MAD Calculator
The mean absolute deviation calculator finds the average distance between each value in your data and the mean. For the set 9, 13, 43, 55, the MAD is 19. Enter any numbers below to get the MAD, the mean, a full working table, and a step by step solution you can follow or check your homework against.
What is mean absolute deviation?
Mean absolute deviation, or MAD, is the average distance between each value in a data set and the mean. It tells you how spread out your numbers are in a single, easy figure. A small MAD means the values sit close together, and a large MAD means they are spread far apart.
The word absolute is the key. When you measure how far each value is from the mean, some gaps are above it and some are below. Normally those positives and negatives would cancel out. Taking the absolute value strips the minus signs, so every distance counts as positive and nothing cancels. That is what makes MAD a true measure of spread.
MAD shows up in middle school math as an introduction to variation, and again in statistics, finance, and forecasting. It is often taught before variance and standard deviation because it uses plain distances instead of squared ones, which makes it easier to picture.
Mean absolute deviation formula
The formula is:
MAD = Σ|xᵢ − x̄| ÷ n
Reading each symbol: xᵢ is each value in the set, x̄ is the mean, |xᵢ − x̄| is the absolute distance of a value from the mean, Σ means add all of those up, and n is how many values you have.
In plain words: subtract the mean from every value, drop the minus signs, add the results, and divide by the number of values. The calculator above runs exactly these steps and shows each one, but doing it by hand once makes the idea stick.
How to calculate mean absolute deviation step by step
Take the data set 9, 13, 43, 55 as an example, since it is one people look up often.
Step 1: Find the mean. Add the values and divide by how many there are. (9 + 13 + 43 + 55) ÷ 4 = 120 ÷ 4 = 30. The mean is 30.
Step 2: Find each distance from the mean. Subtract 30 from each value, then drop the sign. 9 is 21 away, 13 is 17 away, 43 is 13 away, and 55 is 25 away.
Step 3: Average those distances. Add them and divide by 4. (21 + 17 + 13 + 25) ÷ 4 = 76 ÷ 4 = 19.
So the mean absolute deviation of 9, 13, 43, 55 is 19. On average, each value sits 19 units away from the mean of 30. That is a wide spread, which fits, because the numbers range from 9 all the way to 55.
Worked examples
Three common data sets, fully worked, so you can match the method to your own numbers.
MAD of 9, 13, 43, and 55
Mean = (9 + 13 + 43 + 55) ÷ 4 = 30. Distances = 21, 17, 13, 25. MAD = 76 ÷ 4 = 19.
MAD of 1, 4, 6, 7, 7
Mean = (1 + 4 + 6 + 7 + 7) ÷ 5 = 5. Distances = 4, 1, 1, 2, 2. MAD = 10 ÷ 5 = 2. A small MAD, because these values are tightly grouped.
MAD of 16, 14, 2, 4, 18, 20, 12, 10
Mean = (16 + 14 + 2 + 4 + 18 + 20 + 12 + 10) ÷ 8 = 12. Distances = 4, 2, 10, 8, 6, 8, 0, 2. MAD = 40 ÷ 8 = 5.
Mean absolute deviation chart
Small sample sets and their MAD, to show how spread changes the value:
| Data set | Mean | MAD |
|---|---|---|
| 10, 12, 14 | 12 | 1.33 |
| 5, 6, 10 | 7 | 2 |
| 1, 4, 6, 7, 7 | 5 | 2 |
| 23, 45, 32, 34, 23 | 31.4 | 6.72 |
| 16, 14, 2, 4, 18, 20, 12, 10 | 12 | 5 |
| 9, 13, 43, 55 | 30 | 19 |
What does MAD tell you about a data set?
MAD answers one question: on average, how far do the values fall from the center? The bigger the number, the more spread out the data.
A MAD of zero means every value is identical, with no spread at all. A small MAD means the data is consistent and clustered near the mean. A large MAD means the values are scattered. This makes MAD useful for comparing two sets: if data set A has a MAD of 2 and set B has a MAD of 10, set A is far more consistent, even if their means are the same.
MAD vs variance vs standard deviation
All three measure spread, but they handle the distances differently.
| Measure | How it works | Best for |
|---|---|---|
| Mean absolute deviation | Averages the plain distances from the mean | Simple, intuitive spread |
| Variance | Averages the squared distances | Calculations, not interpretation |
| Standard deviation | The square root of the variance | Advanced statistics |
The difference comes down to squaring. Variance and standard deviation square each distance before averaging, which makes large outliers weigh much more heavily. MAD treats every distance equally, so a single extreme value pulls on it far less. That is why MAD is often called the more robust or steadier measure, while standard deviation is the one you will meet more often in advanced work. You can compare them directly with the standard deviation calculator and the variance calculator.
Mean vs median absolute deviation
You can measure deviation from the mean or from the median. The standard MAD uses the mean. The median version, sometimes called robust MAD, measures each distance from the median instead.
The median version resists outliers even better, because the median itself is not dragged around by extreme values the way the mean is. The calculator above lets you switch between the two with the dropdown, so you can see both for the same data. For heavily skewed data, the median version often describes the spread more honestly.
Where mean absolute deviation is used
Education: MAD is a standard middle school topic, used to introduce the idea of spread before variance and standard deviation.
Finance: analysts use it to measure how much returns swing around their average, as a plain-language read on risk.
Forecasting: MAD measures forecast accuracy by averaging how far predictions land from actual results.
Quality control: factories track MAD to see whether a measurement stays consistent across a production run.
Common mistakes when calculating MAD
Forgetting the absolute value. This is the biggest one. If you leave the minus signs in, the positives and negatives cancel and your total collapses toward zero. Always take |xᵢ − mean|.
Using the wrong mean. Every step after the mean depends on it, so a wrong average makes the whole answer wrong. Check it first.
Dividing by n − 1. That belongs to sample variance and sample standard deviation, not MAD. Mean absolute deviation always divides by n, the full count.
Confusing MAD with standard deviation. They are different. MAD averages plain distances, standard deviation squares them first. The two rarely give the same number.
Frequently asked questions
What does MAD stand for in math?
MAD stands for mean absolute deviation. It is the average of the absolute distances between each value in a data set and the mean.
What is the MAD of 9, 13, 43, and 55?
The MAD is 19. The mean is 30, the absolute distances are 21, 17, 13, and 25, and their average is 76 ÷ 4 = 19.
Is a high MAD good or bad?
Neither on its own. A high MAD means the data is spread out, and a low MAD means it is clustered. Whether that is good depends on context. In quality control you want a low MAD, but in a diverse data set a high MAD is simply expected.
How is MAD different from standard deviation?
MAD averages the plain distances from the mean. Standard deviation squares those distances first, then takes a square root at the end. Squaring makes standard deviation more sensitive to outliers, while MAD stays steadier.
Do you divide by n or n − 1 for MAD?
Always n, the total number of values. Dividing by n − 1 applies to sample variance and sample standard deviation, not to mean absolute deviation.
What does a large MAD tell you?
That the values are spread widely around the mean. The larger the MAD, the less consistent the data. It is a quick signal that your numbers vary a lot from one to the next.
Can MAD be zero?
Yes, but only when every value in the set is identical. If there is no distance between any value and the mean, the MAD is zero, meaning no spread at all.
Try these next
- Mean Median Mode Range Calculator — the core averages behind every spread measure.
- 5 Number Summary Calculator — another way to describe how data is distributed.
- Coefficient of Variation Calculator — spread expressed as a percentage of the mean.
- Outlier Calculator — find the extreme values that stretch your MAD.