what does mean absolute deviation mean (MAD) is a statistical measure that tells us how far, on average, the values in a data set are from the mean.
In simple words, it answers this question:
“On average, how far are the data values from the average?”
A small mean absolute deviation means the values are generally close to the mean. A larger MAD means the values are more spread out.
Mean absolute deviation is also called mean deviation or average absolute deviation. It is calculated by finding the distance of every value from the mean, ignoring negative signs, adding those distances, and dividing by the number of values.
Table of Contents
Quick Answer: What Is Mean Absolute Deviation?
Mean absolute deviation is the average of the absolute distances between each data value and the mean.
The basic formula is:
MAD=∑∣xi−xˉ∣nMAD=\frac{\sum |x_i-\bar{x}|}{n}
Where:
- MAD = Mean Absolute Deviation
- xix_i = Each value in the data set
- xˉ\bar{x} = Mean of the data
- ∣xi−xˉ∣|x_i-\bar{x}| = Absolute deviation from the mean
- nn = Number of values
The absolute value is important because distances cannot be negative. Without taking absolute values, positive and negative deviations from the mean would cancel each other out.
What Does “Absolute” Mean in Mean Absolute Deviation?
The word absolute means that we ignore the negative sign.
For example:
∣−5∣=5|-5|=5
and
∣5∣=5|5|=5
Both represent a distance of 5.
Suppose the mean is 10.
If a value is 7:
7−10=−37-10=-3
Its absolute deviation is:
∣−3∣=3|-3|=3
If another value is 13:
13−10=313-10=3
Its absolute deviation is also:
∣3∣=3|3|=3
Both values are therefore 3 units away from the mean.
Why Do We Need What Does Mean Absolute Deviation Mean?
The mean tells us the center of a data set, but it doesn’t tell us how spread out the values are.
Consider two data sets:
Data Set A:
9,10,10,119,10,10,11
Data Set B:
2,6,14,182,6,14,18
Both may have similar centers, but their values are spread out differently.
A measure such as MAD helps describe that spread.
A small MAD means:
Values tend to be close to the mean.
A large MAD means:
Values tend to be farther away from the mean.
This makes MAD a useful and intuitive measure of variability.
What Does Mean Absolute Deviation Mean Formula
The standard formula for mean absolute deviation around the mean is:
MAD=∑∣xi−xˉ∣n\boxed{MAD=\frac{\sum |x_i-\bar{x}|}{n}}
Let’s understand each part.
xix_i
This represents each individual value in the data set.
xˉ\bar{x}
This represents the arithmetic mean.
xi−xˉx_i-\bar{x}
This calculates how far a value is from the mean, including its direction.
∣xi−xˉ∣|x_i-\bar{x}|
The absolute-value signs remove the negative sign and give the distance from the mean.
∑\sum
The sigma symbol means to add all the absolute deviations.
nn
This is the total number of observations.
So the process is:
Find mean → Find each distance → Make distances positive → Add → Divide by number of values.
Curious about another abbreviation? See what does OTD mean. 🔎 What Does OTD Mean in Text? Slang, TikTok, Snapchat & Instagram
How to Calculate Mean Absolute Deviation
There are four basic steps.
Step 1: Find the Mean
Add all the values and divide by the number of values.
Mean=Sum of valuesNumber of valuesMean=\frac{\text{Sum of values}}{\text{Number of values}}
Step 2: Find Each Deviation
Subtract the mean from every value.
xi−xˉx_i-\bar{x}
Step 3: Find the Absolute Deviations
Ignore the negative signs.
∣xi−xˉ∣|x_i-\bar{x}|
Step 4: Find the Average
Add all the absolute deviations and divide by the number of observations.
MAD=Sum of absolute deviationsnMAD=\frac{\text{Sum of absolute deviations}}{n}
These steps are also the standard procedure described by Khan Academy.
Mean Absolute Deviation Example
Let’s calculate MAD for:
4,7,9,12,134,7,9,12,13
Step 1: Find the Mean
Add the values:
4+7+9+12+13=454+7+9+12+13=45
There are 5 values.
Therefore:
Mean=455=9Mean=\frac{45}{5}=9
The mean is 9.
Step 2: Find Each Deviation
| Value | Mean | Deviation |
| 4 | 9 | -5 |
| 7 | 9 | -2 |
| 9 | 9 | 0 |
| 12 | 9 | 3 |
| 13 | 9 | 4 |
Step 3: Find the Absolute Deviations
Now remove the negative signs:
| Value | Deviation | Absolute Deviation |
| 4 | -5 | 5 |
| 7 | -2 | 2 |
| 9 | 0 | 0 |
| 12 | 3 | 3 |
| 13 | 4 | 4 |
Add them:
5+2+0+3+4=145+2+0+3+4=1
Step 4: Divide by the Number of Values
There are 5 values:
MAD=145MAD=\frac{14}{5} MAD=2.8\boxed{MAD=2.8}
Therefore, the mean absolute deviation is 2.8.
Interpretation
A MAD of 2.8 means that a typical value in this data set is about 2.8 units away from the mean of 9.
This interpretation—average distance from the mean—is the key idea behind MAD.
Another Easy Example
Consider:
2,4,6,82,4,6,8
Step 1: Find the mean
Mean=2+4+6+84Mean=\frac{2+4+6+8}{4} Mean=204=5Mean=\frac{20}{4}=5
Step 2: Find absolute deviations
| Value | Distance from 5 |
| 2 | 3 |
| 4 | 1 |
| 6 | 1 |
| 8 | 3 |
Step 3: Add the distances
3+1+1+3=83+1+1+3=8
Step 4: Divide by 4
MAD=84=2MAD=\frac{8}{4}=2
Therefore:
MAD=2\boxed{MAD=2}
So the values are, on average, 2 units away from the mean.
Mean Absolute Deviation With a Larger Data Set
Consider these test scores:
10,12,13,15,18,20,2210,12,13,15,18,20,22
Find the mean
10+12+13+15+18+20+22=11010+12+13+15+18+20+22=110
There are 7 values.
Mean=1107≈15.71Mean=\frac{110}{7}\approx15.71
Now calculate the absolute distances from 15.71.
| Score | Absolute Deviation |
| 10 | 5.71 |
| 12 | 3.71 |
| 13 | 2.71 |
| 15 | 0.71 |
| 18 | 2.29 |
| 20 | 4.29 |
| 22 | 6.29 |
The sum is approximately:
25.7125.71
Therefore:
MAD=25.717≈3.67MAD=\frac{25.71}{7}\approx3.67
So:
MAD≈3.67\boxed{MAD\approx3.67}
The scores are approximately 3.67 points away from the mean on average.
Why Don’t We Keep Negative Deviations?
This is one of the most important ideas to understand.
Suppose the mean is 10 and the values are:
7,137,13
Their deviations are:
7−10=−37-10=-3
and
13−10=313-10=3
If we simply average these deviations:
−3+32=0\frac{-3+3}{2}=0
That would incorrectly suggest there is no variation.
But clearly, both values are 3 units away from the mean.
That’s why we use absolute values:
∣−3∣=3|-3|=3 ∣3∣=3|3|=3
Then:
MAD=3+32=3MAD=\frac{3+3}{2}=3
The absolute value prevents positive and negative deviations from canceling each other.
Mean Absolute Deviation vs. Standard Deviation
MAD and standard deviation are both measures of variability, but they calculate spread differently.
Mean Absolute Deviation
MAD uses absolute deviations:
MAD=∑∣xi−xˉ∣nMAD=\frac{\sum |x_i-\bar{x}|}{n}
Standard Deviation
Standard deviation uses squared deviations and then takes a square root.
For a population:
σ=∑(xi−μ)2N\sigma=\sqrt{\frac{\sum(x_i-\mu)^2}{N}}
The biggest conceptual difference is:
MAD uses absolute distances.
Standard deviation uses squared distances.
Because squaring gives relatively large deviations more weight, standard deviation can be more strongly affected by extreme values.
MAD vs. Standard Deviation Example
Consider:
2,4,6,82,4,6,8
We already found:
MAD=2MAD=2
The population standard deviation is approximately:
2.242.24
The sample standard deviation is approximately:
2.582.58
So the values are different even though both measures describe spread.
| Measure | Result |
| Mean | 5 |
| MAD | 2 |
| Population SD | 2.24 |
| Sample SD | 2.58 |
MAD is generally easier to explain because it directly represents an average distance from the mean.
Mean Absolute Deviation vs. Range
The range is:
Maximum−Minimum Maximum-Minimum
For:
4,7,9,12,134,7,9,12,13
the range is:
13−4=913-4=9
But the MAD is:
2.82.8
The two measures answer different questions.
Range
“How far apart are the smallest and largest values?”
MAD
“How far from the mean are the values on average?”
The range is very simple, but it only uses the smallest and largest observations. MAD considers every value in the data set.
Mean Absolute Deviation vs. Variance
Variance calculates the average of squared deviations from the mean.
MAD calculates the average of absolute deviations.
MAD
∑∣xi−xˉ∣n\frac{\sum |x_i-\bar{x}|}{n}
Population variance
∑(xi−μ)2N\frac{\sum(x_i-\mu)^2}{N}
Because variance squares the deviations, its units are squared.
For example, if your data is measured in meters:
- MAD is measured in meters.
- Standard deviation is measured in meters.
- Variance is measured in square meters.
This is one reason MAD can be easier to interpret.
Is Mean Absolute Deviation Always About the Mean?
Not necessarily in the broad mathematical sense.
“Mean absolute deviation” can sometimes be used more generally for the average absolute distance from a chosen central point, such as the median or mode. However, in many introductory statistics courses, MAD specifically means the mean of the absolute deviations from the arithmetic mean.
Therefore, always check the definition being used in your textbook or course.
Mean Absolute Deviation From the Median
A related calculation uses the median instead of the mean.
Suppose:
2,2,3,4,142,2,3,4,14
The median is:
33
The absolute distances from the median are:
1,1,0,1,111,1,0,1,11
Their average is:
145=2.8\frac{14}{5}=2.8
This is an absolute deviation around the median, not necessarily what an introductory problem means when it simply says “mean absolute deviation.”
This distinction matters because the center you choose changes the result.
MAD and the Median Absolute Deviation Are Not the Same
This is a common source of confusion.
The abbreviation MAD is sometimes also used for Median Absolute Deviation.
These are different statistics.
Mean Absolute Deviation
Usually:
1n∑∣xi−xˉ∣\frac{1}{n}\sum|x_i-\bar{x}|
It uses the mean as the center.
Median Absolute Deviation
Typically:
- Find the median.
- Find each value’s absolute distance from the median.
- Find the median of those distances.
The median-based measure is especially useful when robustness against extreme values is important.
So always check whether your course or software means absolute deviation or median absolute deviation.
What Does a Small MAD Mean?
A small MAD means the data values are generally close to the mean.
For example:
9,10,10,119,10,10,11
has relatively little spread.
The values cluster around their mean.
Therefore, its MAD is relatively small.
Interpretation
The observations tend to stay close to the average.
What Does a Large MAD Mean?
A large MAD means the values tend to be farther from the mean.
For example:
1,5,15,201,5,15,20
has much greater spread than:
9,10,10,119,10,10,11
Therefore, its MAD will also be larger.
Interpretation
The observations are more spread out around the average.
Does MAD Have Units?
Yes.
MAD has the same units as the original data.
For example:
Test scores
MAD = 2.5 points
Temperature
MAD = 3°C
Distance
MAD = 4 kilometers
Time
MAD = 2 minutes
This makes MAD relatively straightforward to interpret because it remains in the original measurement units.
Why Is Mean Absolute Deviation Useful?
MAD is useful because it gives a simple description of variability.
It can help answer questions such as:
- How consistent are test scores?
- How much do daily temperatures vary?
- How much do measurements differ from an average?
- How consistent are production measurements?
- How spread out are observations?
- How close are values to a typical average?
Its interpretation is particularly intuitive:
MAD tells you the average distance from the mean.
Advantages of Mean Absolute Deviation
1. Easy to Understand
The result directly represents an average distance.
2. Uses Every Observation
Unlike the range, MAD considers every data point.
3. Same Units as the Data
This makes the result easier to interpret.
4. Easier Calculation
It doesn’t require squaring and then taking a square root.
5. Useful for Learning Statistics
MAD provides a straightforward introduction to the concept of statistical variability. Khan Academy presents it as a direct way to measure how much values vary around their mean.
Limitations of Mean Absolute Deviation
MAD also has limitations.
1. It Is Less Common Than Standard Deviation
Standard deviation is much more widely used in many statistical procedures.
2. It Still Uses the Mean
If the data contains extreme values, the mean can be pulled toward those values.
3. Different Definitions Exist
“MAD” can sometimes refer to median absolute deviation, so terminology should be checked carefully.
4. Less Convenient for Some Advanced Statistical Methods
Many statistical formulas and models are built around squared deviations rather than absolute deviations.
How Does an Outlier Affect MAD?
An outlier is an unusually large or small value compared with the rest of the data.
Suppose we have:
10,11,12,13,1410,11,12,13,14
Now add:
100100
The mean changes considerably because of the extreme value.
Since MAD is calculated relative to the mean, MAD also changes.
MAD is therefore not completely resistant to outliers.
However, because MAD uses absolute deviations rather than squared deviations, extreme observations do not receive the additional squaring weight used by standard deviation.
Mean Absolute Deviation in Real Life
MAD can be useful in many situations.
Education
A teacher can use variability measures to understand how consistently students perform.
For example:
Average score = 75
MAD = 4 points
This suggests that scores are typically about 4 points from the mean.
Weather
Suppose the average daily temperature is 28°C and MAD is 2°C.
That means daily temperatures are typically about 2°C away from the average.
Manufacturing
A factory may measure the dimensions of products.
A low MAD indicates measurements are closely grouped around the average.
Business
A company can analyze variations in sales, delivery times, or customer wait times.
Science
Researchers can describe how much measurements vary around an average.
Mean Absolute Deviation in Frequency Tables
MAD can also be calculated when data is presented with frequencies.
Suppose:
| Value | Frequency |
| 2 | 2 |
| 4 | 3 |
| 6 | 1 |
The total number of observations is:
2+3+1=62+3+1=6
First calculate the mean using the frequencies:
xˉ=(2)(2)+(4)(3)+(6)(1)6\bar{x}=\frac{(2)(2)+(4)(3)+(6)(1)}{6} xˉ=4+12+66\bar{x}=\frac{4+12+6}{6} xˉ=226\bar{x}=\frac{22}{6} xˉ≈3.67\bar{x}\approx3.67
Then calculate each absolute deviation and account for its frequency.
The general idea remains the same:
MAD=∑fi∣xi−xˉ∣∑fiMAD=\frac{\sum f_i|x_i-\bar{x}|}{\sum f_i}
where fif_i is the frequency of each value.
A Simple Shortcut for Exam Problems
When solving a basic MAD question, use this checklist:
1. Find the mean.
2. Subtract the mean from every value.
3. Take absolute values.
4. Add the absolute deviations.
5. Divide by the number of observations.
Remember:
Mean → Difference → Absolute → Add → Divide
This sequence can help prevent calculation mistakes.
Common Mistakes When Finding MAD
Mistake 1: Forgetting Absolute Values
Incorrect:
−3+(−1)+1+3=0-3+(-1)+1+3=0
Correct:
3+1+1+3=83+1+1+3=8
Always use:
∣xi−xˉ∣|x_i-\bar{x}
Mistake 2: Finding the Median Instead of the Mean
If the question asks for mean absolute deviation from the mean, calculate the arithmetic mean first.
Mistake 3: Dividing by the Wrong Number
For the standard basic calculation:
MAD=sum of absolute deviationsnMAD=\frac{\text{sum of absolute deviations}}{n}
Use the number of observations.
Mistake 4: Squaring the Deviations
Squaring deviations is part of calculating variance and standard deviation.
MAD uses:
∣xi−xˉ∣|x_i-\bar{x}|
not:
(xi−xˉ)2(x_i-\bar{x})^
Mistake 5: Forgetting the Mean
You cannot calculate the usual mean absolute deviation from the mean without first finding the mean.
Mean Absolute Deviation Formula for Quick Revision
Remember this formula:
MAD=∑∣xi−xˉ∣n\boxed{MAD=\frac{\sum|x_i-\bar{x}|}{n}}
In words:
Mean Absolute Deviation = Sum of all absolute deviations from the mean ÷ Number of observations.
Mean Absolute Deviation Worked Example for Exams
Find the MAD of:
5,7,8,105,7,8,10
Step 1: Mean
xˉ=5+7+8+104\bar{x}=\frac{5+7+8+10}{4} xˉ=304=7.5\bar{x}=\frac{30}{4}=7.5
Step 2: Absolute deviations
∣5−7.5∣=2.5|5-7.5|=2.5 ∣7−7.5∣=0.5|7-7.5|=0.5 ∣8−7.5∣=0.5|8-7.5|=0.5 ∣10−7.5∣=2.5|10-7.5|=2.5
Step 3: Add
2.5+0.5+0.5+2.5=62.5+0.5+0.5+2.5=6
Step 4: Divide
MAD=64MAD=\frac{6}{4} MAD=1.5\boxed{MAD=1.5}
Answer:
The mean absolute deviation is 1.5.
Frequently Asked Questions About Mean Absolute Deviation
What does absolute deviation mean?
Mean absolute deviation is the average distance between each data value and the mean.
What is the formula for MAD?
The standard formula is MAD=∑∣xi−xˉ∣nMAD=\frac{\sum|x_i-\bar{x}|}{n}.
What does MAD tell us?
MAD tells us how far the observations are from the mean on average.
What does a high MAD mean?
A high MAD generally indicates that the data values are more spread out around the mean.
What does a low MAD mean?
A low MAD indicates that the values tend to be closer to the mean.
Why do we use absolute values?
Absolute values prevent positive and negative deviations from canceling each other out.
Is MAD the same as standard deviation?
No. MAD averages absolute deviations, while standard deviation is based on squared deviations.
Can MAD be zero?
Yes. MAD is zero when every value is exactly equal to the mean, which means all values are identical.
Is mean absolute deviation important in statistics?
Yes. It is a useful descriptive measure of variability and is especially helpful for understanding the idea of average distance from a central value.
Final Summary
Mean Absolute Deviation (MAD) is a measure of how spread out a data set is.
The key idea is simple:
MAD tells us the average distance between the data values and their mean.
To calculate it:
- Find the mean.
- Find each value’s difference from the mean.
- Take the absolute value of each difference.
- Add the absolute deviations.
- Divide by the number of values.
The formula is:
MAD=∑∣xi−xˉ∣n\boxed{MAD=\frac{\sum|x_i-\bar{x}|}{n}}
For example, for:
4,7,9,12,134,7,9,12,13
the mean is 9, the sum of absolute deviations is 14, and therefore:
MAD=145=2.8MAD=\frac{14}{5}=\boxed{2.8}
So the values are, on average, 2.8 units away from the mean.
The most important thing to remember is:
MAD = average distance from the mean.
Once you understand that idea, calculating mean absolute deviation becomes a straightforward process.
