Group Project Dynamics— how to present advanced ideas without offending others? For my research masters, one graded activity is to undertake a piece of research and present findings.

We collected data and the analysis part was self-assigned to one member and myself. The analysis and results ultimately presented by the team member was very basic, and included nothing more than a sum, count and percentage of responses presented in an excel graph.

The supervisor responded that he was unable to follow findings and suggested advanced analysis.

I then undertook this work, redid the analysis in R, and presented the findings in an advanced/visual form, using regression analysis and correlation matrices and suggested the group explore data/findings and suggest additional hypotheses to test.

The member who originally did the analysis feels let down and asked meaningless, trivial and baseless questions when I presented findings, in an effort to discredit my work. She felt I was trying to better her. 

The supervisor is yet to comment on the validity/usefulness of my analysis work.

In my earlier degree, I gained a good understanding of research methods and statistics. However, this person, from what I’ve seen, does not seem familiar with fundamental research concepts or statistical methods.

How do I handle this situation?

How important is it to regress to ideas that everyone is comfortable with to avoid challenges like these?
You don't have to do anything with this situation any more - your colleague is clearly feeling uncomfortable about it and as you said in comments, reaching out to her now might very easily come across as patronising.

For the next time, I'd advise going through the analysis with her before presenting to the group. Ideally, you'd do this a couple of times as your analysis progresses, and take her initial results as a starting point.

In analyses and presentations, you should never shy away from using complex methods if you feel they are necessary. However, you should also be able to explain them in such a way that your whole audience can understand what you are doing and why.