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July 15th, 2014, 09:38 AM   #1
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Annoying Chi Square Problem

Math aficionados

I have a pretty basic knowledge of Chi Square. I can understand how to test whether or not data is significant e.g. if there is anything significant about the number of people who litter and people who use bins against e.g. their male/female gender etc. I can do the whole chi square and the formulate that requires us to look things up on a T table.

However I was reading a paper in which the author uses Chi Square and its just not clicking in my head. I was hoping someone here could explain his use of Chi Square and how they came up with the numbers. The problem itself isn't that important so ill summarise it and where possible I will show pics of his actual work.

THE PROBLEM - the author of the article has undertaken a research project where he has chosen 5 classic musical composers (Mozart, Bach, Beethoven, Brahms and Haydn). Using two scores from each composer he has counted every time the composer makes a transition from one musical note to another (so a 2-note transition). You don’t need to know anything about music here. Below is a link to the picture of his results.

http://i.imgur.com/Y4C3Rhx.jpg[/IMG]

As you can see there are 25 variations of how one can move from one musical note to another. Remember that the author used two samples from each composer hence there being two results of each composer.

So far so good for me. Then he creates a Chi Square Goodness-of-Fit and at this point I'm lost. Below is the link to that picture.

http://i.imgur.com/bvLkezs.jpg[/IMG]

The general idea here is that the two samples of each composer was aggregated together and instead of 25 variations he aggregated the last 3 because of their low numbers to make it 23 variations.

** You DONT need to think about the 4 unknown samples on the right of that table. That's the main argument of his essay and is not relevant here. We are only concerned with the five composers running along the top and the left.

MY PROBLEM - I have no idea how he got the numbers on that table. There are 3 significant points of data I need to understand.

1. I can understand the names running along the top horizontal columns but what do the same names mean running down the left hand side? For me this confusion is compounded because I don’t understand how one composer is related to themselves in this table e.g. Bach and Bach is 16.2. How was this calculated? Author describes it as a “intersample error”!?? How did he get 16.2.

2. How is Bach related to other composers e.g. Bach along the horizontal (column) related to Mozart 324.3? For the author this means that it is unlikely that Bach’s music can be confused for Mozart’s music (if we only consider their 2-note transitions). I understand the significance of the number but I don’t know how he got 324.3.

3. How then is Bach (now as a row along the left hand side) is related back to Mozart (horizontal columns) with 376.7. Again this means that Mozart’s music also cannot be confused for Bach’s music. How do we get the number 376.7?

This is probably a very simple process but I'm not really a mathematician I study Music Theory and I would really appreciate if anyone on this site could help clear up and explain this result.

Thanking you in advance.

Immy
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