Descriptive statistics standard deviation

The degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

Standard deviation, 227—228 Standard error of the difference, 230 Standard values, 249—251 Statistical power, 253 Statistical significance, 227 Statistics descriptive... 

The flowsheet shown in the introduction and that used in connection with a simulation (Section 1.4) provide insights into the pervasiveness of errors at the source, random errors are experienced as an inherent feature of every measurement process. The standard deviation is commonly substituted for a more detailed description of the error distribution (see also Section 1.2), as this suffices in most cases. Systematic errors due to interference or faulty interpretation cannot be detected by statistical methods alone control experiments are necessary. One or more such primary results must usually be inserted into a more or less complex system of equations to obtain the final result (for examples, see Refs. 23, 91-94, 104, 105, 142. The question that imposes itself at this point is how reliable is the final result Two different mechanisms of action must be discussed ... 

In order to determine the optimal number of compartments, literature information on small intestinal transit times was utilized. A total of over 400 human small intestinal transit time data were collected and compiled from various publications, since the small intestinal transit time is independent of dosage form, gender, age, body weight, and the presence of food [70]. Descriptive statistics showed that the mean small intestinal transit time was 199 min with a standard deviation of 78 min and a 95% confidence interval of 7 min. The data set was then analyzed by arranging the data into 14 classes, each with a width of 40 min. Figure 9 shows the distribution of this data set. 

The Student s (W.S. Gossett) /-lest is useful for comparisons of the means and standard deviations of different analytical test methods. Descriptions of the theory and use of this statistic are readily available in standard statistical texts including those in the references [1-6]. Use of this test will indicate whether the differences between a set of measurement and the true (known) value for those measurements is statistically meaningful. For Table 36-1 a comparison of METHOD B test results for each of the locations is compared to the known spiked analyte value for each sample. This statistical test indicates that METHOD B results are lower than the known analyte values for Sample No. 5 (Lab 1 and Lab 2), and Sample No. 6 (Lab 1). METHOD B reported value is higher for Sample No. 6 (Lab 2). Average results for this test indicate that METHOD B may result in analytical values trending lower than actual values. 

The number of subjects per cohort needed for the initial study depends on several factors. If a well established pharmacodynamic measurement is to be used as an endpoint, it should be possible to calculate the number required to demonstrate significant differences from placebo by means of a power calculation based on variances in a previous study using this technique. However, analysis of the study is often limited to descriptive statistics such as mean and standard deviation, or even just recording the number of reports of a particular symptom, so that a formal power calculation is often inappropriate. There must be a balance between the minimum number on which it is reasonable to base decisions about dose escalation and the number of individuals it is reasonable to expose to a NME for the first time. To take the extremes, it is unwise to make decisions about tolerability and pharmacokinetics based on data from one or two subjects, although there are advocates of such a minimalist approach. Conversely, it is not justifiable to administer a single dose level to, say, 50 subjects at this early stage of ED. There is no simple answer to this, but in general the number lies between 6 and 20 subjects. 

The description of large data tables by the usual univariate statistics (mean, standard deviation, range,. ..) and by histograms is still used in recent literature. Comparison between categories is made by the use of category means and ran s. Sometimes, the correlation coefficients are considered. The discussion of the extracted information can be wide-ranging and difficult to understand immediately. 

In chemical engineering, there are many situations in which the response is montiform. Chromatography comes immediately to mind as a process in which sharp peaks are desirable for analysis or preparation. Such peaks can be very roughly characterized by their means and standard deviations, and if, as is quite often the case, we have reason to suppose that a distribution is asymptotically Gaussian, then these two statistics suffice for its description. If fa) is the distribution in question, its moments about the origin are... 

It is useful to see what effect retaining or not retaining a data point has on the mean and standard deviation for a set of data. The table below shows descriptive statistics for this data. 

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. 

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. 

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 ... 

Descriptive statistics. A series of physical measurements can be described numerically. If for example, we have recorded the concentration of 1000 different samples in a research problem, it is not possible to provide the user with a table giving all 1000 results. In this case, it is normal to summarize the main trends. This can be done not only graphically, but also by considering the overall parameters such as mean and standard deviation, skewness etc. Specific values can be used to give an overall picture of a set of data. 

In tables where the dispersion of each data set is shown by an appropriate statistical parameter, you must state whether this is the (sample) standard deviation, the standard error (of the mean) or the 95% confidence limits and you must give the value of n (the number of repUcates). Other descriptive statistics should be quoted with similar detail, and hypothesis-testing statistics should be quoted along with the value of P (the probabiUty). Details of any test used should be given in the legend, or in a footnote. 

A random sample of product containers may be used for this testing. Alternately, the product may originate from various defined areas within the lyophilizer. The data gathered should be interpreted in terms of descriptive statistics. For each analytical attribute, the mean, the standard deviation, the percentiles, the extreme values, and the normality of the distribution can be determined. 

Each measure of an analysed variable, or variate, may be considered independent. By summing elements of each column vector the mean and standard deviation for each variate can be calculated (Table 7). Although these operations reduce the size of the data set to a smaller set of descriptive statistics, much relevant information can be lost. When performing any multivariate data analysis it is important that the variates are not considered in isolation but are combined to provide as complete a description of the total system as possible. Interaction between variables can be as important as the individual mean values and the distributions of the individual variates. Variables which exhibit no interaction are said to be statistically independent, as a change in the value in one variable cannot be predicted by a change in another measured variable. In many cases in analytical science the variates are not statistically independent, and some measure of their interaction is required in order to interpret the data and characterize the samples. The degree or extent of this interaction between variables can be estimated by calculating their covariances, the subject of the next section. 

Spreadsheet Summary In Chapter 2 oi Applications of Microsoft Excel in Analytical Chemistry, we introduce the use of Excel s Analysis ToolPak to compute the mean, standard deviation, and other quantities. In addition, the Descriptive Statistics package finds the standard error of the mean, the median, the range, the maximum and minimum values, and parameters that reflect the symmetry of the data set. 

The heights of the bars or columns usually represent the mean values for the various groups, and the T-shaped extension denotes the standard deviation (SD), or more commonly, the standard error of the mean (discussed in more detail in Section 7.3.2.3). Especially if the standard error of the mean is presented, this type of graph tells us very litde about the data - the only descriptive statistic is the mean. In contrast, consider the box and whisker plot (Figure 7.2) which was first presented in Tukey s book Exploratory Data Analysis. The ends of the whiskers are the maximum and minimum values. The horizontal line within the central box is the median, fhe value above and below which 50% of the individual values lie. The upper limit of the box is the upper or third quartile, the value above which 25% and below which 75% of fhe individual values lie. Finally, the lower limit of the box is the lower or first quartile, the values above which 75% and below which 25% of individual values lie. For descriptive purposes this graphical presentation is very informative in giving information about the distribution of the data. 

Descriptive statistics Used to summarize information and for the comparison of numbers in different sets of data mean, median, mode, range, variance, standard deviation are descriptive statistics. 

Many LlMSs capture the study data and contain simple programs that calculate group mean and standard deviation (or standard mean error) values—sometimes termed descriptive statistical data. The user should recognize that these statistical programs work with the assumptions that the groups are adequately sized and all distributions are Gaussian. These assumptions may be incorrect and thus all individual data should be examined. 

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