# Arithmetic Mean (Average) - GMAT Math Study Guide

## Table of Contents

## 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)

x

_{1}= 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

N = 5 since there are 5 students

S = 96 + 94 + 92 + 87 + 81 = 450

Another Example:

N = 4 since there are four batters

S = 3 + 2 + 1 + 0 = 6

### More Complex Examples

N = 60 + 12 = 72

S = 50%(60) + (75%)(12) = 30 + 9 = 39

A = 39/72

Another Example:

Let A

_{new}= New Average Number of Books per Week = 4

Let A

_{old}= Old Average Number of Books per Week = 4 - 1 = 3

Let S

_{old}= total number of books read before this past week

Let S

_{new}= total number of books read including this past week = S

_{old}+ 6 [since the student read six books this week]

Let N

_{old}= total number of weeks elapsed before this past week

Let N

_{new}= total number of weeks elapsed including this past week = N

_{old}+ 1 [since 1 week has elapsed]

The question asks for S

_{new}, which equals S

_{old}+ 6

We can write two equations and then solve for S

_{old}:

Isolate N

_{old}

Plug in and solve for S

_{old}:

With S

_{old}= 6, S

_{new}= 6 + 6 [i.e., six new books this past week] = 12 total books

As a check, plug in S

_{old}and solve for N

_{old}:

As a further check: