# What is the correlation of two variables that have an exponential relationship?

Assume that you have datapoints $(x_i,y_i)$ that have an exponential relationship: $(x_i,\log(y_i))$ approximate a straight line. This means that the statistical variables $X$ and $\log(Y)$ are positively correlated. But what can you say about $X$ and $Y$ in this case?

• Is $Y$ a deterministic function of $X$ or is there some random error? What is the range of $X$ & its mean value? Commented Mar 16, 2016 at 17:18
• @gung - X and Y are two statistical variables that are measured. The goal is to derive a statistical relationship between the two of them. Commented Mar 17, 2016 at 8:16

I understand your question is asking around the relationship between $X$ and $Y$, which you have some understanding of already, but would like to explore the statistical basis of.

I don't know of a statistical process that will enable you to test this effectively. One problem is often in the assumptions about distribution of the variables. If $X$ and $Y$ are related by a exponential relationship, then even if the distribution of $log(Y)$ is normal, the distribution of $Y$ cannot be.

I think best is one of two directions (I would choose the former, personally)

• Establish a relationship between $X$ and $Log(Y)$; or
• Establish a relationship between $10^X$ and $Y$

Then once you have statistically proved the relationship, then you can extrapolate the statistical basis to the underlying data.

• Thanks. If there is an exponential relationship between $X$ and $Y$, there is a linear relationship between $X$ and $\log Y$. So if the correlation between $X$ and $\log Y$ is +1, what can you say about the correlation between $X$ and $Y$? Commented Mar 20, 2016 at 0:08
• If you mean the gradient $y=mx+c$ type relationship for a $log Y$, then I would suggest that each unit of change for $x$ is a factor of 10 change for $y$. I cant think of the maths at the moment Commented Mar 20, 2016 at 18:49