Averages are a statistics that are used to get information on a set of data.
The arithmetic mean of a set of n numbers is equal to the sum of the numbers divided by n. SAT math questions that use the word average refer to the arithmetic mean.
For example, the arithmetic mean of 4, 5 and 9 will be (4 + 5 + 9)/3 = 6.
Example of SAT question using the arithmetic mean: The average of the integers 24, 6, 12, x and y is 11. What is the value of the sum x + y?
(a) 11
(b) 17
(c) 13
(d) 15
Answer: The average of the 5 numbers is (24 + 6 + 12 + x + y)/5. This should be equal to 11 according to the problem, so
24 + 6 + 12 + x + y = 11 · 5 = 55
x + y = 55 - 24 - 6 - 12 = 13
The median of a set of numbers is the number in the middle when the numbers are ordered. For example, the median of (1, 5, 2, 7, and 10) is 5 because when the numbers are ordered, 5 is in the middle.
When there is an even number of values, the median is the same as the mean of the two middle numbers.
The mode of a set of numbers is the number that occurs most often in the list. For example, 75 is the mode of 24, 75, 55, 84, 75, and 12. A list of numbers can have more than one mode, for example (22, 24, 22, 25, 24, 27, 27, 29) has 3 modes, 22, 24 and 27.
Example of SAT question using statistics: A parameter is measured during a scientific experiment and the results are -12, 4, -8, 10, 2, 2, 0, -4 and 2. The median and the average of these numbers are:
(a) median = 0, average = -4/9
(b) median = 0, average = 4/9
(c) median = 2, average = -4/9
(d) median = 2, average = 4/9
(e) median = 2, average = 2/9
Answer: You may arrange the numbers in order: -12, -8, -4, 0, 2, 2, 2, 4, and 10. The median will be number in the middle of the row, that is 2
The average will be (-12-8-4+0+2+2+2+4+10)/9 = -4/9. The correct answer is (c)
Probability is the likelihood or chance that something is the case or will happen. A probability may be expressed as either a decimal or a fraction.
If the probability is 0, the event can never occur and if the probability is 1 the event is certain to occur. If the probability is between 0 and 1, that number expresses the likelihood of occurrence.
Example of SAT exercise with probability: If a is an integer chosen randomly from the set {3, 4, 5, 9} and b is an integer chosen randomly from the set {3, 8, 12, 15}, what is the probability that a/b is an integer?
(a) .25
(b) .2
(c) .125
(d) .375
(e) .4
Answer: We have 4 possible integers for a and 4 for b, so the number of possible combinations for (a / b) is 16
a/b is an integer only for 2 combinations:
1. a = 3 and b = 3
1. a = 9 and b = 3
The probability that a/b is an integer is 2/16 = 1/8 = .125. (c) is the correct answer.