1. Degrees of Freedom

Statisticians use the terms "degrees of freedom" to describe the number of values in the final calculation of a statistic that are free to vary.  To put it another way, "degrees of freedom" can also be thought of as opportunities for change. In the context of correlation analysis of two time series, one must have two values of the variables x and y to answer the question "is the third set of values of variables x and y consistent with the implied correlation suggested by the relation of the first two sets?"

In other words, suppose Henry and I were flipping a coin. Say I throw a heads and Henry throws a heads as well.  One more try and we both throw heads.  So based upon two tries, it appears that there is a 100% correspondence.  We both need to toss the coins one more time to see if the correspondence suggested by the first two flips maintains consistency.

Or, if I threw a tails and Henry threw a heads, and on the second try, I threw a heads and he threw a tails (or vice versa) to show a 100% negative correspondence (that whatever I threw, Henry would throw the opposite).  Again, we would need to toss the coins one more time at least to determine if the correlation suggested by the first two flips is consistently opposite.

In short we need at least two flips of the coin to compare future flips against, so to speak.  Thus, the first two flips are not free to vary.

Let's say that we flipped the coins 100 times (the number of tosses is given the symbol "n").  Then the degrees of freedom (df) would be (100 - 2), or 98.  For the analyses we are doing, the df = n - 2, where df is degrees of freedom, and n  is the number of seasons*. Statistical significance is related to the relationship of the number of events (or tries) and the degrees of freedom. For a large degrees of freedom, the smaller the number of events, the less statistically significant the given correlation coefficient would be.

*In the context of correlation, remember that what is happening in the background is that a regression is taking place. The initial regression line is fit to the first two sets of observations, and then each additional pair of observations is compared to the regression line. If the second pair fits on the regression line you are on your way to a correlation coefficient of 1.0. The extent to which the second pair of observations does not fall onto the line (called variance), the correlation coefficient will be less than 1.0. You need a regression line, defined by the first two observation pairs, to compare future observations against. Hence, you lose two degrees of freedom, the first two observation pairs.