Central tendency measures statistical

The first group of descriptive statistics are the measures of central tendency. These statistics describe where the middle of the data falls. There are three measures of central tendency. They are the mean, median, and mode (Hays 1998,155-56). 

The idea behind measures of location and central tendency is contained within the notion of the average. There are predominantly three summary statistics that are commonly used for describing this aspect of a set of data the arithmetic mean - normally shortened to the mean, the mode and the median. 

In general, bias refers to a tendency for parameter estimates to deviate systematically from the true parameter value, based on some measure of the central tendency of the sampling distribution. In other words, bias is imperfect accuracy. In statistics, what is most often meant is mean-unbiasedness. In this sense, an estimator is unbiased (UB) if the average value of estimates (averaging over the sampling distribution) is equal to the true value of the parameter. For example, the mean value of the sample mean (over the sampling distribution of the sample mean) equals the mean for the population. This chapter adheres to the statistical convention of using the term bias (without qualification) to mean mean-unbiasedness. 

Sensitivity, specificity, odds ratio, and relative risk Types of data and scales of measurement Measures of central tendency and dispersion Inferential statistics Students s t-distribution Comparing means Comparing more than two means Regression and correlation Nonparametric tests The x2-test Clinical trials INTRODUCTION... 

Descriptive Statistics involves the presentation of summary statistics, which are concise yet meaningful summaries of large amounts of data. One category of descriptive statistics is the measurement of central tendency. 

Robust statistics are a set of methods that are largely unaffected by the presence of extreme values. Commonly used statistics of this type are the median and the median absolute deviation (MAD). The median is a measure of the central tendency of the data and can be used to replace the mean value. If the data are normally... 

The analytical plan of epidemiological studies should use descrip tive and analytical techniques in describing the sample and results. Descriptive statistics, such as frequency distributions, cross-tabulations, measures of central tendency, and variation, can help explain underlying distributions of variables and direct the assessment of appropriateness of more advanced statistical techniques. Careful weighing of study findings with respect to the design and methods helps to ensure the validity of results. 

Non-parametric statistical techniques (i.e. those that make minimal assumptions about the error distribution) can be used to handle the raw data. Such methods are generally resistant towards the effects of extreme values, often because they use the median (see Section 6.2) as a measure of central tendency or measure of location. Such methods have the further advantage of extreme simplicity of calculation in many cases, but while popular in the behavioural sciences they are less frequently used in the analytical sciences. 

We first review the fundamentals of small particle statistics as these apply to synthetic polymers. This is mainly concerned with the use of statistical moments to characterize molecular weight distributions. One of the characteristics of such a distribution is its central tendency, or average, and the following main topic shows how it is possible to determine various of these averages from measure-... 

One of the primary goals of Statistics is to use data from a sample to estimate an unknown quantity from an underlying population, called a population parameter. In general, we typically use the arithmetic mean as the measure of central tendency of choice because the sample mean is an unbiased estimator of the population mean, typically represented by the symbol p. The main conceptual point about unbiased estimators is that they come closer to estimating the true population parameter, in this case the population mean, than biased estimators. When extreme observations influence the value of the mean such that it really is not representative of a typical value, use of the median is recommended as a measure of central tendency. 

The unknown quantities of interest described in the previous section are examples of parameters. A parameter is a numerical property of a population. One may be interested in measures of central tendency or dispersion in populations. Two parameters of interest for our purposes are the mean and standard deviation. The population mean and standard deviation are represented by p and cr, respectively. The population mean, p, could represent the average treatment effect in the population of individuals with a particular condition. The standard deviation, cr, could represent the typical variability of treatment responses about the population mean. The corresponding properties of a sample, the sample mean and the sample standard deviation, are typically represented by x and s, which were introduced in Chapter 5. Recall that the term "parameter" was encountered in Section 6.5 when describing the two quantities that define the normal distribution. In statistical applications, the values of the parameters of the normal distribution cannot be known, but are estimated by sample statistics. In this sense, the use of the word "parameter" is consistent between the earlier context and the present one. We have adhered to convention by using the term "parameter" in these two slightly different contexts. 

As we have seen, several summary measures of central tendency can be used for continuous outcomes. The most common of these measures is the mean. In clinical trials we calculate sample statistics, and these serve to estimate the unknown population means. When developing a new drug, the estimated treatment effect is measured by the difference in sample means for the test treatment and the placebo. If we can infer (conclude) that the corresponding population means differ by an amount that is considered clinically important (that is, in the positive direction and of a certain magnitude) the test treatment will be considered efficacious. 

Arithmetic Mean The arithmetic mean is the same as the arithmetic mean or average of a distribution - the sum of all the data points divided by the number of data points. The arithmetic mean is a good measure of the central tendency of roughly normal distributions, but may be misleading in skewed distributions. In cases of skewed distributions, other statistics such as the median or geometric mean may be more informative. 

Statistical indices are fundamental numerical quantities measuring some statistical property of one or more variables. They are applied in any statistical analysis of data and hence in most of Q S AR methods as well as in some algorithms for the calculation of molecular descriptors. The most important univariate statistical indices are indices of central tendency and indices of dispersion, the former measuring the center of a distribution, the latter the dispersion of data in a distribution. Among the bivariate statistical indices, the correlation measures play a fundamental role in all the sciences. Other important statistical indices are the diversity indices, which are related to the injbrmationcontentofavariahle,the —> regressiowparameters, used for regression model analysis, and the —> classification parameters, used for classification model analysis. 

Similarly, statistics are everywhere—in news reports, sports, and on your favorite websites. Mean, median, and mode are three common statistics that give information on a group of numbers. They are called measures of central tendency because they are different ways of finding the central trend in a group of numbers. In the first section, you will learn all about mean, median, and mode. 

Fundamental to statistical measurement are two basic parameters the population mean, /r, and the population standard deviation, cr. The population parameters are generally unknown and are estimated by the sample mean, x, and sample standard deviation, s. The sample mean is simply the central tendency of a sample set of data that is an unbiased estimate of the population mean, /r. The central tendency is the sum of values in a set, or population, of numbers divided by the number of values in that set or population. For example, for the sample set of values 10, 13, 19, 9, 11, and 17, the sum is 79. When 79 is divided by the number of values in the set, 6, the average is 79 6 = 13.17. The statistical formula for average is... 

The current bioequivalence rule is not to use the ratio of means, as above, but instead to use the 90% confidence interval around the ratio of medians (another statistical measure of central tendency). When this confidence interval is calculated, it must fall within 80-125% of the standard or the products are considered to lack bioequivalence (that is, bioequivalence has not been proven). 

The software Microsoft Excel for Windows and the Statistical Package for Social Sciences 12.0 (SPSS) were used to create a database and to conduct statistical analysis. Measures of central tendency and dispersion were used to describe data, with means and standard deviations for quantitative variables. The Student t test for paired samples was used for comparisons between groups. The level of significance was established at p<0.05. 

Empirical evidence indicates that the majority of the respondents locate their service quality score at the right-hand side of the scale (Brown et al. 1993 Parasuraman et al. 1988 1991 Peterson and Wilson 1992). This distribution is referred to as negatively skewed. A skewed distribution contains several serious implications for statistical analysis. To begin with, the mean might not be a suitable measure of central tendency. In a negatively skewed distribution, the mean is typictilly to the left of the median and the mode and thus excludes considerable information about the variable under study (Peterson and Wilson 1992). Skewness also attenuates the correlation between variables. Consequently, the true relationship between variables in terms of a correlation coefficient may be understated (Peterson and Wilson 1992). Einally, parametric tests (e.g., t-test, F-test) assume that the population is normally or at least symmetrically distributed. 

As before, the mean and variance of the probability distribution for the present worth statistic (PW) are denoted by fXp and 0-2, respectively. These are measures of central tendency and variability, or... 

You should have a good grasp of statistical measures of central tendency and variation. 

In industry the term variance can be used two different ways. First, there is variance between planned performance and actual results. Second, there is a statistical measure of central tendency of a set of data, which is called variance. It is this second or statistical definition of variance that is discussed here. For example, if the mean flow time in a factory is 10 days, but some jobs are completed as early as 3 days and others take as long as 25 days, there is variance in the flow-time data. Equation 4.1 is used to calculate the variance of a set of data. First, the mean is calculated (shown here as X). Then the difference between the mean and each piece of data in the set is calculated (X - X) and then this difference is squared. The sum of these n pieces of data squared is then divided by (n - 1), which is one less than the number of data points. 

The distribution of random errors should follow the Gaussian or normal curve if the number of measurements is large enough. The shape of Gaussian distribution was given in Chapter 3 (Fig. 3.4). It can be characterized by two variables—the central tendency and the symmetrical variation about tjie central tendency. Two measures of the central tendency are the mean, X, and the median. One of these values is usually taken as the correct value for an analysis, although statistically there is no correct value but rather the most probable value. The ability of an analyst to determine this most probable value is referred to as his accuracy. 

Commonly used descriptive statistics include measures that describe where the middle of the data is. These measures are sometimes called measures of central tendency and include the mean, median, and mode. Another category of measures describes how spread out the data is. These measures are sometimes called measures of variability and include the range, variance, and standard deviation. Additional descriptive measures can include percentages, percentiles, and frequencies. In safety performance measurement, the safety professional must determine the format of the data (i.e., ratio, interval, ordinal, or categorical) that will be collected and match the data format to the appropriate statistic. As will be discussed in the following sections, certain descriptive statistics are appropriate for certain data formats. 

Descriptive Statistics statistical techniques that are used to describe the population or sample. Commonly used descriptive statistics include measures of central tendency mean, median and mode and measures of variability range, variance and standard deviation. Additional descriptive measures can include percentages, percentiles and frequencies. 

As its name suggests, this type of summary statistic is used to indicate the mid-point of a set of numbers. Usually it is referred to as the average but this term can be used for a number of different types of measure of central tendency and it is not normally used by statisticians. Instead, they use terms which describe specific measures of central tendency, and three of these measures, the arithmetic mean, the median and the mode, are of particular interest. 

The most basic step in statistical analysis of a data set is to describe it descriptively, that is, to compute properties associated with the data set and to display the data set in an informative manner. A data set consists of a finite number of samples or data points. In this book, a data set will be denoted using either set notation, that is, jti, X2,..., x or vector notation, that is, as if = (x],jC2,..., x ). Set notation is useful for describing and listing the elements of a data set, while vector notation is useful for mathematical manipulation. The size of the data set is equal to n. The most common descriptive statistics include measures of central tendency and dispersion. 

Levene s test is an alternative to Bartlett s test, the former being much less sensitive than the latter to departures from normality. Nevertheless, unless you have strong evidence that your data do not in fact come from a nearly normal distribution, Bartlett s test has better performance. Levene s test checks indirectly whether the variances of the different levels of concentrations are statistically the same. First, for each level of standards i.e. for each nominal concentration), the absolute differences between the signals of the repKcates and their central tendency is calculated and then a one-way ANOVA (analysis of variance) on the absolute values of the deviations is performed. In the original work, Levene used the mean as a measure of the central tendency. Following the work of Brown and Forsythe, the median is currently used as a robust estimator. Levene s test is based on a comparison of Levene s experimental statistic with a tabulated F value. 

---