# Correlation and causal relationship math

### Correlation and dependence - Wikipedia

Correlation, as a statistical term, is the extent to which two numerical variables have a linear relationship (that is, a relationship that increases or decreases at a . In statistics, dependence or association is any statistical relationship, whether causal or not, .. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson . This is called a causal relationship. The positive correlation between the number of churches and the number of deaths from cancer is an example of a.

### Statistical Language - Correlation and Causation

The packaging material might influence shelf life, but the shelf life cannot influence the packaging material used. The relationship is therefore causal. A bank manager is concerned with the number of customers whose accounts are overdrawn. Half of the accounts that become overdrawn in one week are randomly selected and the manager telephones the customer to offer advice. Any difference between the mean account balances after two months of the overdrawn accounts that did and did not receive advice can be causally attributed to the phone calls.

If two variables are causally related, it is possible to conclude that changes to the explanatory variable, X, will have a direct impact on Y. Non-causal relationships Not all relationships are causal. In non-causal relationships, the relationship that is evident between the two variables is not completely the result of one variable directly affecting the other. In the most extreme case, Two variables can be related to each other without either variable directly affecting the values of the other.

The two diagrams below illustrate mechanisms that result in non-causal relationships between X and Y. If two variables are not causally related, it is impossible to tell whether changes to one variable, X, will result in changes to the other variable, Y.

Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they are kind of implying causality. They're implying cause and effect, but really what the study looked at is correlation. The whole point of this is to understand the difference between causality and correlation because they're saying very different things.

And, as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time.

Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very, very different conclusions.

### Clearing up confusion between correlation and causation

So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So the study does show a correlation.

It does show, if we believe all of their data, that breakfast skipping correlates with obesity and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast and breakfast correlates with activity-- that all of these correlate. What they don't say-- and there's no data here that lets me know one way or the other-- what is causing what or maybe you have some underlying cause that is causing both.

So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning.

And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast-- and I'm not telling you that you should. I have no data one way or the other-- maybe you'll lose even more weight.

Maybe it's even a healthier thing to do. So they're trying to say, look, if you have breakfast it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Likewise they say breakfast skipping, or they're implying breakfast skipping, can cause obesity. But maybe it's the other way around. Maybe people who have high body fat-- maybe, for whatever reason, they're less likely to get hungry in the morning.

So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that.

One could be the physical activity. And these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude.

That we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning, so you're more likely to eat breakfast.

## Australian Bureau of Statistics

And obviously being physically active also makes it so that you burn calories. You have more muscle. So that you're not obese. So notice if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity.

Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person-- let's say this is the reality, that physical activity is causing both of these things.

And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active.

I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast?

Maybe that'll make things worse. And based on that, that the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe something other than physical activity. Maybe you have sleep, maybe people who sleep late and they're not getting enough sleep, maybe that leads to obesity.

And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment-- or they have to run to school in the case of students-- and maybe that's why they skip breakfast.

So once again, if you find someone that's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it will become even worse because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever.