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## Measures of Dispersion

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**Learning Objectives**• In this chapter you will learn • about the importance of variation • how to measure variation • range • variance • standard deviation**Variation**• Variation • is the heart of statistics • no variation, no need to do statistical analysis • the mean would describe the distribution**Quest for Variation**• Measures of dispersion • consider the spread between scores • Calculations include • range • variance • standard deviation**The Range**• Range • distance between the highest and lowest score in a distribution. • Calculation • Range = H minus L • H = the highest score in the data set • L = the lowest score in the data set**Variance and Standard Deviation**• Most commonly used measures of dispersion • based upon the distance of scores in a distribution from the mean • the mean is used as the central point • first step is calculation of deviation scores • how each score stands in relation to the mean**Calculation**• Formula: x = (X minus the Mean) • Table 4.2 reveals how this process works • we have the number of prior drug arrests for five clients who appeared in Tuesday’s drug court**X**f fX x ( x – X) 6 1 6 2 5 1 5 1 4 1 4 0 3 1 3 - 1 2 1 2 - 2 N = 5 ∑fX = 20 ∑x = 0 Table 4.2: Number of Prior Arrests for Drug Offenses Among Tuesday’s Drug Court Clients**Calculation**• The mean number of prior drug arrests was four • next step is to calculate how far each score (person in this case) stood in relation to this mean of four. • The deviation score (x) is calculated by subtracting the mean from each score in the distribution**Calculation**• Next step • on the first line of Table 4.2 is the score 6. If we subtract the mean, the deviation score is 2 • this person had two more prior drug arrests than the average person appearing before the drug court on Tuesday • repeat this process from each score in the distribution**Check the Math**• One characteristics of the mean is that the sum of the deviations from it equals zero (x = 0) • The sum of the deviations from the mean equal zero**Characteristic**• Positive deviation scores of • the values above the mean • are cancelled out • by the negative deviation scores of the values that fall below the mean • We are left with zero • when we sum up the deviation scores in a distribution • except for rounding or calculation errors**Squaring the Distance**• In order to remove the negative signs • square every deviation score and thus cancel out the negative numbers • remember that a negative value times a negative value equals a positive value • squaring the deviation scores gives us a total number we can work with – the variance**The Variance**• A new frequency distribution • based upon how each case stands in relation to the mean • We can calculate another average score – the variance • The variance is the mean of the squared deviations from the mean**Variance**• The previous definition is • a verbal formula • a description of how the variance is calculated • The variance is a mean • it represents the average squared deviations • that each score stands in relation to its mean**Variance**• The variance is • a measure of the spread of scores in a distribution around its mean • The larger the variance • the greater the spread of scores around the mean • The smaller the variance • the more closely the scores are distributed around the mean**Standard Deviation**• We squared the deviation scores around the mean in order to clear the negative numbers • The standard deviation is the square root of the variance**Standard Deviation**• Standard deviation • is a measure of dispersion of the scores around the mean • The higher the standard deviation • the greater the spread in the scores • The lower the standard deviation • the closer the scores are on average from the mean of the distribution**Formulae**Population variance Population standard deviation Sample variance Sample standard deviation**Summary**• Explaining variation is the basis for statistical analysis • We begin with basic measures of dispersion • range • variance • standard deviation**SPSS**Open the SPSS data file Select ANALYZE Select DESCRIPTIVE STATISTICS Select FREQUENCIES**SPSS**Select the variable from the column on the right Highlight it Mouse click the ARROW button to move it in the VARIABLES window**SPSS**Select the appropriate measures of central tendency and, shape and dispersion Mouse click CONTINUE