# GRE Data Analysis - Averages

The **average**, or the arithmetic mean, is defined as the sum of all measurements divided by the number of observations. For example, the average of the numbers 10, 12, and 22 is 14.6, because 10+12+22=44, and there are three 'observations,' so 44 divided by three is 14.6. When there are only numbers involved, an average problem is quite simple. Don't be too surprised to see a question on the math section of the GRE exam that tests your understanding of the concept **mean**.

Related to the notion of average are the following two other concepts that you may need to use on the GRE: **mode** and **median**. The mode of a set of numbers is the number that appears most often, while the median of a set of numbers is the number that is exactly half way between the largest and smallest number, when the numbers are arranged from smallest to largest. For example:

Original question | What is the mode, mean, and median for 2, 3, 4, 5, 6, 6, and 10? |

Mean: | (2+3+4+5+6+6+10) / 7 = 36/7 = 5.14 |

Median: | The median is 5, because it is the number "in the middle" when the numbers are arranged in order from smallest to largest. |

Mode: | The mode is 6 because it appears more than any other number. |

When the count of numbers is even, then there is no "middle" number, so the median is the average of the two middle numbers. For example, if you are given four numbers: 1, 5, 8, 3, then arranged in order they are 1, 3, 5 and 8, and the "middle" numbers are 3 and 5, so the *median* is 3+5 = 8 ÷ 2 = 4.

Another example with slightly more complicated median and mode values:

Original question | What is the mode, mean, and median for 4, 2, 2, 5, 7, and 5? |

Mean: | (4+2+2+5+7+5) / 6 = 25/6 = 4.16 |

Median: | First, in order to compute the median, rearrange the numbers from lowest to highest to get: 2, 2, 4, 5, 5, and 7. Notice that we have an even number of digits, so there is no "middle" number. The concept of median implies that there are an equal number of digits that are greater and smaller than the median number. In this case, the median number is (4+5)/2 = 4.5 |

Mode: | Here, both numbers 2 and 5 appear twice, so the mode is NOT unique. Both 2 and 5 are modes of this series. |