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## Nội dung Text: The gre quatitative section 4

1. – THE GRE QUANTITATIVE SECTION – M easures of Dispersion Measures of dispersion, or the spread of a number set, can be in many different forms. The two forms covered on the GRE test are range and standard deviation. R ANGE The range of a data set is the greatest measurement minus the least measurement. For example, given the fol- lowing values: 5, 9, 14, 16, and 11, the range would be 16 – 5 = 11. S TANDARD D EVIATION As you can see, the range is affected by only the two most extreme values in the data set. Standard deviation is a measure of dispersion that is affected by every measurement. To ﬁnd the standard deviation of n meas- urements, follow these steps: 1. First, ﬁnd the mean of the measurements. 2. Subtract the mean from each measurement. 3. Square each of the differences. 4. Sum the square values. 5. Divide the sum by n. 6. Choose the nonnegative square root of the quotient. Example: 10)2 (x x 10 x 4 16 6 3 9 7 9 7 3 1 9 1 25 15 5 36 16 6 96 In the first column, the mean is 10. ¯¯¯ 96 STANDARD DEVIATION = 6 =4 When you ﬁnd the standard deviation of a data set, you are ﬁnding the average distance from the mean for the n measurements. It cannot be negative, and when two sets of measurements are compared, the larger the standard deviation, the larger the dispersion. 205
2. – THE GRE QUANTITATIVE SECTION – F REQUENCY D ISTRIBUTION The frequency distribution is essentially the number of times, or how frequently, a measurement appears in a data set. It is represented by a chart like the one below. The x represents a measurement, and the f repre- sents the number of times that measurement occurs. x f total: To use the chart, simply list each measurement only once in the x column and then write how many times it occurs in the f column. For example, show the frequency distribution of the following data set that represents the number of students enrolled in 15 classes at Middleton Technical Institute: 12, 10, 15, 10, 7, 13, 15, 12, 7, 13, 10, 10, 12, 7, 12 x f 7 3 4 10 12 4 2 13 15 2 total: 15 Be sure that the total number of measurements taken is equal to the total at the bottom of the frequency distribution chart. D ATA R EPRESENTATION I NTERPRETATION AND The GRE will test your ability to analyze graphs and tables. It is important to read each graph or table very carefully before reading the question. This will help you process the information that is presented. It is extremely important to read all the information presented, paying special attention to headings and units of measure. On the next page is an overview of the types of graphs you will encounter. Circle Graphs or Pie Charts This type of graph is representative of a whole and is usually divided into percentages. Each section of the chart represents a portion of the whole, and all of these sections added together will equal 100% of the whole. 206
3. – THE GRE QUANTITATIVE SECTION – 25% 40% 35% B ar Graphs Bar graphs compare similar things by using different length bars to represent different values. On the GRE, these graphs frequently contain differently shaded bars used to represent different elements. Therefore, it is important to pay attention to both the size and shading of the graph. Comparison of Road Work Funds of New York and California 1990–1995 Money Spent on New Road Work 90 80 in Millions of Dollars 70 60 50 KEY 40 New York 30 California 20 10 0 1991 1992 1993 1994 1995 Year Broken-Line Graphs Broken-line graphs illustrate a measurable change over time. If a line is slanted up, it represents an increase, whereas a line sloping down represents a decrease. A ﬂat line indicates no change as time elapses. se De Unit of Measure rea cr Inc eas D se e ec rea re as Inc e No Change Change in Time 207
4. – THE GRE QUANTITATIVE SECTION – P ercentage and Probability Part of data analysis is being able to calculate and apply percentages and probability. Further review and exam- ples of these two concepts are covered further in the following sections. P ERCENTAGE P ROBLEMS There is one formula that is useful for solving the three types of percentage problems: # % = 100 When reading a percentage problem, substitute the necessary information into the previous formula based on the following: 100 is always written in the denominator of the percentage-sign column. ■ If given a percentage, write it in the numerator position of the number column. If you are not given a ■ percentage, then the variable should be placed there. The denominator of the number column represents the number that is equal to the whole, or 100%. ■ This number always follows the word of in a word problem. For example: “ . . . 13 of 20 apples . . . ” The numerator of the number column represents the number that is the percent. ■ In the formula, the equal sign can be interchanged with the word is. ■ Example: Finding a percentage of a given number: What number is equal to 40% of 50? # % x 40 __ = ___ 50 100 Solve by cross multiplying. 100(x) = (40)(50) 100x = 2,000 100x 2,000 = 100 100 x = 20 Therefore, 20 is 40% of 50. Example: Finding a number when a percentage is given: 208
5. – THE GRE QUANTITATIVE SECTION – 40% of what number is 24? # % 24 40 __ = ___ 100 x Cross multiply: (24)(100) = 40x 2,400 = 40x 2,400 40x = 40 40 60 = x Therefore, 40% of 60 is 24. Example: Finding what percentage one number is of another: What percentage of 75 is 15? # % 15 x __ = ___ 75 100 Cross multiply: 15(100) (75)(x) 1,500 75x 1,500 75x 75 75 20 x Therefore, 20% of 75 is 15. Probability Probability is expressed as a fraction; it measures the likelihood that a speciﬁc event will occur. To ﬁnd the probability of a speciﬁc outcome, use this formula: Number of speciﬁc outcomes Probability of an event = Total number of possible outcomes E xample: If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, ﬁnd the probability of selecting a red marble: Number or speciﬁc outcomes 3 Probability of an event = = 5+3+6 Total number of possible outcomes 3 Therefore, the probability of selecting a red marble is 14 . 209
6. – THE GRE QUANTITATIVE SECTION – M ULTIPLE P ROBABILITIES To ﬁnd the probability that two or more events will occur, add the probabilities of each. For example, in the problem above, if we wanted to ﬁnd the probability of drawing either a red or blue marble, we would add the probabilities together. The probability of drawing a red marble = 134 . And the probability of drawing a blue marble = 154 . Add the two together: 134 + 154 = 184 = 4 . 7 So, the probability for selecting either a blue or a red would be 8 in 14, or 4 in 7. Helpful Hints about Probability If an event is certain to occur, the probability is 1. ■ If an event is certain not to occur, the probability is 0. ■ If you know the probability an event will occur, you can ﬁnd the probability of the event not occurring ■ by subtracting the probability that the event will occur from 1. Special Symbols Problems The last topic to be covered is the concept of special symbol problems. The GRE will sometimes invent a new arithmetic operation symbol. Don’t let this confuse you. These problems are generally very easy. Just pay atten- tion to the placement of the variables and operations being performed. Example: 3)2, ﬁnd the value of 1 Given a b (a b 2. Solution: Fill in the formula with 1 being equal to a and 2 being equal to b. (1 2 3)2 (2 3)2 (5)2 25. So, 1 2 25. Example: b a−b a−c b−c = _____ + _____ + _____ If c b a a c 2 Then what is the value of . . . 1 3 Solution: Fill in variables according to the placement of number in the triangular ﬁgure: a 1, b 2, and c 3. 1–2 1–3 2–3 –1 1 + + = + –1 + –1 = –2 3 3 2 1 3 210 