i.i.d. observations taken from the joint probability distribution over V. The geometrical interpretation comes from considering the N-dimensional vectors x (formed by the successive values of X over the observations), y (formed by the values of Y), and z (formed by the values of Z). It can be shown that the residuals eX,i coming from the linear regression of X on Z, if also considered as an N-dimensional vector eX (denoted rX in the accompanying graph), have a zero scalar product with the vector z generated by Z. This means that the residuals vector lies on an (N–1)-dimensional hyperplane Sz that is perpendicular to z. The same also applies to the residuals eY,i generating a vector eY. The desired partial correlation is then the cosine of the angle φ between the projections eX and eY of x and y, respectively, onto the hyperplane perpendicular to z. As conditional independence test With the assumption that all involved variables are multivariate Gaussian, the partial correlation ρXY·Z is zero if and only if X is conditionally independent from Y given Z. This property does not hold in the general case. To test if a sample partial correlation implies that the true population partial correlation differs from 0, Fisher's z-transform of the partial correlation can be used: The null hypothesis is , to be tested against the two-tail alternative . can be rejected if where is the cumulative distribution function of a Gaussian distribution with zero mean and unit standard deviation, is the significance level of , and is the sample size. This z-transform is approximate, and the actual distribution of the sample (partial) correlation coefficient is not straightforward. However, an exact t-test based on a combination of the partial regression coefficient, the partial correlation coefficient, and the partial variances is available. The distribution of the sample partial correlation was described by Fisher. Semipartial correlation (part correlation) The semipartial (or part) correlation statistic is similar to the partial correlation statistic; both compare variations of two variables after certain factors are controlled for. However, to calculate the semipartial correlation, one holds the third variable constant for either X or Y but not both; whereas for the partial correlation, one holds the third variable constant for both. The semipartial correlation compares the unique variation of one variable (having removed variation associated with the Z variable(s)) with the unfiltered variation of the other, while the partial correlation