Scheefheid en exces bij frequentie‐ en kansverdeelingen

An empirical frequency distribution, belonging to a certain quantity x, can be considered as the realization of a {theoretical) probability distribution. This latter furnishes i. the mathematical expectation (theoretical mean), 2. the (theoretical) moments with respect to, i.e. the math. exp. of the different powers of t = x —, denoted by m = tk, in particular m2— t2=s̀2, the square of the standard deviation. The deviations of a probability distribution from‐ the normal type are often characterized by the Skewness S and the Excess E, defined by Plotting the probability density y = ‐ as an ordinate against the abscissa x, we obtain the density curve. The author shows how far, under certain simplifying conditions, the algebraic sign of S and E can be predicted from the course of the density curve. Skewness: If an asymmetrical density curve ascends more steeply to the left of the top than it descends to the right, and if, in particular, at any pair of points with equal y, the curve is left steeper than right, the Skewness S is proved to be positive. If, on the other hand, the curve is right steeper than left, S is negative. Excess: With a symmetrical, not normal curve, S = o, whilst E can differ from zero. Such a curve can be confronted with a normal curve having the same centre and equal a. Then these two curves must have at least two points of intersection on both sides of the centre (top). Provided that there are precisely two points of intersection, the sign of the Excess E is connected with the height of the top of the given curve. If this top is higher than that of the corresponding normal curve (leptokurtosis), E is positive; if it is lower (platykurtosis), E is negative. In case there are more than two points of intersection on either side of the centre, there is no close connection between the top‐height and the sign of E. This is illustrated by the behaviour of density curves of the type y = C (1 — t2)3 (1 +αt2+βt4).