How many observations required to meaningful correlation? I am researching on a minimum number of observations required for meaningful correlation. I have found a few discussion on various discussion forums, but I am looking for some paper or proven article. Can you share it with me?
 A: See BBR section 8.5.2.  You'll see that n=400 is required to estimate the correlation coefficient to within a margin of error of +/- 0.1 with 0.95 confidence, if you know nothing about the true correlation.
A: "Meaningful" is a tricky term. Frank gave a good answer about how many you need for a certain level of precision, but that's not exactly the same as "meaningful".
Going only on my own intuition about "meaningful" I would say that the answer is "it depends". What does it depend on? Probably lots of things.Two that spring to mind are:


*

*How large the correlation is and

*How close to linear the relationship between the variables is (assuming you are talking about Pearson's correlation).
Sometimes, N = 3 would be enough. E.g. suppose you were investigating the ideal gas law. This is about the relation between pressure and temperature of a gas. The relationship is very strong (because we can measure both these things very well) and strictly linear (because that's how physics works). So, plot these at 0 degrees, 50 degrees and 100 degrees and you will get three points very close to a line and a high $R^2$ and a pretty low standard error - the exact value depending on how precisely you measured temp and pressure and on things like imperfections in the apparatus.
But, even if you measure pretty inaccurately, you get pretty good results e.g.:
x <- c(0, 50, 100) #Temp
y <- c(65, 75, 90)  #Pressure, rounding to 5 units

cor(x,y)
cor.test(x,y)

cor.test.plus <- function(x) {
  list(x, 
       Standard.Error = unname(sqrt((1 - x$estimate^2)/x$parameter)))
}

cor.test.plus(cor.test(x,y))

The $R^2$ is 0.993 and the SE is 0.11.
That seems meaningful to me.
The SE of $R^2$ is given by $\sqrt{\frac{1-R^2}{n-2}}$ so, you can define for yourself what "meaningful" is. 
