Correlation Doesn’t Mean Causation! But What Does It Mean?

acquired into information science, there was a phrase that we’d all heard; everybody is aware of it, younger and outdated:

“Correlation doesn’t indicate causation.”

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

Listed below are two very fascinating information:

• International locations that eat extra pizza are likely to have larger math scores.
• The extra sun shades bought, the extra shark assaults happen.

Now, if that have been all the data you’ve got… 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 “in all probability not”.

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

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

That’s the place issues get fuzzy.

As a result of we are likely to deal with correlation like a obscure thought, we consider it as if it means “They’re form of associated”, or “They transfer collectively by some means”. However correlation isn’t only a feeling; it’s a exact mathematical measurement of how two variables transfer collectively.

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

So, let’s get into it!

What’s correlation?

When folks say two issues are “correlated,” they often imply considered one of three issues:

• “These two issues appear associated.”
• “These two issues transfer collectively.”
• “There’s some connection between these two issues.”

On a floor degree, all three of those should not improper, however they’re lacking some nuances.

Correlation will not be a vibe. It’s a measurement! And like all measurement, it solutions a really particular query.

Taking a step again, think about you accumulate 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.

If you take a look at this plot, you discover that the factors have a tendency to maneuver upward. So that you conclude, “As research time will increase, scores have a tendency to extend 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 means?

That query has considered one of three solutions:

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

The Math Behind Correlation

Let’s attempt to make interested by correlation extra intuitive. We are going to try this utilizing the Pearson correlation coefficient, which we will 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 persist with me and let’s unpack it with out turning it right into a lecture.

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

Covariance seems to be 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 destructive covariance.

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

Step 2: Normalize It

Covariance alone is tough to interpret as a result of it is dependent upon scale. To beat that, we divide by the usual deviations: $sigma_{X}$ and $sigma_{Y}$. This rescales all the things right into a clear vary: -1 to 1. That provides us frequent floor for evaluating variable values.

After these two steps, we will now calculate the Pearson coefficient! If we get:

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

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

What Totally different Correlations Look Like

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

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

What Correlation Truly Tells You

Correlation tells you: these variables transfer collectively in a structured means. 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.

In truth, we will plot the variety of ice cream gross sales and drowning incidents to get:

We are able to 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 actual driver is temperature: sizzling climate means extra ice cream gross sales, extra folks going to the seashore, and extra swimming.

So, although we will clearly see that correlation is actual, the reason is hidden.

Correlation and Nonlinearity

Now think about this relationship:

y = x²

That is clearly a robust relationship, as x will increase or decreases, y will increase! However for those who 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 matches the connection. It is a essential limitation. If the connection is curved, correlation might fail, even when a robust relationship exists.

So, as an alternative of considering: “Correlation = relationship”, it’s higher to assume: “Correlation = how nicely a straight line explains the connection.”

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 necessary nonetheless?

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

“One thing fascinating could be occurring right here.”

From there, you examine additional. Correlation measures alignment; additional investigation supplies a proof.

Closing Takeaway

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

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

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

“Hey… this seems to be fascinating.”

Now, the actual work begins, as we examine why that is actually fascinating.