Correlation vs causation: are all Rick Astley’s fans proud iPhone owners?

Did you know that the annual Google search interest for Rick Astley’s “Never Gonna Give You Up” strongly correlates with Apple’s (AAPL) opening price on the first trading day of the year?

Believe it or not, the world is full of these kinds of unexpected correlations. There’s even a webpage dedicated to finding them: https://www.tylervigen.com/spurious-correlations.

However, one very common mistake is to assume that whenever there is a correlation between two random variables, there must be an underlying causal relationship between them. In reality, that’s generally not the case.

Correlation: when two random variables travel together

Take two random variables, X and Y. To continue with our fun example from before, let’s say:

• X = the annual Google search interest for “Never Gonna Give You Up”.
• Y = Apple’s (AAPL) opening stock price on the first trading day of the year.

Mathematically, the correlation between two random variables is measured by the correlation coefficient:

🚨Formula alert! Don’t worry, we’ll break it down.

The denominator is the easy part: it’s simply the product of the standard deviations of X and Y. As you may remember, the standard deviation of a random variable measures how much its observations tend to spread out around its expected value.

The numerator is a little trickier: it’s the covariance of X and Y. The formula for the covariance (last one today, I promise!) is:

Let’s read it together. The covariance between X and Y is the expected value of the product of two quantities:

1. the difference between X and its expected value, and
2. the difference between Y and its expected value.

In other words, we first look at how far X and Y are from their respective expected values, multiply those two differences together, and then take the expected value of the result.

If X and Y tend to rise above their respective expected values at the same time, and fall below them at the same time, then they tend to move together. This is exactly the kind of relationship that covariance captures.

Let’s visualize this with the data from our recurring example:

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('ggplot')

array_1 = np.array([2.16667,11.1667,26.9167,20.9167,18.9167,20.1667,15.75,14,13.4167,19.0833,19.75,19.9167,17.8333,15.6667,31.0833,42.6667,60.5,39.4167])
array_2 = np.array([2.59,3.08,7.12,3.07,7.62,11.63,14.62,19.78,19.85,27.85,25.65,28.95,42.54,38.72,74.06,133.52,177.83,130.28])
array_1_name = "Searches for 'never gonna give you up'"
array_2_name = "Apple's stock price (AAPL)"

mean_1 = np.mean(array_1)
mean_2 = np.mean(array_2)
tx = np.linspace(2016, 2023, len(array_1))

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

ax1.plot(tx, array_1, color='black', marker='o', markersize=4)
ax1.axhline(y=mean_1, color='black', linestyle='--', alpha=0.5)
ax1.fill_between(tx, array_1, mean_1, where=(array_1 > mean_1), color='green', alpha=0.3, interpolate=True)
ax1.fill_between(tx, array_1, mean_1, where=(array_1 <= mean_1), color='red', alpha=0.3, interpolate=True)
ax1.set_title(array_1_name)
ax1.set_xlabel('Year')

ax2.plot(tx, array_2, color='black', marker='o', markersize=4)
ax2.axhline(y=mean_2, color='black', linestyle='--', alpha=0.5)
ax2.fill_between(tx, array_2, mean_2, where=(array_2 > mean_2), color='green', alpha=0.3, interpolate=True)
ax2.fill_between(tx, array_2, mean_2, where=(array_2 <= mean_2), color='red', alpha=0.3, interpolate=True)
ax2.set_title(array_2_name)
ax2.set_xlabel('Year')

plt.tight_layout()
plt.show()

You can clearly see that whenever searches for “Never Gonna Give You Up” are above their mean, Apple’s stock price tends to be above its mean as well. Likewise, when searches fall below their mean, Apple’s stock price tends to fall below its mean too. That is what we imply when we say that the two random variables travel together.

Correlation isn’t causation: Rick Astley’s fans own Android smartphones too

As we anticipated in the introduction, people often assume that whenever two random variables are correlated, there must be a causal relationship between them. But that conclusion doesn’t necessarily follow.

The source of the confusion is that correlation and causation are related, but they are not the same thing.

Let’s start with a real causal relationship. Suppose Apple sells more iPhones in a given quarter. All else being equal, this will increase the revenue Apple generates from iPhone sales. Here, we have a clear causal relationship: selling iPhones causes Apple’s iPhone revenue to increase.

We would therefore expect the two variables to be correlated. More iPhone sales generally mean more iPhone revenue, while fewer sales generally mean less revenue.

This gives us an important rule of thumb:

Causation often implies correlation, but correlation does not imply causation.

The second part is the one we need to be careful about. In our Rick Astley example, searches for “Never Gonna Give You Up” and Apple’s stock price are correlated, but that doesn’t mean that one causes the other. The correlation only tells us that the two variables tend to move together. It doesn’t tell us why.

Rickrolling is an internet prank where someone is tricked into clicking a link that unexpectedly leads to Rick Astley’s 1987 hit “Never Gonna Give You Up.”

So, perhaps Rick Astley fans really are enthusiastic iPhone owners. Perhaps they prefer Android. Perhaps they don’t own a smartphone at all. Our correlation cannot tell us.

One possible explanation is that both variables are being influenced by a third factor. For example, broader technological and internet adoption could simultaneously contribute to Apple’s growth and to the popularity of online memes, including Rickrolling. As more people spend time online, there may simply be more opportunities for “Never Gonna Give You Up” to be searched, shared, and Rickrolled into unsuspecting browsers.

Closing thoughts

So, are Rick Astley’s fans proud iPhone owners? Maybe. But again, maybe not. With the data we have, we simply can’t say. At least we’ve uncovered an interesting correlation, and, more importantly, learned why correlation and causation don’t always go hand in hand.