Significance test for skewed distribution I have evaluated two methods on a test set with $n=4252$ samples. Each sample gets a score from the two methods and the scores are in the range $[0,1]$.
The distribution of the scores of method 1 and method 2 are plotted in the following picture, along with the mean and median values:

The sample means of method 1 and 2 are calculated as $\mu_1=0.75$, $\mu_2=0.824$ and the sample standard deviations are $\sigma_1=0.236$, $\sigma_2=0.194$.
I have not done many significance tests before, so I am a bit lost how to proceed here and which method to apply; especially because the distributions do not seem to follow a normal distribution.
My null hypothesis $H_0$ is that "Method 1 and Method 2 do not differ significantly"; the alternative hypothesis $H_a$ is that "Method 2 is better than Method 1".
How can I test this?

Edit:
I think I need to be a bit more explicit here. I have the ground truth data for all the $n$ samples. Method 1 and Method 2 do something to every single sample and use a score function $f$ to compare the result with the ground truth. A score of 1 means that the method worked perfectly on this sample, a score of 0 means that the method failed completely. I think the details of how $f$ works and that it needs ground truth data for the comparison with the method's result is not relevant here. If it helps, let us just say that some wise entity tells us with absolute certainty for each sample how well method 1 and method 2 worked, that is, this entity gives us for each sample two numbers between 0 and 1 and this is all we work with.
Now I run the two methods separately on all $n$ samples so that I get $n$ scores for method 1 and $n$ scores for method 2. The distribution of the scores is plotted in the figure above.
One of the two methods should be used in a production environment, so I have to decide which one is better. The ideal method would return a score of 1 for all samples. So if for example method 1 would return a score of 0.1 for all samples and method to would always give 0.9 it would be easy to say that method 2 is favorable. But how can I go on here? I honestly dont care if medians, means or whole distributions are compared. All I want is to find out is which method performs better.
Link to scores of method 1:
https://pastebin.com/jkdBEJM1
Link to scores of method 2:
https://pastebin.com/1S4zYAaa

Edit 2
Here is the difference plot @NickCox suggested:

For each sample $i\in[1,n]$ the two scores $s_1^{(i)}$ and $s_2^{(i)}$ of method 1 and method 2 are used to calculate the difference $d_i=s_2^{(i)} - s_1^{(i)}$ and the mean $m_i=(s_1^{(i)}+s_2^{(i)})\cdot 0.5$ The point $(m_i/d_i)$ is then plotted.
However, this plot doesn't make me much smarter.
 A: One way to do significance testing is to measure the tails not by the x axis variable, but by probabilities. So you take 2.5% from each tail to get 5% significance. 
Instead of measuring the distance from mean in standard deviations you simply build CDF then look at 2.5% and 97.5% quantiles which give you the 95% confidence interval.
Example. Let's say you have 1000 data points from a standard lognormal distribution. You look at the 2.5% percentile, and it comes up at 0.1442; then you look at 97.5%, which is 7.2485. This is your confidence interval that you can use for inference.
x = lognrnd(0,1,1000,1);
p = prctile(x, [2.5 97.5])
hist(x,100)
line([p(1) p(1) ],[0 250])
line([p(2) p(2) ],[0 250])

p =

    0.1442    7.2485


UPDATE:
One more thing: the mean is often not a good measure of centrality for heavily skewed distribution. That's one reason why you were struggling when trying to apply the mean and standard deviation here. Hence, the method that I suggested works better with the median, which happens to be symmetrical in relation to the way the tails are defined
