Descriptive median

The use of the mean with either the SD or SEM implies, however, that we have reason to believe that the sample of data being summarized are from a population that is at least approximately normally distributed. If this is not the case, then we should rather use a set of statistical descriptions which do not require a normal distribution. These are the median, for location, and the semiquartile distance, for a measure of dispersion. These somewhat less familiar parameters are characterized as follows. 

Visit www.aaamath.com and, under Math Topics on the right-hand side of the screen, scroll down to click on Statistics. There are brief descriptions, followed by interactive practice problems dealing with mean, median, mode, and range. 

In terms of summary statistics, means are less relevant because of the inevitable skewness of the original data (otherwise we would not be using non-parametric tests). This skewness frequently produces extremes, which then tend to dominate the calculation of the mean. Medians are usually a better, more stable, description of the average . 

Another graphical description of the data is used when comparing the results of several trials is the box plot (also called box-and-whisker plot). A box represents the range of the middle 50% of the data, and whiskers extend to the maximum and minimum values. A line is drawn at the median value. A glance a this plot allows one to assess the symmetry and spread of the data. Figure 5.4 is a box plot for the carbonate data of figure 5.2. Specific plots, such as Youden two-sample plots for method performance studies, are discussed below. 

Application of robust statistics, especially methods of median statistics, for quantitative description of widely varying values may give information which can often be interpreted better than the results from normal parametric statistical methods. 

The methods of robust statistics have recently been used for the quantitative description of series of measurements that comprise few data together with some outliers [DAVIES, 1988 RUTAN and CARR, 1988]. Advantages over classical outlier tests, such as those according to DIXON [SACHS, 1992] or GRUBBS [SCHEFFLER, 1986], occur pri-marly when outliers towards both the maximum and the minimum are found simultaneously. Such cases almost always occur in environmental analysis without being outliers in the classical sense which should be eliminated from the set of data. The foundations of robust statistics, particularly those of median statistics, are described in detail by TUKEY [1972], HUBER [1981], and HAMPEL et al. [1986] and in an overview also by DANZER [1989] only a brief presentation of the various computation steps shall be given here. 

Within the USEPA s EPI Suite, descriptions of basic physical-chemical properties for 2,4-dichloroaniline (DCA CAS No. 554007), pentachlorophenol (PCP CAS No. 87865), nonylphenol (NP CAS No. 104405), and linear alkylbenzenesulfonate (C12 LAS CAS No. 25155300) were obtained using the EPI Suite (Table 3.1), whereas acute and chronic toxicity estimates for the median effective and chronic effect concentrations (the geometric mean between chronic lowest-observed-effect concentration [LOEC] and no-observed-effect concentration [NOEC]) and for fish, daphnids, and algae for each substance were estimated from ECOSAR (Table 3.2). These compounds were chosen based on their widespread use. 

Textual descriptions of the exposure assessment results might be useful if statements about the mean, the central tendency estimate (median) or a selected quantile of the exposure distribution are given without a description of uncertainty. However, each of the point estimates mentioned will have a different level of uncertainty with respect to model assumptions, database and calculation method. A typical wording to describe results might be, for example ... 

To use the Data Analysis tool, enter the data as above and then proceed through the menus Tools then Data Analysis, then select Descriptive Statistics. In the box labelled Input Range , enter A2 B11 and tick the box for Summary statistics . The mean, median and standard deviation will be shown for both data sets, but you will probably need to widen the columns to make the output clear. 

Detailed instructions are provided for the calculation of the mean, median and SD (but not quartiles) using Microsoft Excel. Readers are referred to the accompanying web site for detailed instructions on generating all these descriptive statistics (including quartiles) using Minitab or SPSS. Generalized instructions that should be relevant to most statistical packages are provided in the book. 

The early chapters (1-5) are fairly basic. They cover data description (mean, median, mode, standard deviation and quartile values) and introduce the problem of describing uncertainty due to sampling error (SEM and 95 per cent confidence interval for the mean). In theory, much of this should be familiar from secondary education, but in the author s experience, the reality is that many new students cannot (for example) calculate the median for a small data set. These chapters are therefore relevant to level 1 students, for either teaching or revision purposes. 

Individual plasma concentrations of XYZ1234 were tabulated together with standard descriptive statistics. Individual and median profiles were presented graphically. 

All bioanalytical data, derived PK data, and safety data were listed and descriptive statistics calculated. Individual and median data were plotted. The log-transformed PK parameters AUC and Cmax were analyzed for dose proportionality. The PK parameters AUC(0-24) and Cmax were also descriptively analyzed for accumulation ratio. 

The mean data for Cmax,mit (and to a lesser extent AUCx init) on day 1 confirm the descriptive observation already made that the increases with increasing dose were slightly higher than strict dose proportionality would predict. Median Tmax,init was very constant (6 h) throughout the dose range studied. 

Due to the small sample size, all variables were only presented descriptively for the different bioanalytical data and pharmacokinetic parameters calculated number of relevant observations, geometric mean, geometric standard deviation, arithmetic mean, standard deviation, coefficient of variation, median, minimum and maximum. 

Where appropriate, individual data were presented together with descriptive statistics including mean, standard deviation, standard error of the mean, coefficient of variation (in %), median, minimum, maximum, and the number of relevant observations. 

Serum progesterone Descriptive statistics and comparison of serum progesterone concentrations in cycles 1 and 2. Descriptive comparison of the proportion of subjects who ovulated whilst receiving Drug XYZ and contraceptive concomitantly and the proportion of subjects who ovulated whilst receiving contraceptive alone. Ovulation was assumed if serum progesterone levels exceeded 1.4 ng/mL on day 20 of a menstrual cycle. Individual and mean/median profiles were presented graphically. 

Descriptive statistics (number of observations (n), mean, standard deviation, coefficient of variation in percent (CV %) or median and range) were calculated for each parameter. Statistical tests using SPSS software were as follows ... 

For pharmacokinetics in plasma Individual concentrations of XYZ1234 will be tabulated together with descriptive statistics and plotted. Median profiles will be presented graphically by CYP 2C19 metabolizer status and gender. Pharmacokinetic parameters (at least Cmax, tmax, AUC(o-t) [t = 24 h and last > LOQ ], AUCinf, ti/2z, MT, as well as CL/f and Vz/f) will be determined based on plasma concentrations of X YZ1234 using non-compartmental procedures. 

For references and descriptions, see Section 2,4.3. Phase Q is of variable composition the formula given is a median. 

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