■ Notice that the graph is broken into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs, based on their signs: LENGTHS OF HORIZONTAL AND VERTICAL SEGMENTS Two points with the same y-coordinate lie on the same horizontal line, and two points with the same x-coordinate lie on the same vertical line. Find the distance between a horizontal or vertical segment by taking the absolute value of the difference of the two points. Example: Find the length of the line segment AB and the line segment BC. Points Sign of Coordinates Quadrant (2,3) (–2,3) (–3,–2) (3,–2) (+,+) (–,+) (–,–) (+,–) I II III IV II I III IV (−2,3) (2,3) (−3,−2) (3,−2) – THE GRE QUANTITATIVE SECTION– 200 Solution: | 2 – 7 | = 5 = AB | 1 – 5 | = 4 = BC DISTANCE OF COORDINATE POINTS To fine the distance between two points, use this variation of the Pythagorean theorem: d = ͙(x 2 – x ෆ 1 ) 2 + (y ෆ 2 – y 1 ) 2 ෆ Example: Find the distance between points (2,3) and (1,–2). Solution: d = ͙(1 – 2) ෆ 2 + (–2 ෆ – 3) 2 ෆ d = ͙(1 + –2 ෆ ) 2 + (– ෆ 2 + –3 ෆ ) 2 ෆ d = ͙(–1) 2 + ෆ (–5) 2 ෆ d = ͙1 + 25 ෆ d = ͙26 ෆ (1,–2) (2,3) (2,1) (7,5) C BA (7,1) – THE GRE QUANTITATIVE SECTION– 201 MIDPOINT To find the midpoint of a segment, use the following formula: Midpoint x = ᎏ x 1 + 2 x 2 ᎏ Midpoint y = ᎏ y 1 + 2 y 2 ᎏ Example: Find the midpoint of the segment AB. Solution: Midpoint x = ᎏ 1+ 2 5 ᎏ = ᎏ 6 2 ᎏ = 3 Midpoint y = ᎏ 2+ 2 10 ᎏ = ᎏ 1 2 2 ᎏ = 6 Therefore the midpoint of A – B — is (3,6). Slope The slope of a line measures its steepness. It is found by writing the change in y-coordinates of any two points on the line over the change of the corresponding x-coordinates. (This is also known as the rise over the run.) The last step is to simplify the fraction that results. Example: Find the slope of a line containing the points (3,2) and (8,9). (5,10) Midpoint (1,2) B A – THE GRE QUANTITATIVE SECTION– 202 Solution: ᎏ 9 8 – – 2 3 ᎏ = ᎏ 7 5 ᎏ Therefore, the slope of the line is ᎏ 7 5 ᎏ . NOTE: If you know the slope and at least one point on a line, you can find the coordinate point of other points on the line. Simply move the required units determined by the slope. In the example above, from (8,9), given the slope ᎏ 7 5 ᎏ , move up seven units and to the right five units. Another point on the line, thus, is (13,16). IMPORTANT INFORMATION ABOUT SLOPE The following are a few rules about slope that you should keep in mind: ■ A line that rises to the right has a positive slope and a line that falls to the right has a negative slope. ■ A horizontal line has a slope of 0 and a vertical line does not have a slope at all — it is undefined. ■ Parallel lines have equal slopes. ■ Perpendicular lines have slopes that are negative reciprocals. Data Analysis Review Many questions on the GRE will test your ability to analyze data. Analyzing data can be in the form of sta- tistical analysis (as in using measures of central location), finding probability, and reading charts and graphs. All these topics, and a few more, are covered in the following section. Don’t worry, you are almost done! This is the last review section before practice problems. Sharpen your pencil and brush off your eraser one more time before the fun begins. Next stop…statistical analysis! (3,2) (8,9) – THE GRE QUANTITATIVE SECTION– 203 Measures of Central Location Three important measures of central location will be tested on the GRE. The central location of a set of numeric values is defined by the value that appears most frequently (the mode), the number that represents the middle value (the median), and/or the average of all the values (the mean). MEAN AND MEDIAN To find the average, or the mean, of a set of numbers, add all the numbers together and divide by the quan- tity of numbers in the set. Average = ᎏ nu s m um be o r f o v f a v lu al e u s es ᎏ Example: Find the average of 9, 4, 7, 6, and 4. ᎏ 9+4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 The denominator is 5 because there are 5 numbers in the set. To find the median of a set of numbers, arrange the numbers in ascending order and find the middle value. ■ If the set contains an odd number of elements, then simply choose the middle value. Example: Find the median of the number set: 1, 5, 3, 7, 2. First, arrange the set in ascending order: 1, 2, 3, 5, 7. Then, choose the middle value: 3. The answer is 3. ■ If the set contains an even number of elements, simply average the two middle values. Example: Find the median of the number set: 1, 5, 3, 7, 2, 8. First, arrange the set in ascending order: 1, 2, 3, 5, 7, 8. Then, choose the middle values 3 and 5. Find the average of the numbers ᎏ 3+ 2 5 ᎏ = 4. The answer is 4. MODE The mode of a set of numbers is the number that occurs the greatest number of times. Example: For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most num- ber of times. – THE GRE QUANTITATIVE SECTION– 204 Measures 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. RANGE 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. STANDARD DEVIATION 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: 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. x 6 7 7 9 15 16 x Ϫ 10 Ϫ4 Ϫ3 Ϫ3 Ϫ1 5 6 (x Ϫ 10) 2 16 9 9 1 25 36 96 STANDARD DEVIATION = Ί ¯¯¯ 96 6 = 4 In the first column, the mean is 10. – THE GRE QUANTITATIVE SECTION– 205 . is the number that occurs the greatest number of times. Example: For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most num- ber of times. – THE GRE. 5, 3, 7, 2, 8. First, arrange the set in ascending order: 1, 2, 3, 5, 7, 8. Then, choose the middle values 3 and 5. Find the average of the numbers ᎏ 3+ 2 5 ᎏ = 4. The answer is 4. MODE The. order: 1, 2, 3, 5, 7. Then, choose the middle value: 3. The answer is 3. ■ If the set contains an even number of elements, simply average the two middle values. Example: Find the median of the number