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## Statistics and probability

### Course: Statistics and probability > Unit 3

Lesson 8: Other measures of spread# Mean absolute deviation (MAD) review

## Mean absolute deviation

The mean absolute deviation of a dataset is the average distance between each data point and the mean. It gives us an idea about the variability in a dataset.

Here's how to calculate the mean absolute deviation.

**Step 1:**Calculate the mean.

**Step 2:**Calculate how far away each data point is from the mean using positive distances. These are called absolute deviations.

**Step 3:**Add those deviations together.

**Step 4:**Divide the sum by the number of data points.

Following these steps in the example below is probably the best way to learn about mean absolute deviation, but here is a more formal way to write the steps in a formula:

### Example

Erica enjoys posting pictures of her cat online. Here's how many "likes" the past 6 pictures each received:

10, 15, 15, 17, 18, 21

**Find the mean absolute deviation.**

**Step 1:**Calculate the mean.

The sum of the data is 96 total "likes" and there are 6 pictures.

The mean is 16.

**Step 2:**Calculate the distance between each data point and the mean.

Data point | Distance from mean |
---|---|

10 | open vertical bar, 10, minus, 16, close vertical bar, equals, 6 |

15 | open vertical bar, 15, minus, 16, close vertical bar, equals, 1 |

15 | open vertical bar, 15, minus, 16, close vertical bar, equals, 1 |

17 | open vertical bar, 17, minus, 16, close vertical bar, equals, 1 |

18 | open vertical bar, 18, minus, 16, close vertical bar, equals, 2 |

21 | open vertical bar, 21, minus, 16, close vertical bar, equals, 5 |

**Step 3:**Add the distances together.

**Step 4:**Divide the sum by the number of data points.

start text, M, A, D, end text, equals, start fraction, 16, divided by, 6, end fraction, approximately equals, 2, point, 67 likes

On average, each picture was about 3 likes away from the mean.

*Want to learn more about mean absolute deviation? Check out this video*.

### Practice problem

*Want to practice more problems like these? Check out this exercise*.

## Want to join the conversation?

- I've learned how to find out the answers of variances, deviations, MADs. But, I don't understand what are these answers "saying", they're meaningless to me. Can anyone tell me what are these answers about?(8 votes)
- Range, MAD, variance, and standard deviation are all measures of spread. They tell you how spread out the data are. Data that are very similar will have a small spread, whereas data that are wildly different from each other will have a large spread.

Range and MAD are very basic measures. Since the variance takes the square of each deviation, large deviations (>1) will cause the variance to become very large indeed.(11 votes)

- Why is the MAD a part of so many everyday activities (Grocery store sales, average daily likes for a clip, etc.), but it isn't actually used everyday?(3 votes)
- It is so you can relate to what happens and aren't drowning in aerospace technicalities while learning statistics. While you would not actually calculate the MAD for fun, it is just so you have a bit more interest, as just having a set like Data Set A: [#, #, #, #]

would make people even more bored than how often the likes on a cat video differ.(6 votes)

- i like suger cookies(5 votes)
**What is Sample MAD**

Should we use the concept of*dividing by n-1*for calculating the*Mean Absolute Deviation of a Sample Data too?*Why in the formula above in this article divides the sum of differences of individual data points from the sample mean*by n*,*not n-1?*(4 votes)- In real life, what's the actual purpose of finding the MAD? What would you use it for? I don't get how the mean is related to the data set. Is it the average?(3 votes)
- The median is the middle point of the data set, the mean is the arithmetic average of all points (sum of all points divided by number of points)(3 votes)

- How can you tell if a data set is more widespread or clustered based off of the MAD?(3 votes)
- i used to think MAD was the same as the mean... i was probably getting many questions incorrect haha(2 votes)
- see? if you practice and practice, your going to get better and better, even fortnite or blox fruits!🎡🧨(2 votes)

- what does the like E shaped symbol means in the above rule?(2 votes)
- ∑ is the Greek capital letter sigma, and represents a sum.

Sal explains the sigma notation here:

https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/v/sigma-notation-sum

Notice that in the video, Sal says we need to define an index.

In this case the index isn't explicitly stated, instead it's to be understood that we sum over 𝑖 as it goes from 1 through 𝑛,

and if we were to write the sum out it would look like this:

MAD = (|𝑥₁ − 𝑥̅| + |𝑥₂ − 𝑥̅| + |𝑥₃ − 𝑥̅| + ... + |𝑥_𝑛 − 𝑥̅|)∕𝑛(2 votes)

- Ok, I'm a pretty fast learner and I even answer questions, but what is the formula in plain English? I always write formulas on sticky notes so I understand and remember them but I can't find a way to simplify the formula! HALP

Thank you in advance

~Green Bear(1 vote)- Hi!

Here's the written-out version of the formula:

(|x1 - mean| + |x2 - mean| + ... + |xn - mean|) / n

In words, MAD is the mean of the absolute values of (each data point minus the original mean).

Hope this clears things up!😊(4 votes)

- what sort of situations could we use mad in?(2 votes)
- The mean absolute deviation (MAD) is the mean (average) distance between each data value and the mean of the data set. It can be used to quantify the spread in the data set and also be helpful in answering statistical questions in the real world.

Many professionals use mean in their everyday lives. Teachers give tests to students and then average the results to see if the average score was high, in between, or too low. Each average tells a story. Absolute deviation can further help to see the distance between each of the scores and the beginning average scores, since A small mean absolute deviation tells us that most of the data values are very close to the mean (since the expected distance from each data value to the mean is small).

Got from Google. I am no expert.

But I hope this helps!(1 vote)