Question

"Two measured variables make a study correlational."

Unfortunately, this conflates (mistakenly treats as the same) the types of claims we can make with the types of statistical tests we can use. We pick out statistical tests based on the levels of measurement in our data, and while the measured/manipulated distinction is important for interpretation (manipulated variables allow for stronger arguments for causality), this doesn't effect our choice of test.

To make this clear, first give me an example that uses two measured variables but isn't tested using a correlation (a different test is right choice).

Second, give me an example where two variables aren't both just measured (at least one is manipulated) and yet a correlation is the proper test.

Answer 1

"Two measured variables make a study correlational."- this happens due to mathematical coincidence, mostly we are interested to test the causation. Actually Causation always implies corelation but the reverse is not true.

Example 1. Let us assume the two measured variables are a. number of new car sold in each of last 30 years in USA and b. Numbers of divorces in each of last 30 years in USA. One can observe a very clear positive corelation between these two measured variables. But there is not direct causality. Most probabily both the variables are related to some other factors.

Example 2. Let us assume the measured variable is a. number of new car sold in each of last 30 years in USA and the manipulated variable is b. Average wage of IT employee in each of last 30 years in USA . One can observe a very clear positive corelation between these two variables. There is causal relationship though the average wage is a manipulated variable.