2.18 All measures of variation are nonnegative in value for all sets of data. a. What does it mean for a value to be "nonnegative"? b. Describe the conditions necessary for a measure of variation to have the value zero. c. Describe the conditions necessary for a measure of variation to have a positive value. 2.19 Consider the sample 2, 4, 7, 8, 9. Find the following: a. Range b. Variance s2, using formula (2.5) c. Standard deviation, s 2.20 Fifteen randomly selected college students were asked to state the number of hours they slept the previous night. The resulting data are 5, 6, 6, 8, 7, 7, 9, 5, 4, 8, 11, 6, 7, 8, 7. Find the following: a. Variance s2, using formula (2.5) b. Variance s2, using formula (2.9) c. Standard deviation, s 2.21 Consider the following two sets of data: Set 1 45 80 50 45 30 Set 2 30 80 35 30 75 Both sets have the same mean, which is 50. Compare these measures for both sets: _(x - x), SS(x), and range. Comment on the meaning of these comparisons relative to the distribution. 2.22 Comment on the statement: "The mean loss fortile mean? The difference nd the mean x, or x - x, is called a deviation from the ean. Each individual value x deviates from the mean y an amount equal to (x - x). This deviation (x - x) zero when x is equal to the mean x. The deviation c - x) is positive when x is larger than x and negative Then x is smaller than x. When you square the deviations and take an aver- ge of those, you get something called the sample vari- nce, s2. It is calculated using n - 1 as the divisor: sample variance: sum of (deviations squared) s-squared = number - 1 $2 = E( x - X) 2 n - 1 (2.5) here n is the sample size-that is, the number of data lues in the sample. The variance of the sample 6, 3, 8, 5, 3 is calculated Table 2.12 using formula (2.5). OTES: The sum of all the x values is used to find x. The sum of the deviations, E(x - x), is always zero, provided the exact value of x is used. Use this heck in vo latinlations will play Sandra Fau director of com- Affect, a social media firm in N R staffs are in the Combining formulas (2.7) and (2.8) yields the "shortcut formula" for sample variance: ( sum of x ? ) (sum of x) 2 s-squared number number - 1 (Ex) 2 sample variance: $2 = n n- 1 (2.9) Formulas (2.8) and (2.9) are called "shortcuts" be- cause they bypass the calculation of x. The computa- tions for SS(x), s2, and s using formulas (2.8), (2.9), and (2.6) are performed as shown in Table 2.13. The unit of measure for the standard deviation is the same as the unit of measure for the data. For example, if our data are in pounds, then the stan- dard deviation s will also be in pounds. The unit of measure for variance might then be thought of units