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Double Cumulative and Lorenz Curves in Weather Modification

Edwin L. CrowNational Center for Atmospheric Research, Boulder, CO 80307

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Abstract

A graphical presentation of precipitation data has been used for some years in which the cumulative percentage of the total mass falling on various days of a sample of precipitation days, ordered from the largest to the smallest mass, is plotted against the cumulative percentage of days. This type of graph is called a “double cumulative curve” (DCC), but it is essentially the same as the Lorenz curve in economics. The paper reviews the literature, summarizes the properties, shows the DCC's for uniform, log-normal, exponential, gamma and degenerate distributions, studies the average effect of sample size, and presents a formula for testing the significance of the difference between two DCC's. This formula is applied to compare hail data of various types. It is concluded that a sample DCC is a biased estimate of the population DCC, but the bias becomes negligible as the sample size increases beyond ∼30.

Abstract

A graphical presentation of precipitation data has been used for some years in which the cumulative percentage of the total mass falling on various days of a sample of precipitation days, ordered from the largest to the smallest mass, is plotted against the cumulative percentage of days. This type of graph is called a “double cumulative curve” (DCC), but it is essentially the same as the Lorenz curve in economics. The paper reviews the literature, summarizes the properties, shows the DCC's for uniform, log-normal, exponential, gamma and degenerate distributions, studies the average effect of sample size, and presents a formula for testing the significance of the difference between two DCC's. This formula is applied to compare hail data of various types. It is concluded that a sample DCC is a biased estimate of the population DCC, but the bias becomes negligible as the sample size increases beyond ∼30.

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