## Statistics and Probability Review

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.