Pearson, Spearman & Kendall Correlation

Pearson's r when the relationship looks linear and there are no strong outliers; it measures how well a straight line fits. Spearman's ρ when you care about any monotonic relationship or the data are skewed or ordinal. Kendall's τ when samples are small or have many ties — its value is smaller than ρ for the same data but its p-value and interpretation, the probability that a random pair is ordered the same way on both variables minus the probability it is not, are more robust.

Because they answer different questions. Pearson uses the values, so a single extreme point can pull it a long way; Spearman uses only ranks, so the same point counts no more than any other. A curved but steadily increasing relationship also gives a high ρ and a lower r. A large gap is a signal to look at the plot before choosing.

Through Fisher's z transformation: atanh(r) is close to normally distributed with standard error 1/√(n − 3), so the interval is built there and transformed back with tanh. That is why it is asymmetric around r and never passes ±1. It needs at least four pairs, and for a one-sided alternative it is one-sided too.

When neither variable has ties and there are fewer than fifty pairs, the rule R's cor.test follows. The exact distribution comes from counting how many permutations of n items have each number of inversions. With ties or more pairs, the normal approximation with the tie-corrected variance is used, matching scipy's asymptotic method.

No. A correlation can come from a common cause, from reverse causation, from how the sample was selected, or from chance across many comparisons. And a correlation of zero does not mean independence: these coefficients only detect relationships that go consistently one way, so a U-shaped pattern can score near zero.