## Averages

**This section looks at Averages. Covering Mean, Median, Mode and Range.**

**The Median Value**

The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6:

1, 2, 4, 6, 6, 7, 8 (6 is the middle value when the numbers are in order)

If you have n numbers in a group, the median is the (n + 1)/2 th value. For example, there are 7 numbers in the example above, so replace n by 7 and the median is the (7 + 1)/2 th value = 4th value. The 4th value is 6.

On a histogram, the median value occurs where the whole histogram is divided into two equal parts. An estimate of the median can be found using algebraic methods. However, an easier method would be to use the data to draw a cumulative frequency polygon and estimate the median using that.

**Mean**

There are four types of average: mean, mode, median and range. The mean is what most people mean when they say "average". It is found by adding up all of the numbers you have to find the mean of, and dividing by the number of numbers.

So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .

When you are given data which has been grouped, the mean is Sfx / Sf , where f is the frequency and x is the midpoint of the group (S means "the sum of").

**Example**

Work out an estimate for the mean height.

Height (cm) | Number of People (f) | Midpoint (x) | fx (f multiplied by x) |

101-120 | 1 | 110.5 | 110.5 |

121-130 | 3 | 125.5 | 376.5 |

131-140 | 5 | 135.5 | 677.5 |

141-150 | 7 | 145.5 | 1018.5 |

151-160 | 4 | 155.5 | 622 |

161-170 | 2 | 165.5 | 331 |

171-190 | 1 | 180.5 | 180.5 |

Sfx = 3316.5

Sf = 23

mean = 3316.5/23 = __144cm__ (3s.f.)

**Mode**

The mode is the number in a set of numbers which occurs the most. So the modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5, because there are more 5s than any other number.

On a histogram, the modal class is the class with the largest frequency density.

**Range**

The range is the largest number in a set minus the smallest number. So the range of 5, 7, 9 and 14 is (14 - 5) = 9.