The Normal Curve Doesn't End - It Just Pretends to !!

What this means for Marketing Measurement

Many people (even some data scientists) look at a normal distribution and assume its tails meet the x-axis at some neat tidy “end.”

They don’t.

The Normal curve never touches the axis. It is asymptotic - meaning it keeps approaching infinity, run very close parallel to the x axis but never quite touches it.

Fun Fact - Normal Distribution ranges from -∞ to +∞.

Why does this matter?

Because this simple mathematical subtlety has big consequences in Marketing Measurement and MMM.

When you assume the curve “ends” you are implicitly saying:

“Extreme outcomes have zero probability.”

Consequence in Marketing Mix Modeling (MMM).

Just this week, I read a comment saying “we chose Normal distribution as a prior for our Bayesian MMM and thus it didn’t restrict the coefficients signs”.

While the statement is not wrong, people are very poor judges of what distribution to apply. Most importantly they don’t understand the full properties of a probability distribution.

The thinking is only - does it have only positive support? ok then I am using it for my marketing cases.

Or like mentioned above - does it have both + and - support? then it serves well for my no sign constraining case.

If your prior distribution is Normal and you think the tails die off completely, you will underestimate the probability of outliers - like unusually high ROIs or extreme coefficient values.

Applying normal also means believing that 68% of your data is in and around the mean !! Remember the 68-95-99.7% rule.

Further most people don’t think about probability convolutions. Think many ripples in water coinciding.

The Fat Tails

Sometimes to account for improbable events, one needs to provide provision for it. T distribution has fatter tails than normal distribution because it accounts for more uncertainty. This remarkable property is why it is used in T-SNE as well. I wrote about it few years ago.

Also see the links about Richard’s Quartet.

Bottomline :

Before you apply any probability distribution, pls know thoroughly about its properties. Whether a distribution has positive or negative support is good starting point. But one needs to go far beyond that.

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Thanks for reading.

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