Inverse Normal Function Research Articles

Standardized precipitation index based on pearson type III distribution

The initial step in calculating the Standardized Precipitation Index (SPI) is to determine a probability density function (pdf) that describes the precipitation series under analysis. Once this pdf is determined, the cumulative probability of an observed precipitation amount is computed. The inverse normal function is then applied to the cumulative probability. The result is the SPI. This article assessed the changes in SPI final values, when computed based on Gamma 2-parameters (Gam) and Pearson Type III (PE3) distributions (SPIGam and SPIPE3, respectively). Monthly rainfall series, available from five weather stations of the State of São Paulo, were chosen for this study. Considering quantitative and qualitative assessments of goodness-of-fit (evaluated at 1-, 3-, and 6-months precipitation totals), the PE3 distribution seems to be a better choice than the Gam distribution, in describing the long-term rainfall series of the State of São Paulo. In addition, it was observed that the number of SPI time series that could be seen as normally distributed was higher when this drought index was computed from the PE3 distribution. Thus, the use of the Pearson type III distribution within the calculation algorithm of the SPI is recommended in the State of São Paulo.

On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP

The problems that rendered Excel 97 unfit for use as a statistical package have not been fixed in either Excel 2000 or Excel 2002 (also called “Excel XP”). Microsoft attempted to fix errors in the standard normal random number generator and the inverse normal function, and in the former case actually made the problem worse.

APL approximations for common statistical critical values

Evans and Gilfillan (1986) presented APL functions ZTAB, FTAB, CHlSQ, and BINOM, which produce quantile values from z scores, F values, x2 values, and frequencies, respectively. Quantiles of t scores may be obtained from function FTAB. Statistical analyses frequently requirethatcriticalvalues be obtained fromprobabilityvaluesunder the z, t, X , andF distributions when the requiredprobability valuesare not tabled. Oftensuch requirements arise under Bonferroni multiple comparison procedures, where in order to maintaina given level of familywise Type I error over many comparisons, the researcher mayneeda t or F criticalvaluecorresponding to a nontabled probability value. Other uses stem from the analysisof simpleeffects in factorial analysisof variance. This paper presents four APL functions that compute Z, t, X , and F critical values from upper-tailprobability values. The functions are ideal for constructing tables of critical values of the z, t, X , F, and incomplete beta distributions for given probability levels. The function that computes the F values (PTOF) depends on functions for the natural logarithmof the complete beta function (LNBETA), the incomplete beta function (BETAIN), and the inverse incomplete beta function (XINBTA). These three functions may be used for other applications. The functions are presented in Listing 1. Each of the seven functions has been written and tested in APL* PLUS/PCfromSTSC, Inc. mM APLdoesnotacceptthe diamond separator; enter the expressions separated by the diamonds in order from left to right. The execution times were obtained on an mM PC-AT compatible with an 80287 internal arithmetic coprocessor. Inverse Normal Function. Monadic function PTOZ acceptsa list of one or more upper-tailprobabilityvalues in its right argument and returns a vector of the corresponding approximate signed normal deviates. Thus, the result for rightargument [0.950.05 0.025] is [-1.645 1.645 1.960]. PTOZ uses the approximation given by Odeh and Evans (1974). Execution time for a random sampleof 100upper-tailprobability values was0.94 sec. Inverse t Function. Dyadic function PTOT accepts a list of one or more degrees-of-freedom values in its left argument and a list of associated upper-tail probability values in its right argument. PTOT returns a vector of the corresponding approximateStudent's t values. Thus,