The Wikipedia article Spearman's rank correlation coefficient contains an example for calculating ρ. At the end of the sections is the statement "...with a P-value = 0.6864058 (using the t distribution)." The author does not describe how the P-value was calculated from the data in the example.
How was the P-value derived for the article's specific example?
Note: The included links to Wikipedia's P-value and T-distribution entries are too generic to provide a clear answer.
In the next section in the article, "determining significance", there is the equation
$t=r\sqrt{\frac{n-2}{1-r^2}}$
If you plug in their estimate of r into that equation you get a t statistic of -0.505, which you can compare to a standard t distribution via a table or a computer. For example (EDITED- thanks to @whuber for correcting my earlier version):
> spearmentt <- function(r,n){r*sqrt((n-2)/(1-r^2))}
> test <- spearmentt(-0.17575757575,10)
> test
[1] -0.5049782
> 1- pt(test,8)
[1] 0.6864058
It's worth noting that this is a one sided test for whether r is significantly larger than zero. Probably more appropriate test would be
> pt(test,8)
[1] 0.3135942
which is a one-sided t-test for whether r is significantly less than zero, i.e., is there evidence of a negative correlation between TV watching and IQ.
r <- 1 - 6 * 194/(10*(10^2-1)); n <- 10; t <- r * sqrt((n-2)/((1-r)*(1+r))); 1 - pt(t, n-2) returns 0.6864058 as given by Wikipedia.
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In R I would suggest using the functions rcorr from the Hmisc package and the custom function flattenCorrMatrix (http://www.sthda.com/english/wiki/print.php?id=78) which you can find online. rcorr calculates the correlation matrix and the flatten function will give you easy access to correlation and p values.
Something along these lines is nice for multiple variables that you want to test for correlations:
#extract columns you want to test
y<-x[,c(6:7,9:14,16,17,20,21)]
#calculates spearman correlations of all possible variable pairings
z<-rcorr(as.matrix(y),type=c("spearman"))
#flattens the matrix and gives you a nice table of all results
z1<-flattenCorrMatrix(zf$r, z$P)
# write your results in csv file
write.csv(z1, file = "z1results.csv")
Cheers
Spearman correlation coefficient is a Pearson correlation coefficient calculated on the ranks of observations. Hence, you can apply p-value formulae of Pearson correlation $r$ to Spearman $\rho$.
For instance, the estimator of the variance is $\sigma^2_\rho=\frac{(n-2)}{1-\rho^2}$, you can use it in usual way to calculate the t-stat $t=\rho/\sigma_\rho$, then lookup the p-value from a table or from the CDF of t-distribution with $n-2$ degrees of freedom.
