Statistics Demystified, 2nd edition Stan Gibilisco Explanations for Quiz Answers in Chapter 2 1. Statisticians define a discrete variable as a quantity that can attain only certain, well-defined values corresponding to points on a number line, such as all the positive whole numbers less than 500. The correct choice is B. 2. In a discrete distribution, the mode is the value that the discrete variable attains with the greatest frequency. The correct choice is B. Some discrete distributions have more than one mode; we call them multimodal distributions. In a discrete distribution where each value of the discrete variable occurs exactly as often as all the other values do, we say that the distribution has no mode, or that the mode is undefined. 3. In order to determine the mean score µ in this situation, we can simply find the average (also known as the arithmetic mean) of all the nonnegative integers between, and including, 0 and 20, because each of the 21 students got a different score, and every possible score was accounted for. First, we must add up all the possible scores, getting 0 + 1 + 2 + 3 + ... + 18 + 19 + 20 = 210 Then we divide by the number of values (in this case 21), getting a mean score of µ = 210 / 21= 10 The correct choice is D. 4. As defined at the bottom of page 59, the term measure of central tendency refers to either the mean, the median, or the mode. Based on that strict and straightforward standard, the answer is clearly A. 5. As defined in the "Tip" on page 62, variance and standard deviation both constitute measures of dispersion. Based on that rigid criterion, the correct choice is D. 6. In a discrete distribution, we define the median as the value (or the average of two competing values) such that the number of elements greater than or equal to it is the same as the number of elements less than or equal to it. In the situation described here, we have 37 elements, which we can describe as the set S = {0, 1, 2, 3, ..., 34, 35, 36} If we choose values one by one and test them for adherence to this definition, we'll find that the median must be 18. We have 19 elements less than or equal to 18, which we can denote as the set S18- = {0, 1, 2, 3, ..., 16, 17, 18} and 19 elements greater than or equal to 18, which we can portray as the set S18+ = {18, 19, 20, ..., 34, 35, 36} The correct choice is C. 7. In this situation, the only two scores that got any "hits" are 8 correct and 10 correct. Both of these scores occurred with the same frequency, so they both represent modes (we have a bimodal distribution). The correct choice is A. 8. Recall the definition of the term median as it applies to a discrete distribution. It's the value (or the average of two competing values) such that the number of elements greater than or equal to it is the same as the number of elements less than or equal to it. In this case, that's the score of 9 (even though no one actually got that score). The correct choice is B. 9. Before we can calculate the variance, we must know the mean score µ. In this case, we have 22 scores of 8 correct, 22 scores of 10 correct, and no other scores whatsoever. The total number of scores equals 44. Therefore µ = (22 x 8 + 22 x 10) / 44= (176 + 220) / 44= 396 / 44= 9.000 expressed to three decimal places. If we call our discrete variable x, and if we let n represent the total number of elements (in this case 44), then we can calculate the variance of x as Var (x) = (1/n) [(x1 - µ)2 + (x2 - µ)2 + ... + (xn - µ)2] where x1, x2, ..., and xn represent the individual scores. When we input the values and then do the arithmetic (a tedious process indeed), we end up with a variance of exactly 1, so we can say that Var (x) = 1.000 carried out to three decimal places. We can intuitively see that this figure makes sense when we recall the fact that in a discrete distribution, the variance equals the average of the squares of the "distances" of each value from the mean. All the scores in this distribution lie either "1 below the mean" or else "1 above the mean," and we observe an equal number of lower scores and higher scores. The correct choice is C. 10. By definition, the standard deviation equals the square root of the variance. We've calculated the variance as exactly equal to 1 (or 1.000, carried out to three decimal places). Therefore, the standard deviation also equals 1.000 (carried out to three decimal places). The answer is C.