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Chapter 3

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Sociology 2205A/B
William Marshall

Statistics – Chapter 3 Introduction: - frequency distributions, graphs, and chart summarize overall shape of distribution of scores in a comprehensive way - need to show the average or typical case in the distribution – measures of central tendency - also how much variety or heterogeneity there is in distribution – measures of dispersion - three common measures of central tendency – mean (average score), mode (most common score), median (middle case) - they reduce data - measures of dispersion provide information about variety, diversity, or heterogeneity Nominal level measures: - mode is the score that appears the most - most useful when you want a “quick and easy” indicator at central tendency or when you are working with nominal level variables - some have no modes at all, or so many that it loses meaning - with ordinal and interval – ratio data, mode may not be most common/typical Table 3.2 - to conveniently measure dispersion of variable we can use a statistic based on the mode called the variation ratio (v) - the longer the proportion of cases in mode of a variable, the less the dispersion among cases of that variable fm = # of cases in mode n = total # of cases Ordinal Level Measures: - median is a measure of central tendency that represents the exact centre of a distribution of scores - cases must be places highest to lowest first - then find centre - when “n” is odd, find the middle case by adding 1 to “n” and then dividing the sum by 2 Example: 1 (added in), 2, 4, 6, 6, 8, 10, 10 6 + 6/2 = 6 Table 3.4 - the range (R) is the difference of interval between the highest score (H) and lowest score (L) in a distribution, provides a quick and general notion of variability of variables measured at either the ordinal-or-interval-ratio level - R will be misleading as a measure of dispersion if just are of these scores is either exceptionally high or low  there are called outliers R = H – L - the interquartile range (Q) is a type of range that avoids this problem Q = Q˅3 - Q˅1 Example: 2,3,8,9 R = 9 – 2 = 7 Q = 3 + 8/2 = 5.5 Q˅1 = 2 + 3/2 = 2.5 Q˅3 = 8 + 9/2 = 8.5 Q˅3 - Q˅1 = 6 Table 3.5 - the range and interquartile range provide a measure of dispersion based on just two scores from a set of data Interval-Ratio-Level Measures: - mean reports the average score of a distribution = mean = sum of all scores = number of cases - this gives mean of sample - means of population: = sum of all scores = number of cases in the population Some Characteristics of the Mean: - mean is center of any distribution of scores in the sense that all points around it cancel out - if you take each score, subtract the mean from it, and add all of the differences, it will always be zero - this indicated that the mean is a good descriptive measure of the centrality of scores Least Squares Principle: - if you square and sum these differences, the result sum will be less than the sum of the squared difference between scores and any other point in the distribution Table 3.7 - mean will always have a greater value than the median when it has really high scores this - vice versa when it has really low numbers Figure 3.2 – 3.4 Variance and Standard Deviation: - deviations are the distances between scores and means - this quantity will increase in value as the scores increase in their variety or heterogeneity - the sum of the deviations is a logical basis for a statistic that measures the amount of variety in a set of scores - to eliminate the negative
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