# What is the mean absolute deviation?

Computes the mean absolute deviation (MAD) of a list of numbers: the average distance of each number from the mean.

- Page: https://www.acalculator.org/statistics/mean-absolute-deviation-calculator
- JSON spec: https://www.acalculator.org/statistics/mean-absolute-deviation-calculator.json
- Version: 947f1c879a45

## Default answer

Example with the default inputs (Your numbers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]): The mean absolute deviation of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 is 2.5, around a mean of 5.5.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| input | Your numbers | The numbers, separated by commas, spaces, semicolons, or new lines. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| mad | Mean absolute deviation | The average of the distances \|x − mean\| of each number from the mean. |
| mean | Mean | The sum divided by the count. |
| count | Count | How many numbers are in the list. |
| sum | Sum | All the numbers added up. |
| min | Smallest | The smallest number in the list. |
| max | Largest | The largest number in the list. |
| range | Range | The largest number minus the smallest. |

## Method

MAD = Σ|x − mean| ÷ n, where mean = Σx ÷ n.

## Assumptions

- The deviations are measured from the mean, not the median.
- The sum of distances is divided by n, the count of numbers.

## Worked examples

1. input = 1 or 2 gives mad = 2.5, mean = 5.5, range = 9. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (variance, standard deviation, average absolute deviation). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm.
2. input = 2 or 4 gives mad = 1.5, mean = 5. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (variance, standard deviation, average absolute deviation). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm.
3. input = 1.5 or 2.5 gives mad = 2.90625, mean = 4.4375. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (variance, standard deviation, average absolute deviation). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm.
4. input = 100 or undefined gives mad = 0, mean = 100. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (variance, standard deviation, average absolute deviation). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm.

## FAQ

### What is Mean Absolute Deviation (MAD) and why is it important?

Mean Absolute Deviation (MAD) is a measure of variability that shows the average distance between each data point and the mean. Unlike variance, MAD uses absolute values instead of squared differences, making it more intuitive to interpret. It's useful for understanding how spread out your data is from the average.

### How is MAD calculated?

MAD is calculated by: 1) Finding the mean of all values, 2) Subtracting the mean from each value and taking the absolute value, 3) Finding the average of these absolute differences. Formula: MAD = Σ|x - x̄| / n, where x̄ is the mean and n is the number of data points.

### What's the difference between MAD and standard deviation?

Both measure variability, but MAD uses absolute differences while standard deviation uses squared differences. MAD is more robust to outliers because it doesn't square the differences, making extreme values less influential. Standard deviation is more commonly used in statistical tests, but MAD is easier to interpret.

### When should I use MAD instead of variance or standard deviation?

Use MAD when you want a measure of variability that's less sensitive to outliers, easier to interpret, or when you need to explain the concept to non-technical audiences. MAD is also useful in robust statistics and when you want to avoid the influence of extreme values.

### Can MAD be negative?

No, MAD cannot be negative. Since MAD involves taking absolute values of differences from the mean, all terms in the calculation are non-negative, making the final result always positive or zero. Zero MAD means all values are identical.

### What does a high MAD tell me about my data?

High MAD indicates that data points are widely spread out from the mean, suggesting high variability or dispersion. This could mean the data is diverse, has outliers, or comes from a population with high variability. Low MAD suggests data points are clustered close to the mean.

### What's the relationship between MAD and the range of data?

Both MAD and range measure spread, but differently. Range is simply max - min, while MAD considers all data points and their distances from the mean. MAD is more robust because it's not affected by just two extreme values like range is, and it provides a more representative measure of typical deviation. MAD is never more than half the range.

### How many decimal places should I use when reporting MAD?

Use 2-4 decimal places for MAD, depending on your data precision and context. For most practical purposes, 2-3 decimal places are sufficient. The calculator shows up to 6 decimal places but you can round as needed for your specific use case.

### What are some common applications of MAD in real life?

MAD is used in quality control (measuring product consistency), finance (risk assessment), education (analyzing test scores), weather forecasting (temperature variability), and many other fields where understanding data spread is important and you want a measure that's less sensitive to outliers.

### How does MAD compare to other measures of variability?

MAD is more robust than variance/standard deviation because it's less affected by outliers. It's easier to interpret since it's in the same units as your data. However, it's less commonly used in statistical tests because many statistical methods are based on squared differences (variance).

### Can I use MAD for comparing variability between different datasets?

Yes, MAD can be used to compare variability between datasets, but be cautious when the means are very different. For datasets with similar means, MAD provides a good comparison. For datasets with very different means, you might want to use the coefficient of variation (MAD/mean) for a relative comparison.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (variance, standard deviation, average absolute deviation). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm