# Arithmetic Mean (Average) - GMAT Math Study Guide

## Definitions

• Mean (aka Arithmetic Mean, Average) - The sum of all of the numbers in a list divided by the number of items in that list.
For example, the mean of the numbers 2, 3, 7 is 4 since 2+3+7 = 12 and 12 divided by 3 [there are three numbers] is 4.

## Average Formula

In working with an average, there is one central formula that is used to answer questions pertaining to an average. This formula can be manipulated in many different ways, enabling test writers to create different iterations on mean problems.

The following is the formal mathematical formula for the arithmetic mean (a fancy name for the average).

A = average (or arithmetic mean)
n = the number of terms (e.g., the number of items or numbers being averaged)
x1 = the value of each individual item in the list of numbers being averaged

The following is the formula for the arithmetic mean, stated in a more readable and understandable form.

A = average (or arithmetic mean)
N = the number of terms (e.g., the number of items or numbers being averaged)
S = the sum of the numbers in the set of interest (e.g., the sum of the numbers being averaged)

One common trap that some students fall into is they automatically divide by 2. However, dividing the sum of numbers by 2 is only correct when there are two terms. When there are more than two terms that are being averaged, dividing by two will give the wrong answer.

### Basic Examples

If a teacher tutored five students and they subsequently scored 96, 94, 92, 87, and 81, what was the average score of the students whom the teacher tutored?
N = 5 since there are 5 students
S = 96 + 94 + 92 + 87 + 81 = 450

Another Example:

If a baseball pitcher throws three straight strikes to the first batter, two strikes to the second batter, one strike to the third batter, and zero strikes to the fourth batter, what is the average number of strikes the pitcher threw to each of the four batters?
N = 4 since there are four batters
S = 3 + 2 + 1 + 0 = 6

### More Complex Examples

A well-respected three point shooter in basketball is shooting 50% from three-point territory (meaning he makes 50% of his three-point shots). If he has attempted 60 three point shots thus far this season, what would his three point percentage be if he made three-fourths of the 12 shots he will attempt during the coming game?
N = 60 + 12 = 72
S = 50%(60) + (75%)(12) = 30 + 9 = 39

A = 39/72

Another Example:

A PhD student in English finished six books during the past week, increasing his average number of books read per week by 1. Assuming that the PhD's new average number of books read per week is 4, what is the total number of books the PhD student has read (including this past week)?

Let Anew = New Average Number of Books per Week = 4
Let Aold = Old Average Number of Books per Week = 4 - 1 = 3
Let Sold = total number of books read before this past week
Let Snew = total number of books read including this past week = Sold + 6 [since the student read six books this week]
Let Nold = total number of weeks elapsed before this past week
Let Nnew = total number of weeks elapsed including this past week = Nold + 1 [since 1 week has elapsed]

The question asks for Snew, which equals Sold + 6

We can write two equations and then solve for Sold:

Isolate Nold

Plug in and solve for Sold:

With Sold = 6, Snew = 6 + 6 [i.e., six new books this past week] = 12 total books

As a check, plug in Sold and solve for Nold:

As a further check: