The gre quatitative section 4 pps

The x represents a measurement, and the f repre-sents the number of times that measurement occurs.. ■ If given a percentage, write it in the numerator position of the number column.. ■ T

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  10

4

3

3

1 5 6

(x  10)2

16 9 9 1 25 36 96

STANDARD DEVIATION =  ¯¯¯96

6 = 4

In the first column, the mean is 10

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

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

Be sure that the total number of measurements taken is equal to the total at the bottom of the frequency distribution chart

DATA REPRESENTATION AND INTERPRETATION

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

total:

7 10 12 13 15

3 4 4 2 2 15

total:

Bar 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

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

Increase

Decrease

No Change

Increase Decrease

Change in Time

Comparison of Road Work Funds

of New York and California

1990–1995

New York California

KEY

0 10 20 30 40 50 60 70 80 90

1991 1992 1993 1994 1995

Year

25%

40%

35%

ples of these two concepts are covered further in the following sections.

PERCENTAGE PROBLEMS

There is one formula that is useful for solving the three types of percentage problems:

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?

Solve by cross multiplying

100(x) = (40)(50)

100x = 2,000

110000x= 21,00000

x = 20

Therefore, 20 is 40% of 50

Example:

Finding a number when a percentage is given:

40

x

40% of what number is 24?

Cross multiply:

(24)(100) = 40x

2,400 = 40x

2,44000= 4400x

60 = x

Therefore, 40% of 60 is 24

Example:

Finding what percentage one number is of another:

What percentage of 75 is 15?

Cross multiply:

15(100)  (75)(x)

1,500  75x

1,

7

5

5

00

7

7

5 5

x



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:

Probability of an event =

Example:

If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, find the probability of selecting a red marble:

Probability of an event = Number or specific outcomes = 5 +33 + 6

Total number of possible outcomes

Number of specific outcomes



Total number of possible outcomes

x

15

40 24

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= 47

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 operaatten-tions being performed

Example:

Given a  b  (a b 3)2, find the value of 1  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:

Solution:

Fill in variables according to the placement of number in the triangular figure: a  1, b  2, and c 3

1 –

3

2

+ 1 –

2

3

+ 2 – 1

3

= – 3

1

+ –1 + –1 = –21

3 

b

c a

2

3 1

If = _ + _ + _a − b a − c b − c

c b a

Then what is the value of