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Variability Statistics means never having to say you're certain. Statistics - Chapter 4 2 Variability The amount by which scores are dispersed or scattered in a distribution. Page 74 graphs Statistics - Chapter 4 3 Range Difference between the largest and smallest scores. Problem: large groups may have large range Variance and Standard Deviation Standard Deviation - The square root of the variance. Or The square root of the mean of all the squared deviations from the mean!!??? The value of a standard deviation can NOT be negative. Standard Deviation A rough measure of the average (or standard) amount by which scores deviate on either side of their mean. Progress Check 4.1 Page 80 Employees of Corporation A earn annual salaries described by a mean of $90,000 and a standard deviation of $10,000. a. The majority of all salaries fall between what two values? b. A small minority of salaries are less than what value? c. A small minority of all salaries are more than what value? c. Answer parts (a), (b), and (c) for Corporation B’s employees, who earn annual salaries described by a mean of $90,000 and a standard deviation of $2,000. Standard Deviation Deviations from the mean. The sum of all the deviations equals the variance. To calculate the variance The sum of squares equals the sum of all squared deviation scores (p. 83) Sum of Squares Calculation example of sample sum of squares (SS) using the computation formula (p. 83) SS=ΣX2 – (ΣX)2 n Standard Deviation Calculation example of sample standard deviation using the computation formula (p. 86) s = √s2 = √ SS2 n-1 Why n-1? (p88) This applies the sample estimate to the variance rather then the population estimate. If we use the population estimate we would underestimate the variability. In other words, this allows a more conservative and accurate estimate of the variance within the sample. Degrees of freedom Degrees of freedom (df) refers to the number of values that are free to vary, given one or more mathematical restrictions, in a sample being used to estimate a population characteristic. (p. 90) The value of the population mean – mu (μ) Most of the time the population mean is unknown so we use the value of the sample mean and the degrees of freedom (df) = n-1. Standard Deviation calculation (p88) 1. 2. 3. 4. 5. 6. 7. 8. Assign a value to n representing the number of X scores. Sum all X scores. Square the sum of all X scores. Square each X score. Sum all squared X scores. Substitute numbers into the formula to obtain the sum of squares, SS. Substitute numbers in the formula to obtain the sample variance, s2. Take the square root of s2 to obtain the sample standard deviation, s. Qualitative data and variance No measures of variability exist for qualitative data! However, if the data can be ordered, then the variability can be described by identifying extreme scores (ranks). Progress Check Calculate the mean, median, mode, and standard deviation for the following height of students in inches. 64, 61, 73, 70, 71, 75, 69, 60, 63, 71, 65, 62 Statistics - Chapter 4 16