What statistical method would I need to employ to predict a school's attendance if I know previous five years' attendances? I have attendances month by month (September-June) for a given school X for the following school years:2011-2012, 2012-2013, 2013-2014, 2014-2015. I also have the attendance for students at school X for the current year 2015-2016 up to the current full month which is December. The attendances are expressed as percentages e.g. for the month of September for the 2013-2014 school year, 97.2% of all students who could show up did show up. 
Schools are most interested in making a prediction about their attendances for each succeeding month in the 2015-2016. For example, school X would like to know given its attendance for this school year so far (September to December), what is our attendance likely to be in January, in February, and most importantly the attendance in the month of June. The attendance in the month of June serves as a kind of benchmark that the school must reach if it gets to keep certain continued funding, doesn't get placed on state-mandated probation etc. 
My initial approach is to do the following. Take the change in attendance from September to October in 2011-2012. Then do the same thing for years 2012-2013, 2013-2014.... Then I would take the average of these differences. I would do the same thing for the difference in attendance from the Month of November compared to October, and December compared to November, etc. Then I would simply take these averages and use that to make predictions about the months of January 2016, February 2016, March 2016 etc. 
For example, after doing the calculations, I found out that the average change in percentage over these available school years from the month of December to January is 1.5%, I would take the value I have for December and add 1.5% to it, which say is now 96.2% after having 1.5% added to it. Then I would take the average change from January to February, which say is a decrease of .6% (-.6%) and subtract it from the previous value that I had obtained of 96.2% to get 95.6% and continue to do so. 
This seems rather cumbersome and extraordinarily crude. What statistical method would be most appropriate to utilize to make these needed predictions? I am sure there are probably very complicated methods that are available to do this. Which method is most tractable and elementary enough that somebody like myself (not too sophisticated in statistics but capable enough hopefully) to make a prediction about attendance for each of the upcoming months in year 2015-2016?
 A: *

*A simple regression will be a good place to start. You can use the previous month's attendance % as a predictor for the current month. You want to check the p-value to see if it is significant (i.e. if your choice of using the previous month was a good one)

*There might be seasonality by month of the year (e.g. attendance may be low in months with holidays), so you can check to see if the month is a good predictor. Alternately you can check to see if the season in which the month falls (i.e. Fall, Winter,Spring, Summer) is a good predictor. Number of school days in a month is another one

*If you decide to try something more involved, Forecasting Principles and Practices by Rob Hyndman & George Athana­sopou­los, is a good online reference for modeling time series data. ARIMA or seasonal-ARIMA may be worth exploring 

A: I think you can try Markov chain methodology here. Start with coming up with Transition Probability Matrix and the if the matrix satisfy Anderson Goodman and Likelihood ratio tests, can be used to forecast.
