> A sailor is sailing her boat across the lake on a windy day. As the wind blows, she counters by turning the rudder in such a way so as to exactly offset the force of the wind. Back and forth she moves the rudder, yet the boat follows a straight line across the lake. A kindhearted yet naive person with no knowledge of wind or boats might look at this woman and say, “Someone get this sailor a new rudder! Hers is broken!” He thinks this because he cannot see any relationship between the movement of the rudder and the direction of the boat.

https://mixtape.scunning.com/01-introduction#do-not-confuse-...

A clear graphical set of illustrations is the bottom row in this famous set: https://en.wikipedia.org/wiki/Correlation#/media/File:Correl...

They have clear dependence; if you imagine fixing ("conditioning") x at a particular value and looking at the distribution of y at that value, it's different from the overall distribution of y (and vice versa). But the familiar linear correlation coefficient wouldn't indicate anything about this relationship.

I mentioned it in another comment, but the most trivial example is:

X ~ Unif(-1,1)

Y = X^2

In this case X and Y have a correlation of 0.

You can check the example described here: https://stats.stackexchange.com/questions/644280/stable-viol...

Judea Pearl’s book also goes into the above in some detail, as to why faithfulness might be a reasonable assumption.

Imagine your data points look like a U. There's no (lineral) correlation between x and y, you are equally likely to have a high value of y when x is high or low. But low values of y are associated with medium values of x, and a high value of y means x will be very high or very low.