, that item just tripled in cost. Yeah, the
relative change is massive. That hurts your wallet.
But if you are buying a $10 ,000 piece of industrial
equipment, and the price goes up by that same
absolute $100... The relative change is miniscule.
Exactly. The absolute $100 bill is physically
identical, but the mathematical importance of
that bill changes depending on the reference
value. I follow the logic, but I'm going to push
back slightly here. Is absolute change strictly
a trap, or is there a functional utility to it?
Oh, there's definitely utility. Because if I
think about my own paycheck... You know, if my
boss tells me I'm getting $100 bonus this week,
I care deeply about that absolute $100. Of course
you do. That is real utility. I can buy groceries
with that. Even if I calculate the relative change
to my total annual salary and realize it's a
fraction of a percent, the relative math doesn't
diminish the absolute utility of the cash in
my pocket. And that's a vital distinction, really.
Absolute numbers are completely necessary when
the real world utility of the unit matters to
you directly. A dollar has a fixed purchasing
power regardless of your total salary. Right.
The mathematical trap only springs when you are
attempting to compare two different scenarios
or, and this is key, when a narrator is attempting
to frame a change as inherently massive or insignificant.
Ah, so when someone is trying to sell you a narrative.
Yes, that is when relying on the absolute number
leaves your perception highly vulnerable to manipulation.
And if relative change is how we judge the importance
of a comparison, percentages are the vocabulary
we use to communicate it. But that vocabulary
is incredibly fragile, isn't it? Fragile is the
perfect word for it. It breaks so easily. Take
the example of two vehicles from our source material.
Let's say car M cost $50 ,000 and car L cost
$40 ,000. Depending entirely on which car you
anchor as your reference value, the grammatical
phrasing of the comparison dictates a completely
different mathematical reality. Walk me through
that. Well, if car L is the reference, the absolute
difference is $10 ,000. You divide 10 ,000 by
car L's price of 40 ,000 and you get 0 .25. So
you would say car M is 25 % more than car L.
Right. Or if you express it as a direct ratio
rather than a difference, car M is 125 % of the
cost of car L. Oh, I see. The specific words
more than versus of signal whether you are citing
the relative difference or the total ratio. Exactly.
The grammar literally dictates the math. But
if you flip the anchor, the math changes completely.
Like, if we start with CAR -M at $50 ,000 as
the reference, the difference is now negative
$10 ,000. Which changes everything. Right. Divide
that by $50 ,000, and you get negative 0 .20.
So suddenly, CAR -L is 20 % less than CAR -M,
or 80 % of what CAR -M costs. Yes. The physical
prices of the car is never moved. Yet you can
truthfully report the gap as either 25 % or 20%,
purely based on which bumper you decided to point
at first. It really requires hypervigilance from
the listener. I mean, if you aren't paying attention
to the specific noun acting as the baseline,
your brain will misinterpret the scale of the
math. Here's where it gets really interesting,
though. The ultimate manipulation of this vocabulary
is the percentages of percentages trap. Oh, this
one drives mathematicians crazy. We see this
constantly with interest rates. Let's say a bank
raises the interest rate on a mortgage from 3
% to 4%. The bank sends out a notice saying,
we have implemented a 1 % increase in your rate.
And that phrasing is fundamentally incorrect.
Completely wrong. Yeah. It is a conflation of
absolute and relative terms. The absolute change
between 3 % and 4 % is one percentage point.
Right. A point, not a percent. Exactly. But to
find the relative change, you must take that
one point absolute difference and divide it by
the original reference value of 3%. Which means
the actual relative change to your interest rate
is 33 and one -third percent. Yes. Your interest
burden didn't go up by one percent, it went up
by a third. It's difficult not to assume that
marketers and financial institutions intentionally
exploit the public's misunderstanding of a percentage
point versus a percent increase. It's a very
convenient confusion for them. Calling a massive
33 % spike in borrowing costs a 1 % increase.
Just brilliantly downplays the bad news. When
the underlying variable you are measuring is
already formatted as a percentage, the linguistic
waters become very muddy. Muddy and expensive.
Very true. It requires strict discipline to separate
absolute percentage points from relative percentage
change. Otherwise, the magnitude of the shift
is completely obscured. So the language of relative
change is easily weaponized? But the actual formula
for relative change has its own mechanical failures,
doesn't it? There are situations where taking
the new value, subtracting the old value, and
dividing by the old value just mathematically
breaks down. It does. The most obvious failure
is when your reference value is zero. You cannot
divide by zero. The function is undefined. Just
a classic math error. Right. But the far more
counterintuitive failure occurs when your reference
value is negative. The denominator sign completely
corrupts the logic of the result. Let's use temperature
to visualize this. Imagine the weather outside
goes from negative 10 degrees to negative 6 degrees.
The temperature got warmer. it increased by a
positive 4 degrees. But if we plug those numbers
into the standard relative change formula, we
take the actual change, which is positive 4,
and we divide it by the reference value, which
is negative 10. OK, so positive 4 divided by
negative 10. Which yields negative 0 .4. So the
formula outputs a relative change of negative
40%. It's telling me the temperature decreased
by 40%, even though I know definitively that
the absolute temperature increased. Yes. The
math is outputting a direct contradiction to
physical reality. That is wild. The classic formula
simply doesn't know how to handle the crossing
of the negative threshold. It yields a negative
relative change for a positive absolute increase.
So how do people fix that? Well... To bypass
this mechanical flaw, professionals use a special
variation called percent error. It's heavily
utilized in scientific fields when comparing
an experimental value, like a measurement taken
in a lab, to a theoretical or widely accepted
value. I like to visualize this using a dartboard.
Treat the theoretical accepted value like the
bullseye. The standard formula for percent error
puts absolute value brackets around the top part
of the fraction. Right. It forces the numerator
to be positive. So if you throw a dart, that
standard formula just measures the pure magnitude
of your miss. It tells you that you missed the
bullseye by two inches, but it strips away the
context of where the dart landed. What's fascinating
here is that scientists will frequently and intentionally
remove those absolute value brackets. Why? Because
the negative or positive sign of the error holds
vital physical context. The directionality matters
just as much as the magnitude. Right, because
if you take the brackets away, the math tells
you whether you overthrew or underthrew the dart.
And there's a brilliant example in the sources
regarding the speed of light. Oh yes, that's
a perfect illustration. If a physicist measures
the speed of light in a vacuum, and they calculate
a negative percent error without the brackets,
it simply means their experimental velocity was
slower than the accepted speed of light. The
dart landed short. They just need to recalibrate
their lasers. But if they calculate a positive
percent error without those brackets... Then
the dart went straight through the backboard.
A positive percent error means they recorded
a speed faster than light. They just mathematically
violated the theory of relativity. Which is huge.
That isn't a measurement error you just file
away in a drawer. That is data that rewrites
the laws of physics. The presence or absence
of that single negative sign completely alters
the interpretation of the data. And this raises
an important question about the rigidity of these
formulas. In percent error, you must adhere to
a strict operational order. It has to be experimental
value minus theoretical value. The order matters.
Yes, the values do not commute. If you flip them,
the sign flips, and a scientist might mistakenly
believe they broke relativity when in reality
they simply subtracted in the wrong direction.
Which introduces a massive mechanical headache
for systems that rely on automated math. Oh,
a nightmare for programmers. Right. If the basic
formula explodes when it hits zero and outputs
contradictions on negative numbers, how do software
programmers or economists stabilize their models?
They can't have a global financial model crash
just because a localized metric dropped below
zero. They stabilize it by fundamentally reinventing
the denominator. You see, the core mechanical
flaw with the classic relative change formula
is that it lacks symmetry. Standard division
inherently anchors to the past. The starting
reference value. Let's walk through why that
asymmetry happens. If a stock goes from $100
up to 125, the absolute change is $25. Yeah.
We divide that by the starting anchor of 100,
which gives us a 25 % increase. But if that same
stock drops from 125 back down to 100, the absolute
change is negative $25. OK, sure. However, your
anchor has now shifted. You are now dividing
that negative 25 by 125, which yields a 20 %
decrease. The journey is physically identical
in distance. But because the denominator changed,
the relative math is totally skewed. Going up
is 25%, coming down is 20%. It feels mathematically
unbalanced. It is. And in data science, that
lack of balance creates compounding errors over
time. A relative change from state A to state
B inherently must have the exact same magnitude
as moving from state B back to state A. So how
do they force this symmetry? Experts use indicators
of relative change. Instead of dividing by the
starting reference value, they replace the denominator
with a mean, an average of the two numbers. Wait,
so if I use the average of the two numbers as
the denominator, the math stabilizes. Let's test
it on the stock. Okay, the mean of 100 to 125
is 112 .5. So if the stock goes up, we divide
the positive $25 change by 1 .12 .5, which is
about 22 .2%. And if the stock goes down, you
divide the negative $25 change by that exact
same mean of 1 .12 .5, yielding negative 22 .2%.
The symmetry is perfectly restored. Whether you're
moving forward or backward, the denominator acts
as a fixed anchor midway between the two states.
Exactly. But mathematicians don't just use a
simple average. They tailor the type of mean
to the specific vulnerabilities of their field.
Like what? For instance, programmers checking
floating point values in computer code are heavily
advised to use maximum mean change. Floating
point values are essentially decimal calculations
in software, right? Why do they specifically
need the maximum mean? Because computers struggle
with dividing by incredibly tiny numbers. If
a programmer uses a standard average and the
numbers are close to zero, the resulting fraction
could become massively inflated. Which causes
a crash. Right, a floating point overflow. By
forcing the denominator to be the maximum, the
larger of the two numbers being compared, they
guarantee the denominator is as robust as possible,
mathematically minimizing the resulting fraction
and keeping the software stable. That is incredibly
clever. They are choosing a mathematical flavor
that actively defends against the hardware's
weakness. And economists do the exact opposite
right. They use minimum mean change. Yes. Economists
modeling massive financial systems will often
use the smaller of the two numbers as their denominator.
Dividing by the minimum value yields the largest
possible relative change. So it exaggerates the
shift. In econometrics, this acts as a built
-in stress test. It forces the economic model
to react to the most volatile possible interpretation
of the data, ensuring the system can survive
extreme variance. It's fascinating how the specific
denominator is chosen not just for accuracy but
for survival. But, you know, averaging the denominator
is still essentially just patching a broken system.
If we want a mathematical framework that is fundamentally
unbroken from the ground up, we have to abandon
standard percentages entirely. We have to change
the scale. Which brings us to logarithmic change.
It is arguably the most elegant solution in mathematics
for tracking relative proportion over time. So
what does this all mean? Let's break down the
mechanics. Logarithmic change uses the natural
logarithm of the ratio of the two numbers. Scientists
scale this result by 100 to create units called
log points, or centinovers. The brilliance of
log points is that for very small fluctuations,
they are virtually indistinguishable from standard
percentages. A 1 % increase in a standard formula
translates to roughly 0 .995 centintopers. So
to the layperson, the day -to -day numbers look
identical. Exactly. But log points possess a
mathematical superpower that standard percentages
lack, true additivity. And to understand why
additivity matters, we have to resolve the trap
I mentioned at the very beginning of this deep
dive, the 50 % trap. Ah, yes. Standard percentages
are inherently multiplicative, but they masquerade
as additive. This is why the human brain gets
wiped out by volatility. Let's run the math on
the casual investor. You put $100 into a stock,
it goes up 50%. You now have $150. Pretty straightforward.
Then the next day, the market corrects and your
stock drops 50%. Your brain naturally adds plus
50 and minus 50 and assumes you're back at zero.
You assume you still have your original $100.
But a 50 % drop on $150 is $75. You are left
with $75. You just lost a quarter of your entire
portfolio, even though the headline percentages
were equal and opposite. Standard percentages
simply cannot be added together over time. A
50 % loss actually requires a 100 % gain just
to break even. But log points mechanically translate
multiplicative relationships like scaling a portfolio
up and down onto a genuinely additive scale,
they calculate the continuous area of change.
Which means if we map that exact same stock market
journey using logarithmic change? If your stock
gains 50 cent in tulpers and then loses 50 cent
in tulpers, Well, you can literally just sum
them up. Plus 50 minus 50 equals zero. You are
left with exactly $100. Log points allow you
to perfectly sum up a series of volatile changes
over time without the math wildly skewing as
you move up and down the scale. It completely
neutralizes the 50 % trap. It's so clean. It
really is. Log change is the unique two variable
function that is both perfectly additive and
matches relative change linearly. It removes
the illusion of volatility and provides a stable
baseline for complex systems. So you now possess
the ultimate decoder ring for the data you encounter
every day. You understand the mechanics of the
unit illusion and why 100 centimeters feels scarier
than one meter. And you know how to parse the
vocabulary of deception, separating absolute
percentage points from relative percent increases.
You've seen how mean denominators force symmetry
in computer models and how the additivity of
log points saves you from the stock market's
50 % trap. And this raises an important question
for you to take forward. The next time you see
a news headline declaring a massive percentage
drop in a market or a huge swing in a political
poll stop and check the math. Always check the
math. What was the reference value? Is the narrator
using a specific unit or a shifted denominator
to artificially inflate the scale of the change?
Look past the math they want you to see and calculate
the math that actually matters. Remember the
mountain. Don't just stare at the photograph
and let the raw numbers induce panic. Find the
scale on the map. Find your reference value.
Thank you for joining us on this custom Taylor
deep dive. Keep questioning the numbers around
you and we'll catch you next time.