What is the definition of the higher quartile?

Answer 1

One can define an upper quartile for a population and a upper quartile for a sample.

Population: Let X be a random variable. The the upper population quartile is the value, x, for which Prob { X <=x } = 0.75. Or, in words, the probability of drawing a value from the population that is less than its upper quartile is 0.75, by definition.

Sample: Let x1,x2, ... xn be a sample drawn independently from the same population. Sort them from smallest to largest to form the so-called order statistics: x(1), x(2), ... x(n). For simplicity let's assume that n=100. Then the smallest three-quarters of the values are x(1), x(2), ... x(75), and we would call x(75) the upper sample quartile. If you had only ten values, say, then you might use some value between x(7) and x(8) as the sample upper quartile. For this and other reasons the sample quartile may only be useful where large samples are involved.