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# Measure of central tendency

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## Measure of central tendency

Don't be put off, **"measure of central tendency"** is just a mathematical and rather posh way of saying **"averages"**.

**The mode is the most popular value or values.**

It is the piece or pieces of data that occur most often.

**Example:**

**You are given the following pieces of data showing the number of songs on 7 CD's:**

**The median is the middle piece of data when the data is in numerical order.**

**Example:**

Using **7** pieces of data:

**In this case, 25 is the median as it is clearly the value in the middle.**

**However, if there is an even number of values, there will not be a single value in the middle:**

**Example:**

With 101 pieces of data, **odd**, we must find just over halfway. In this case, the 51^{st} value. This value will be the **median**.

With 50 pieces of data, **even**, we must find halfway and the next value. In this case, the 25^{th} and 26^{th} values. The **median** will be halfway between these values.

**These rules are especially important with large groups of data.**

**Example:**

This example uses a frequency table showing the rainfall in mm over a period of 2 summer months:

Rainfall (mm): |
0 | 1 | 2 | 3 | 4 | 5 |

Frequency (f): |
4 | 12 | 15 | 16 | 9 | 4 |

The table shows 60 pieces of data ranging from 0 to 5. As there is an even number the median is found by finding the value halfway between the 30^{th} and 31^{st} pieces of data. In this case, both the 30^{th} and 31^{st} pieces of data are 2's.

**Hence the median is 2mm.**

This is the most widely used of all the averages.

The **mean** of a set of data is the sum of all the values divided by the number of values.

The mean is denoted by x with an overbar, and for n pieces of data, it is calculated by:

Don't be worried, ∑x just means the sum of all the x's - for instance, add all the bits of data together.

**Example: **

**Now let's have a look at a couple of trickier examples when we are given larger sets of data in frequency tables:**

**Example:**

Again, the following example uses a frequency table showing the rainfall in mm over a period of 2 summer months:

Rainfall (mm): |
0 | 1 | 2 | 3 | 4 | 5 |

Frequency (f): |
4 | 12 | 15 | 16 | 9 | 4 |

**This gives a mean rainfall of 2.4mm**

**Note:** 2 × 15 as there are 15 days with 2mm of rainfall, so 30mm altogether etc...

When dealing with even larger sets of data, we may be given the data in terms of a **grouped frequency table.**

**Example:**

In this example, the data given is the height (x) of 100 Premiership footballers:

Height (cm): | Frequency (f): |
---|---|

165 ≤ x < 170 | 8 |

170 ≤ x < 175 | 18 |

175 ≤ x < 180 | 34 |

180 ≤ x < 185 | 23 |

185 ≤ x < 190 | 17 |

To find the exact mean of this data is impossible as we don't know the exact data!

We can however, find an **estimate** of the mean by assuming each footballer is the height halfway within his interval - for instance, assume the 18 footballers in the interval are all 172.5cm tall! (Just like the median the quickest and easiest way to find halfway is to add up the 2 values and halve.)

**Lets look what that does to our table:**

Height (cm): | Frequency (f): | x × f |
---|---|---|

167.5 | 8 | 1340 |

172.5 | 18 | 3105 |

177.5 | 34 | 6035 |

182.5 | 23 | 4197.5 |

187.5 | 17 | 3187.5 |

Total: |
100 |
17865 |

**Hence the mean height of the 100 footballers is 178.65cm**

**Note the addition of the last column:** This is just the mid-heights multiplied by the frequencies and is an efficient way of showing your workings.