Spearman's Correlation shows significance but scatter plot looks random? I have some data measured on a continuous scale. The data have been collected from the same subjects and I want to look for correlation between two dependent variables. The data are not normally distributed. 
A cursory look at the raw data suggests correlation, but when I generate a scatter plot I get the following:  
 
That looks pretty random to me or is it? 
However when I run a Spearman's correlation I get the following result:

So can someone confirm if there is actually any correlation or not? 
Edit-Graph with cross to create four quadrants here...

 A: One simple approach here would be to impose a cross on your graph which divided it into four quadrants, perhaps at the 0.5 position on each axis. Then ask yourself: "Are there more points in the North-East and South-West quadrants of the graph compared to the other two?". If the answer is yes then you have a positive correlation. A more precise version could be obtained by imposing a 3 by 3 grid perhaps at 0.33 and 0.67, and so on.
Incidentally if you are comparing measurements from two laboratories as your labelling suggests then there are better ways of doing this than correlation.
A: I think this is just a visual effect. The points which are not on the regression line are very striking because they lay far apart from each other. However, there are many points close to the regression line that are on top of each other -- appearing as one.
A: Scatterplots containing even modest numbers of data points always look like a blur. One approach to uncovering the "signal" summarized in a dependence metric like a Spearman correlation is to create ranked buckets of information based on the feature (or independent) variables. Then, based on those groupings, average both the feature and the target (or dependent) variable. A scatterplot can be formed with the x-axis as the feature buckets and the y-axis as the average of the target variable across those buckets. 
Here's an example based on 1,643 observations:
Spearman correlation between X and Y, rho=0.261, p-value=<.0001
Here's a scatterplot of the raw data:

Here's a scatterplot of the same information after grouping X into 20 buckets:

A: Scatterplots sometimes "hide" points if they have (almost) the same coordinates as others.
Consider using a "heatmap" density plot instead, which may better convey point density.
But also beware: the data shows that strong positive correlation may exist even when humans would consider it to be weak. Here, you simply have several points in the top-right (at 1,1) and bottom-left (0,0) corners, but next to no points in the other corners. By definition, this is a positive correlation. In reality, it may be useless.
If you intend to check if the lab results are correlated, consider splitting the data set into "obvious" cases (where both results are almost exactly 0 or 1) and "difficult" cases. Look at the difficult cases only.
