Standard Deviation

Standard Deviation and Normal Distribution For questions in the Quantitative Comparison format (“Quantity A” and “Quantity B” given), the answer choices are always as follows: (A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given. Where answer choices do not appear on Quantitative Comparison questions in this book, you should choose A, B, C or D based on the above. For questions followed by a numeric entry box | l, you are to enter your own answer in the box. For questions followed by a fraction-style numeric entry box , you are to enter your answer in the form of a fraction. You are not required to reduce fractions. For example, if the answer is —, you may enter or any equivalent fraction. All numbers used are real numbers. All figures are assumed to lie in a plane unless otherwise indicated. Geometric figures are not necessarily drawn to scale. You should assume, however, that lines that appear to be straight are actually straight, points on a line are in the order shown, and all geometric objects are in the relative positions shown. Coordinate systems, such as xy-planes and number lines, as well as graphical data presentations, such as bar charts, circle graphs, and line graphs, are drawn to scale. A symbol that appears more than once in a question has the same meaning throughout the question. 1. SetS: {5, 10, 15} If the number 15 were removed from set S and replaced with the number 1,000, which of the following values would change? Indicate all such values. ] The mean of the set ] The median of the set ] The standard deviation of the set

Dataset W: -9, -3, 3,9 Dataset X: 2, 4, 6, 8 Dataset Y: 100, 101, 102, 103 Dataset 2. 7,7, 7, 7 2. Which of the following choices lists the four datasets above in order from least standard deviation to greatest standard deviation? A) W,X,Y,Z B) W,V X, Z C) W,X,Z Y D) ZYrX W E) Z,X Y, W Set N is a set of x distinct positive integers where x > 2. Quantity B Quantity A The standard deviation of set N if The standard deviation of set N every number in the set were 3. multiplied by —3 The 75th percentile on a test corresponded to a score of 700, while the 25th percentile corresponded to a score of 450. Quantity A Quantity B 4. A 95th percentile score 800 A species of insect has an average mass of 5.2 grams and a standard deviation of 0.6 grams. The mass of the insects follows a normal distribution. S. Quantity A Quantity B The percent of the insects that The percent of the insects that

(A) (B) ©) (D) (E) 54% and 61% 61% and 68% 68% and 75% 75% and 82% 82% and 89% 10. Which of the following sets of data applies to this box-and-whisker plot? (A) (B) ©) (D) (E) Ul el eele— O = ——————d —4,-4,-2,0,0,5 —4,1,1,3,4,4 —4,-4,-3,1,5 -5,3,4,5 —4,4,-2,-2,0,0,0,5 11. If a set of data consists of only the first ten positive multiples of 5, what is the interquartile range of the set? (A) (B) ©) (D) (E) 15 25 27.5 40 45 12. On a given math test with a maximum possible score of 100 points, the vast majority of the 149 students in a class scored either a perfect score or a zero, with only one student scoring within 5 points of the mean. Which of the following logically follows about dataset 7, made up of the SCOrcsS on the test?

Indicate all such statements. [ Dataset 7T'is not normally distributed. [ The range of dataset 7 would be significantly smaller if the scores had been more evenly distributed. L] The mean of dataset T is not equal to the median. Jane scored in the 68th percentile on a test, and John scored in the 32nd percentile. Quantity A Quantity B The proportion of the class that ~ The proportion of the class that received a score less than John’s scored equal to or greater than 13. score Jane’s score In a class with 20 students, a test was administered and was scored only in whole numbers from 0 to 10. At least 1 student got every possible score, and the average score was 7. Quantity A The lowest score that could have Quantity B been received by more than 1 4 14. student A test 1s scored out of 100 and the scores are divided into five quintile groups. Students are not told their scores, but only their quintile group. Quantity A , Quantity B The scores of two students in the _ o The score of a student in the top bottom quintile group, chosen at o quintile group, chosen at random. 15. random and added together

Some rock samples are weighed, and their weights are determined to be normally distributed. One standard deviation below the mean is 250 grams and one standard deviation above the mean is 420 grams. Quantity A Quantity B 20. The median weight, in grams 335 grams | | | | | | | | | [ | [ } 1 500 600 The graph represents the normally distributed scores on a test. The shaded area represents approximately 68% of the scores. Quantity A Quantity B 21. The mean score on the test 550 22. A

A and B are graphical representations of normally distributed random variables X and Y, respectively, with relative positions, shapes, and sizes as shown. Which of the following must be true? Indicate all such statements. [ ¥ has a greater standard deviation than X. [ The probability that Y falls within 2 standard deviations of its mean is greater than the probability that X falls within 2 standard deviations of its mean. [ Y has a greater mean than X. 23. The outcome of a standardized test is an integer between 151 and 200, inclusive. The percentiles of 400 test scores are calculated, and the scores are divided into corresponding percentile groups. Quantity A .. : Quantity B The minimum number of integers . ) The minimum number of between 151 and 200, inclusive, , percentile groups that correspond that include more than one _ to a score of 200 percentile group

[ Both biologists’ and physicists’ salaries are normally distributed. [ The range of salaries is greater than $150,000. Town X TownY 26. The graph on the left above represents the number of family members per family in Town X, while the graph on the right above represents the number of family members per family in Town Y. The median family size for Town X is equal to the median family size for Town Y. The horizontal and vertical dimensions of the boxes above are identical and correspond to the same measurements. Which of the following statements must be true? Indicate all such statements. L] The range of family sizes measured as the number of family members is larger in Town X than in Town Y. L] Families in Town Y are more likely to have sizes within 1 family member of the mean than are families in Town X. L] The data for Town X has a larger standard deviation than the data for Town Y. T 27. The box-and-whisker plot shown above could be a representation of which of the following sets? (A) —2,0,2,4 B) 3,3,3,3,3,3 (©) 1,25, 100

(D) 2,4,8,16,32 (E) 1,13,14,17 28. Which of the following statements must be true about the data described by the box-and-whisker plot above? Indicate all such statements. ] The median of the whole set is closer to the median of the lower half of the data than it is to the median of the upper half of the data. [ The data is normally distributed. [ The set has a standard deviation greater than zero. 29. The earthworms in sample A have an average length of 2.4 inches, and the earthworms in sample B have an average length of 3.8 inches. The average length of the earthworms in the combined samples is 3.0 inches. Which of the following must be true? Indicate all such statements. [ There are more earthworms in sample A than in sample B. [ The median length of the earthworms is 3.2 inches. [ The range of lengths of the earthworms is 1.4.