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Summary Satistics (1)

Summary Satistics (1) - SOME COMMONLY USED SUMMARY...

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SOME COMMONLY USED SUMMARY STATISTICAL MEASURES OF SAMPLED DATA Dr. Harvey A. Singer I. Introduction Consequent with the existence of large numerical data sets of actual observations is the necessity to summarily and briefly describe those data sets in terms of a few numbers which, in their own way, are representative and characteristic of the entire data set. Exploratory data analysis, also known as descriptive statistics, usually employs summary statistical measures to meet this objective. The purpose of data description in terms of summary statistical measures is four-fold: (1) to reduce or compress a data set into its fundamentally representative values, (2) to distill the salient information contained within a data set, (3) to facilitate identification of possible structures existing within the data set, and (4) to formulate an estimate, hypothesis or basis for future prediction. The statistical measures are of two principal types: measures of central tendency and measures of variability. The former endeavors to determine or locate the value(s) about which the data tend to cluster; the latter measures the 'strength' of the clustering by assessing the spread or dispersal of the observations. Comparisons between these two types of measures gauge the representativeness of the central tendency measures on the data set. The choice of which statistical measures to employ to describe the data set are dictated by the data itself, the aim, scope and specific purpose of the description, and self- compatibility of the description. This outline is organized into three sections. Sections II and III list the commonly employed statistical measures of central tendency and variability, respectively. Each measure is defined and annotated with some of its properties and characteristics. Section IV lists some comparisons between these measures. II. Measures of Central Tendency Mean Definition: The mean is the arithmetic average of all the data. Specifically , for a population © 1999 by Harvey A. Singer 1

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of size N consisting of the data x 1 , x 2 , , x N , the population mean µ is defined as μ = 1 N x i i = 1 N Further, for a sample of size n consisting of the data x 1 , x 2 , …, x n drawn from a population of size N, where n ² N, the sample mean x is defined as = = n i i x n x 1 1 Properties & Characteristics: 1. The mean is defined by a simple but formal calculational scheme. 2. The mean is inherently a measure of magnitude. 3. It can always be calculated for any set of numerical data, so the mean always exists. 4. A set of numerical data has one and only one mean, so the mean is always unique. 5. The mean takes into account each and every value contained in the data set. 6. The mean is sensitive to the extremes, or outliers, of the data set 7. The mean is the value to be expected on average and in the long run .
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