Understanding Scatterplot relationships and its causes

Question

I'm trying to get started with statistics and probability and stumbled upon a scatterplot. I just don't understand the answer of the question, it's probably pretty easy but I just don't get it.

Your mathematical understanding can hopefully help me.

Question with Scatterplot

I get why a and c is wrong and why b is correct. But I don't get why d and e is not correct. If x increments, y increments too. So if the head length increments, the skull width increments too. That's what the chart shows, doesn't it?

In addition, would this

• Skull width and head length are positively associated.

be correct? Why, or why not?

Kind regards!

Source: Openintro.org, Page 12

Answer 1

You said, "if the head length increments, the skull width increments too." But does it? Consider (83, 52) and (86, 50) which are two of the points (just eyeballing the coordinates). The head length "went up" from 83 to 86, but the skull width "went down" from 52 to 50. So the scatter plot doesn't support a blanket statement like "if the head length increments, the skull width increments too."

Now if you plotted a best-fit line, $y = mx + b$, then the slope of that line would be positive. The equation is a model of the relationship between head length and skull width. It is true that according to the model, if the head length increments, the skull width increments too.

But even if we are only talking about the model, not the noisy points themselves, the scatterplot still doesn't tell us that a longer head causes the skull to be wider. Note, they are using the word "cause" here in its scientific sense, of one thing actually causing another thing, the way the sun and moon cause the tides. They aren't talking about "mathematical causality," as when an increase in $x$ "causes" an increase in $y$ when $y = mx + b$ holds.

A scatterplot never establishes scientific/real-world causality, although it can be evidence of such. The safe/proper way that statisticians talk about the two variables in a case like this is to say that "an increase in the head length is associated with an increase in the skull width." It's a useful turn of phrase to know.

Answer 2

Something other than head length or skull width (e.g. genes or environment) causes both to be larger or both to be smaller.

An example: It is found that people living in certain neighborhoods are healthier than those in certain other neighborhoods. Does that mean living in those neighborhoods causes them to be healthy? If so, then relocating to one of those neighborhoods might make you healthier. Does it mean that improving your health will cause you to live in one of those neighborhoods? It may be that better health and residence in those neighborhoods are both caused by having a higher income. In that case, just moving there won't make you healthier, and getting healthier won't cause you to relocate.