Correlation Doesn’t Imply Causation! However What Does It Imply?

acquired into information science, there was a phrase that we’d all heard; everybody is aware of it, younger and previous:

“Correlation doesn’t imply causation.”

It’s a catchy phrase, and also you’ve positively mentioned it a few times, and would possibly even have nodded confidently when another person mentioned it. Particularly for datasets that don’t relate to one another, however the place it’s humorous and intriguing to indicate causation!

Listed here are two very attention-grabbing information:

• Nations that eat extra pizza are inclined to have increased math scores.
• The extra sun shades offered, the extra shark assaults happen.

Now, if that had been all the knowledge you’ve… what do you have to conclude?

Does consuming pizza make you higher at math? Will shopping for a brand new pair of sun shades trigger a shark assault?

Although it’s humorous to consider, the reply to these questions is “probably not”.

And but, these are examples of one thing very actual: Correlation.

The query value asking now’s: if correlation doesn’t equal causation, then what does it imply?

That’s the place issues get fuzzy.

As a result of we are inclined to deal with correlation like a obscure thought, we consider it as if it means “They’re kind of related”, or “They move together somehow”. However correlation isn’t only a feeling; it’s a exact mathematical measurement of how two variables transfer collectively.

As an alternative of simply repeating the warning, let’s truly perceive the idea. When you do, these bizarre examples cease being shocking and begin making sense.

So, let’s get into it!

What’s correlation?

When folks say two issues are “correlated,” they normally imply one in all three issues:

• “Those two things seem related.”
• “Those two things move together.”
• “There’s some connection between those two things.”

On a floor degree, all three of those aren’t incorrect, however they’re lacking some nuances.

Correlation is just not a vibe. It’s a measurement! And like every measurement, it solutions a really particular query.

Taking a step again, think about you gather the information on what number of hours college students studied and their examination scores.

You plot it, and also you see one thing like this:

Every level represents one scholar. The x-axis is how lengthy they studied, and the y-axis is their rating.

Once you have a look at this plot, you discover that the factors have a tendency to maneuver upward. So that you conclude, “As study time increases, scores tend to increase too”, which is what we name a optimistic correlation.

However, is that only a pattern or is the information telling you one thing extra?

On this instance, the connection you simply plotted is: when one variable is above its common, the opposite tends to be above its common too.

That’s the important thing thought most individuals miss: correlation isn’t about uncooked values, it’s about how variables transfer relative to their averages.

So, the query correlation solutions is:

Do two variables transfer collectively in a constant manner?

That query has one in all three solutions:

• Up + up → optimistic correlation
• Up + down → adverse correlation
• No constant sample → no correlation

The Math Behind Correlation

Let’s attempt to make occupied with correlation extra intuitive. We’ll try this utilizing the Pearson correlation coefficient, which we are able to outline as:

$r = frac{cov(X, Y)}{ sigma_{X}.sigma_{Y}}$

Okay, I do know that equation isn’t what anybody thinks of once I say “intuitive”… However follow me and let’s unpack it with out turning it right into a lecture.

Step 1: Covariance (AKA Do They Transfer Collectively?)

Covariance appears at how two variables transfer relative to their averages. For instance, if each variables are above their averages, we get optimistic covariance; if one is above and the opposite beneath, we get adverse covariance.

Principally, covariance solutions: “Are these variables aligned in how they deviate from their averages?”

Step 2: Normalize It

Covariance alone is difficult to interpret as a result of it will depend on scale. To beat that, we divide by the usual deviations: $sigma_{X}$ and $sigma_{Y}$. This rescales every little thing right into a clear vary: -1 to 1. That provides us frequent floor for evaluating variable values.

After these two steps, we are able to now calculate the Pearson coefficient! If we get:

1. +1 → good optimistic relationship.
2. 0 → no linear relationship.
3. -1 → good adverse relationship.

This code merely measures how constantly these two variables transfer collectively—not how massive they’re, however how nicely aligned they’re.

What Completely different Correlations Look Like

• Left: sturdy optimistic correlation → clear upward sample
• Center: no correlation → random scatter
• Proper: sturdy adverse correlation → downward sample

Correlation measures consistency of motion, not simply whether or not two variables are associated.

What Correlation Really Tells You

Correlation tells you: these variables transfer collectively in a structured manner. It tells us that there’s a sample right here to concentrate to.

However, it does NOT inform you why or how they do, or whether or not one causes the opposite.

The traditional instance of correlation is that ice cream gross sales and drowning incidents are correlated.

Actually, we are able to plot the variety of ice cream gross sales and drowning incidents to get:

We will see a transparent upward relationship between these two variables… extra ice cream gross sales result in extra drownings?…

However that’s deceptive. As a result of the true driver is temperature: sizzling climate means extra ice cream gross sales, extra folks going to the seashore, and extra swimming.

So, although we are able to clearly see that correlation is actual, the reason is hidden.

Correlation and Nonlinearity

Now contemplate this relationship:

y = x²

That is clearly a robust relationship, as x will increase or decreases, y will increase! However in case you compute correlation:

np.corrcoef(x, y)[0,1]

You’ll get one thing near 0.

That’s as a result of correlation solely measures: How nicely a straight line suits the connection. It is a essential limitation. If the connection is curved, correlation might fail, even when a robust relationship exists.

So, as a substitute of considering: “Correlation = relationship”, it’s higher to suppose: “Correlation = how well a straight line explains the relationship.”

The Misunderstanding

The vagueness of the idea of correlation, and the way in which we’re taught it, results in some misunderstandings. Three quite common ones are:

1. Assuming causation: Simply because two variables transfer collectively doesn’t imply one causes the opposite.
2. Ignoring hidden variables: There could also be a 3rd issue driving each.
3. Lacking nonlinear relationships: Correlation solely sees straight-line patterns.

You be questioning now, if correlation is a quite simple time period that doesn’t inform us a lot, why is it vital nonetheless?

As a result of it’s extremely helpful as a primary sign. It tells you:

“Something interesting might be happening here.”

From there, you examine additional. Correlation measures alignment; additional investigation supplies an evidence.

Closing Takeaway

“Correlation doesn’t imply causation.” That’s true. However right here’s the issue: folks hear this and suppose: “Correlation is meaningless.” That’s not true!

Correlation measures how variables transfer collectively; it ranges from -1 to 1, captures linear relationships, nevertheless it does NOT indicate causation.

Correlation isn’t deceptive. We simply anticipate an excessive amount of from it when it isn’t attempting to clarify the world. It’s only a sign indicating:

“Hey… this looks interesting.”

Now, the true work begins, as we examine why that is actually attention-grabbing.