Comparison of Two Approaces for Determining Ground-Water Discharge and Pumpage in the Lower Arkansas River Basin Colorado 1997-98

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identifies the sources of variation in the data set (Iman and Conover, 1981). A necessary assumption about the analysis of variance model is that the probability distribution of the data is normal. This is a common assumption made when applying statistical models, but it is an assumption that may not be true for many water- resources data sets. One reason a normality assumption is useful is that the normal distribution is characterized by the mean and variance (which is the standard deviation squared). The mean is a measure of central tendency of the random variable, and the vari- ance is a measure of magnitude of random variability. Given the mean and variance, probability statements may be expressed in terms of these parameters; for example, a normally distributed random variable is with probability 0.95 within 1.96 standard deviations of the mean. Another necessary assumption about the analysis of variance model is that the variances are constant. During data analysis, differences for every well discharge and pumpage estimate initially were computed by subtracting the well discharge or pumpage estimates associated with the PCC approach at each well from the well discharge or pumpage asso- ciated with the TFM measured at the same well on the same date. An analysis of the differences computed in this manner indicated that the assumptions of normality and equal variances were not met. There- fore, a transformation of the differences was done by subtracting the natural logarithm of well discharge or pumpage associated with the PCC approach from the natural logarithm of the well discharge or pumpage associated with the TFM. The resulting differences were normally distributed, and the variances were equal for well discharge. However, the differences in pumpage were not normally distributed. Thus, a rank transformation was performed on the differ- ences in pumpage. This consisted of ranking all of the individual differences, and then applying the anal- ysis of variance model to the ranks. The rank transfor- mation for a sample of n observations replaces the smallest observation by the integer 1 (called the rank), the next smallest by rank 2, an so on until the largest observation is replaced by rank n. Using ranks dimin- ishes the influence of the outlying values on the final results. A consequence of doing this is that the final results of the analysis reflect the behavior of the majority of the data points, but the influence of the outlying values has been diminished. An inverse rank transformation (linear approximation) to the results of the analysis of variance was then done, resulting in estimates of the mean or central tendency of the distribution of differences in pumpage. However, data outliers may well have a significant effect in situations for which properties of the proba- bility distribution other than central tendency are important. The natural logarithmic transformation that was applied to the data has another useful property that makes it appropriate for analyzing this data set. Differ- ences in logarithmically transformed variables are equivalent to relative or fractional differences rather than to absolute differences. Relative differences are an informative way to evaluate differences in well discharge and pumpage. In essence, for small differ- ences, the relative differences, which is the difference in natural log transformed variables, multiplied by 100 times, is nearly equivalent to percent difference. Tornqvist and others (1985) provide a more complete discussion of the advantages of using the log transfor- mation to evaluate relative differences. During data analysis, various site characteris- tics, hereinafter called fixed effects (method of discharge measurement, make of TFM, and discharge distribution type) were identified as sources of varia- tion. Additionally, the site, date, and random error, hereinafter called random effects, were identified as sources of variation. Therefore, it was necessary to take these additional sources of variation into consid- eration when making comparisons of well discharge and pumpage. COMPARISON OF INSTANTANEOUS GROUND -WATER DISCHARGE MEASUREMENTS A comparison of the instantaneous discharge measurements using the TFM's to those using the three portable flowmeters was made by evaluating the differences between the measurements and by deter- mining whether the differences are statistically signifi- cant. Because it was determined that the method of discharge measurement, make of TFM, discharge distribution type, and the site, date, and random error were identified as sources of variation, an additional level of data analysis was required. This section of the report presents (1) the magnitude in differences in well discharge; (2) an esti- mate of the overall mean difference in well discharge COMPARISON OF INSTANTANEOUS GROUND -WATER DISCHARGE MEASUREMENTS 11