Applied Statistical Methods, 1st Edition by Irving W. Burr and J. William Schmidt (Auth.)

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By Irving W. Burr and J. William Schmidt (Auth.)

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5* FURTHER DESCRIPTIVE MEASURES OF FREQUENCY DISTRIBUTIONS, THIRD A N D FOURTH MOMENTS The mean and the standard deviation of a sample give a large amount of the information contained in the sample. For most purposes this is all that we need, especially if the data are well behaved or "normal," rising regularly to the maximum frequency and falling away again symmetrically. If there are irregularities such as isolated cases or evidence of nonhomogeneity of conditions, then probably no summary measures are adequate or even appropriate.

Thus by tending to carry the sign of the numerically largest deviations, m3 is capable of measuring lack of symmetry. Now there are two serious shortcomings to m3 as a skewness measure: (1) It carries the physical unit cubed, and hence m3 is not readily com­ parable from one set of data to another. (2) A related defect is that it has no absolute meaning because it does not take into account the natural variability of the data. 18) ™2 = Y(y-y)2ln. 16) 44 3. Summarization of Data by Objective Measures It is to be noted that m2 as just defined is very similar to s2, the sample variance, the difference being division by n.

3. Events Note that (AuB)DA and (AuB)DB. 2) As an example, let A be the event of three to six accidents in a year at an intersection and B the event of one to four accidents there. Then A U B is the event of one to six accidents in the year there. Another important event determined by A and £ , say, is their * 'inter­ section/ ' It consists of all outcomes which are in both A and B at the same time, or common to both A and B. The notation is intersection of A and B = outcomes common to A and B = AnB.

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