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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:
$\text{MAD}=\frac{\sum |{x}_{i}-\overline{x}|}{n}$

### 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.
$\text{mean}=\frac{96}{6}=16$
The mean is $16$.
Step 2: Calculate the distance between each data point and the mean.
Data pointDistance from mean
$10$$|10-16|=6$
$15$$|15-16|=1$
$15$$|15-16|=1$
$17$$|17-16|=1$
$18$$|18-16|=2$
$21$$|21-16|=5$
Step 3: Add the distances together.
$6+1+1+1+2+5=16$
Step 4: Divide the sum by the number of data points.
$\text{MAD}=\frac{16}{6}\approx 2.67$ likes
On average, each picture was about $3$ likes away from the mean.

### Practice problem

The following table shows the number of lemons that grew on Mary's lemon tree each season.
SeasonNumber of lemons
Winter$3$
Spring$15$
Summer$21$
Fall$13$
Find the mean absolute deviation (MAD) of the data set.
lemons

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?
• 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.
• 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?
• 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.
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?
• what is the difference between mean absolute deviation and variance?
• Mean absolute deviation is the average of the deviations from the mean. Variance is the average of the squared deviations from the mean.
• How can you tell if a data set is more widespread or clustered based off of the MAD?
• 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

~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!😊
• Hey um I had this question in class and I had no idea how to do it: For which class would mean be a better indicator of a test score in the class

Class A, Mean: 85.2%, MAD: 14
Class B, Mean: 85.2%, MAD: 2

pls help
(1 vote)
• Class B, because the data points are closer together.
• i used to think MAD was the same as the mean... i was probably getting many questions incorrect haha
• what does the like E shaped symbol means in the above rule?
• ∑ is the Greek capital letter sigma, and represents a sum.

Sal explains the sigma notation here: