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I have two normally distributed samples. I want to know how close or similar it is. I tried few methods to find the similarity, like z-score and bhattacharyya distance.

Bhattacharyya distance didn't work for me. It gives the same distance if the standard deviation of two samples is same. It doesn't change with change in mean.

I want to know whether any method is available that take the samples or its mean and standard deviation to find the similarity or similarity rank something like this.

I am not from mathematics background, so please ignore the terminology mistakes and let me know if any clarification is required.

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The question you are asking can be answered in multiple ways so hard to give a specific answer. However, if you are looking at the similarity between two normal distributions you could consider the overlap coefficient. This takes into account the mean and standard deviation of both distributions. You can find various exercises, examples and calculators online. An article by Inman and Bradley in 1989 provides some background to calculate the overlap coefficient (https://www.tandfonline.com/doi/abs/10.1080/03610928908830127). It is not an overly used measure in statistics. Personally, I am not exactly sure why this is the case as the overlap coefficient inherently intuitive, more so in my opinion, than other statistical meausures such as z-scores or effect sizes.

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