Exercise 2 – statistical observations course

The central limit theorem

Use the same data as for exercise 1 (TAT1000mb.cdf, or TAT1000mb.mat).

1. Calculate and plot the pdf based on daily data, and on pentad averages (running 5 day box averages), running 30 day box averages (monthly), yearly averages. Plot also the running-box averages time series. Does the central limit theorem work for this example? Explain.

2. In the example above, you were asked to take running box averages, which are sequential, for example sequential 5 day averages. We now check what happens when we take random samples of 5-day averages. For this you can use the command randperm. Plot the resulting 5-day averaged histogram, and time series. Does the central limit theorem work for this case? Explain. Calculate the standard deviation, and compare to the theoretical central limit theorem standard deviation. 

Notes: To take running-box averages, it is easiest to reshape the data into a desired matrix, for example, a 5 by 21915/5 matrix for the pentad averages, and then calculate the column averages, and calculate the pdf from the remaining column. Use the command reshape. For the monthly averages, you can use 30 day box averages. Alternatively you can write a loop to calculate each month average, and then it would be easiest to use the leapyear.m routine.

You should hand in plots, some explanation in words and the m files you wrote.