Is there a quantifiable difference and is the cost of private education worthwhile?

All though there are some very good state schools and in some countries, the state education is excellent, in those countries where it isn't it does prove an attractive alternative for parents who can afford it. It is unfortunate that not everyone can have access to the same standards of education. Perhaps the greatest bonus of the private school system is a) class size and b) focus on the individual. A student who is struggling is likely to get 1 to 1 tutelage to try and improve their performance in a particular subject with a view to boosting their confidence and obtaining better results. The same student who may be struggling in the state system, may well get overlooked because there are 40 or 50 other children who all have issues or special needs.
And, inevitably, the range of subjects offered is probably broader enabling students to widen their horizons.