Independent measurements

Quantitatively, this is explained by the following example: Consider a regression model with, say, three unknown parameters β0, β1 and β2. An experimenter performed 10 repeated measurements at exactly the same value of independent variables X. In this case regression analysis fails to give a unique value for the three unknown parameters: the experimenter did not provide enough information. The best one can do is to calculate the average value of the dependent variable Y and its standard deviation.
If the experimenter had performed five measurements at X1, four at X2 and one at X3, where X1, X2 and X3 are different values of the independent variable X then regression analysis would provide a unique solution to unknown parameters β.
In the case of general linear regression (see below) the above statement is equivalent to the requirement that matrix XTX is regular (that is: it has an inverse matrix).