Understanding statistical significance in variance and correlation?

Question

I have little background in statistics. For a dozen items, I have vectors of ratings for these items.

I have calculated the variance, and Pearson correlation for these items (using numpy). Now I would like to compare them.

My adviser asked me to see if the differences in variance and correlation were "statistically significant" using the 95% rule. How can I do this?

How does this statistical significance relate to scipy's p-value which is returned with Pearson correlation (it makes no mention of "95%")?

The p-value roughly indicates the probability of an uncorrelated system producing >datasets that have a Pearson correlation at least as extreme as the one computed from >these datasets. The p-values are not entirely reliable but are probably reasonable for >a datasets larger than 500 or so.

Answer 1

If difference between two numbers lies within two standard deviations above or below 0, then this result is not 95% statistically significant. Roughly 95% of normal distribution is covered by two std's, this is where it comes from. That is, if your values have standard deviations of $\sigma_1$ and $\sigma_2$, the joint is $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$. If $|x_1 - x_2| > 2\sigma$, they are different with 95% probability. This all assumes normality.

p-value relates to the quality of one variable in predicting (correlating to) another, this is a different thing from what you are asked to do.