问询：是否存在关联正态变量标准差与另一变量均值的[-1,1]关联指标及自制方法合理性

Great question—let’s tackle your two queries one by one:

1. Are there existing metrics for this kind of association? #

Absolutely! What you’re exploring is a form of heteroscedasticity-related association (linking the spread of one variable to the central tendency of another), and there are established ways to get a [-1, 1] metric that behaves just like a correlation coefficient:

• Correlation between group-level statistics: If your "second variable" is categorical, split your data into groups based on this variable, calculate the mean of the second variable and the standard deviation of your normal-distributed variable for each group. Then compute the Pearson or Spearman correlation coefficient between these two sets of group-level values. This naturally falls in [-1, 1], where a positive value means higher group averages correspond to larger standard deviations, and vice versa. For continuous second variables, you can use sliding windows or binning (just ensure bins have enough samples to estimate standard deviation reliably).
• Regression-derived signed R-squared: Treat the standard deviation of your normal variable as the dependent variable, and the mean of the second variable as the independent variable, then fit a simple linear regression. Take the square root of the model’s $R^2$, and attach the sign of the regression slope. This gives you a [-1, 1] metric that directly quantifies the linear strength and direction of the association—this is essentially identical to the Pearson correlation approach above.

There are more specialized metrics tied to heteroscedasticity tests (like Levene’s test), but most of these are non-negative (e.g., $\eta^2$ for effect size). The correlation-based methods above are the closest match to your request for a correlation-like [-1, 1] measure.

2. Is my self-conceived method reasonable? #

Without the specific details of your approach, it’s hard to give a definitive yes/no—but here are key checks to evaluate its validity:

• Does it produce a metric bounded strictly between [-1, 1]? And does the sign have clear interpretability (e.g., positive = higher mean → larger standard deviation)?
• How do you handle estimation error? Standard deviations can be noisy with small samples—does your method account for that (e.g., weighting groups by sample size)?
• Is the association you’re measuring linear? If the true relationship is non-linear, a correlation-like metric might not capture it well—does your method allow for that, or is it explicitly targeting linear associations?
• Have you tested its robustness? For example, if you use binning, do different bin sizes change the result drastically? If you use an alternative spread measure (like median absolute deviation instead of SD), does the association hold?

If you share the specific steps or formulas of your method, we can dive deeper into its strengths and potential pitfalls!

内容的提问来源于stack exchange，提问作者Ruben van Bergen