# Mean Deviation

How far, on average, all values are from the middle.

## Calculating It

Find the mean of all values ... use it to work out distances ... then find the mean of those distances!

In three steps:

- 1. Find the mean of all values
- 2. Find the
**distance**of each value from that mean (subtract the mean from each value, ignore minus signs) - 3. Then find the
**mean of those distances**

Like this:

### Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the **mean**:

Mean = \frac{3 + 6 + 6 + 7 + 8 + 11 + 15 + 16}{8} = \frac{72}{8} = 9

Step 2: Find the **distance** of each value from that mean:

Value | Distance from 9 |
---|---|

3 | 6 |

6 | 3 |

6 | 3 |

7 | 2 |

8 | 1 |

11 | 2 |

15 | 6 |

16 | 7 |

Which looks like this:

*(No minus signs!)*

Step 3. Find the **mean of those distances**:

**Mean Deviation** = \frac{6 + 3 + 3 + 2 + 1 + 2 + 6 + 7}{8} = \frac{30}{8} = **3.75**

So, the **mean = 9**, and the **mean deviation = 3.75**

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For **deviation** just think **distance**

## Formula

The formula is:

Mean Deviation = \frac{Σ|x − μ|}{N}

**Σ**is Sigma, which means to sum up- || (the vertical bars) mean Absolute Value, basically to ignore minus signs
**x**is each value (such as 3 or 16)**μ**is the mean (in our example**μ = 9**)**N**is the number of values (in our example**N = 8**)

Let's look at those in more detail:

## Absolute Deviation

Each distance we calculate is called an **Absolute Deviation**, because it is the Absolute Value of the deviation (how far from the mean).

To show "Absolute Value" we put "|" marks either side like this:

**|−3| = 3**

For any value **x**:

Absolute Deviation = |x − μ|

From our example, the value **16** has:

Absolute Deviation = |x − μ| = |16 − 9| = |7| = 7

And now let's add them all up ...

## Sigma

The symbol for "Sum Up" is **Σ** (called Sigma Notation), so we have:

Sum of Absolute Deviations = Σ|x − μ|

Divide by how many values **N** and we have:

Mean Deviation = \frac{Σ|x − μ|}{N}

Let's do our example again, using the proper symbols:

### Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the **mean**:

**μ** = \frac{3 + 6 + 6 + 7 + 8 + 11 + 15 + 16}{8} = \frac{72}{8} = 9

Step 2: Find the **Absolute Deviations**:

x | |x − μ| |
---|---|

3 | 6 |

6 | 3 |

6 | 3 |

7 | 2 |

8 | 1 |

11 | 2 |

15 | 6 |

16 | 7 |

Σ|x − μ| = 30 |

Step 3. Find the **Mean Deviation**:

Mean Deviation = \frac{Σ|x − μ|}{N} = \frac{30}{8} = 3.75

*Note: the mean deviation is sometimes called the Mean Absolute Deviation (MAD) because it is the mean of the absolute deviations.*

## What Does It "Mean" ?

Mean Deviation tells us how far, on average, all values are from the middle.

Here is an example (using the same **data** as on the Standard Deviation page):

### Example: You and your friends have just measured the heights of your dogs (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Step 1: Find the **mean**:

**μ** = \frac{600 + 470 + 170 + 430 + 300}{5} = \frac{1970}{5} = **394**

Step 2: Find the **Absolute Deviations**:

x | |x - μ| |
---|---|

600 | 206 |

470 | 76 |

170 | 224 |

430 | 36 |

300 | 94 |

Σ|x − μ| = 636 |

Step 3. Find the **Mean Deviation**:

Mean Deviation = \frac{Σ|x − μ|}{N} = \frac{636}{5} = 127.2

So, on average, the dogs' heights are **127.2 mm from the mean**.

(Compare that with the Standard Deviation of **147 mm**)

## A Useful Check

The deviations on **one side** of the mean should equal the deviations on the **other side**.

From our first example:

### Example: 3, 6, 6, 7, 8, 11, 15, 16

The deviations are:

6 + 3 + 3 + 2 + 1 | = | 2 + 6 + 7 |

15 | = | 15 |

Likewise:

### Example: Dogs

Deviations left of mean: 224 + 94 = **318**

Deviations right of mean: 206 + 76 + 36 = **318**

If they are not equal ... you may have made a msitake!