multiple correlation: Suppose you calculate the linear regression of a single dependent variable on more than one independent variable and that you include a mean in the linear model. The multiple correlation is analogous to the statistic that is obtainable from a linear model that includes just one independent variable. It measures the degree to which the linear model given by the linear regression is valuable as a predictor of the independent variable. For calculation details you might wish to see the wikipedia article for this statistic.
partial correlation: Let's say you have a dependent variable Y and a collection of independent variables X1, X2, X3. You might for some reason be interested in the partial correlation of Y and X3. Then you would calculate the linear regression of Y on just X1 and X2. Knowing the coefficients of this linear model you would calculate the so-called residuals which would be the parts of Y unaccounted for by the model or, in other words, the differences between the Y's and the values given by b1X1 + b2X2 where b1 and b2 are the model coefficients from the regression. Now you would calculate the correlation between these residuals and the X3 values to obtain the partial correlation of X3 with Y given X1 and X2. Intuitively, we use the first regression and residual calculation to account for the explanatory power of X1 and X2. Having done that we calculate the correlation coefficient to learn whether any more explanatory power is left for X3 to 'mop up'.
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The difference between multicollinearity and auto correlation is that multicollinearity is a linear relationship between 2 or more explanatory variables in a multiple regression while while auto-correlation is a type of correlation between values of a process at different points in time, as a function of the two times or of the time difference.
If measurements are taken for two (or more) variable for a sample , then the correlation between the variables are the sample correlation. If the sample is representative then the sample correlation will be a good estimate of the true population correlation.
Evidence that there is no correlation.
They can be positive correlation, negative correlation or no correlation depending on 'line of best fit'
Yes it can be a correlation coefficient.