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I would like to look at the correlation between different attendance rates over the course of a year.

I have attendance rates from 2013-2014 and 2014-2015 and the attendance rates are calculated for each quarter (Q1-Q4).

My question is would it be appropriate to conduct a correlation analysis to determine if the attendance rates at each quarter are related (i.e. they increase/decrease at the same points each year)?

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You're looking for seasonality in a time series, a standard way to do so is with autocorrelation, but with only 8 points I'm not sure you can do any useful quantitative analysis. All you can really say is attendance is up/down/the same as last year, and (e.g.) it appears highest in Q1.

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