### Values

The table below lists the correlation values that are significant at 4 significance levels and specified degrees of freedom. To perform a 1-tailed test, simply use the values in the list. For example, the 95% significance level correlation value for 40 years is listed under the .950 column for 40 degrees of freedom and is .264. For a 2-tailed test, for the significance level you want (for example 95%) use the value for the
level+((1.000-level)/2.0) significance level.
That is, .950+(1.000-.950)/2 or .975 level
For 40 years, a correlation of +/- .3124. would be significant. These significance levels are local. For a resolution of 144x73 gridpoints, one would assume at least .05X(144*73)=526 grids would be significant by chance at the one-sided 95% level.

### Caveats

Note that determining actual field significance is trickier than this due to the grids being correlated in space so that Monte Carlo or similar tests are usually run. for example, this paper discusses field significance calculations.

For some variables (like SST), there is year to year correlation at a region so that the number of years will be greater than the actual degrees of freedom meaning a higher correlation value is needed for significance.
(Linear) correlation makes certain assumptions about the data which can lead to spurious high correlations (or, hide real relationships). For example, the assumption is that the data is normally distributed which doesn't always hold for variables like precipitation. Also, since the method maximizes linear relationships, variables in quadrature may appear to have 0 correlation when in fact the relationship between them is exactly defined.

NOTE, CORRELATION DOES NOT IMPLY CAUSATION!!

Two variables a and b may be highly correlated but the correlation could mean a causes b, b causes a, the correlation is due to a third factor related to both a and b, or, the correlation could simply arise by chance. The user should be cautious when interpreting results.

A good discussion on the significance of correlations is available at graphpad software page.

### References

Livezey, R.E. and W.Y. Chen, 1983: Statistical field significance and it's determination by Monte Carlo Techniques. Mon. Wea. Review., 111, 46-59.

Diaconis, P. and B. Efron, 1983: Computer Intensive methods in statistics, Sci. Am., 248, 116-130.

```                Significance Level
Degrees of   .950   .975   .990   .995
Freedom
2        1.000  1.000  1.000  1.000
3        0.920  0.954  0.977  0.986
4        0.833  0.891  0.936  0.956
5        0.758  0.829  0.889  0.919
6        0.697  0.774  0.844  0.880
7        0.646  0.727  0.802  0.843
8        0.605  0.685  0.764  0.808
9        0.570  0.650  0.729  0.775
10        0.540  0.619  0.699  0.746
11        0.514  0.592  0.671  0.719
12        0.491  0.567  0.647  0.695
13        0.471  0.546  0.624  0.672
14        0.453  0.526  0.604  0.652
15        0.437  0.509  0.585  0.633
16        0.423  0.493  0.568  0.615
17        0.410  0.478  0.552  0.599
18        0.398  0.465  0.538  0.584
19        0.387  0.453  0.524  0.570
20        0.377  0.441  0.512  0.557
21        0.367  0.431  0.500  0.545
22        0.358  0.421  0.489  0.533
23        0.350  0.411  0.479  0.522
24        0.343  0.403  0.469  0.512
25        0.336  0.395  0.460  0.503
26        0.329  0.387  0.451  0.493
27        0.322  0.380  0.443  0.485
28        0.316  0.373  0.436  0.476
29        0.311  0.366  0.428  0.469
30        0.305  0.360  0.421  0.461
31        0.300  0.354  0.415  0.454
32        0.295  0.349  0.408  0.447
33        0.291  0.343  0.402  0.441
34        0.286  0.338  0.396  0.434
35        0.282  0.333  0.391  0.428
36        0.278  0.329  0.385  0.423
37        0.274  0.324  0.380  0.417
38        0.271  0.320  0.375  0.412
39        0.267  0.316  0.370  0.407
40        0.264  0.312  0.366  0.402
41        0.260  0.308  0.361  0.397
42        0.257  0.304  0.357  0.392
43        0.254  0.300  0.353  0.388
44        0.251  0.297  0.349  0.384
45        0.248  0.294  0.345  0.379
```
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