Median Absolute Deviation Calculator

Q: When should I use MAD instead of Standard Deviation?

Use MAD when: (1) your data may contain outliers or extreme values, (2) the data is not normally distributed, (3) you need a robust measure that won't be skewed by a few unusual observations, (4) you're doing exploratory data analysis and want to identify potential outliers, or (5) working in fields like finance or quality control where robustness is critical.

Q: How can MAD be used for outlier detection?

MAD is excellent for outlier detection using a modified Z-score: M = 0.6745 × (xᵢ - median) / MAD. Values with |M| > 3.5 are typically considered outliers. This method is more reliable than using standard deviation because it doesn't let outliers influence the detection threshold itself.

Calculate the median absolute deviation (MAD) of a dataset with step-by-step formulas, interactive visualization, outlier detection, and robustness comparison to standard deviation.

Median Absolute Deviation Calculator

Try example datasets:

Simple Series With Outlier Quality Control Skewed Data

Enter Your Data

Scale Factor

Custom k

Precision

Tip: If your text contains non-numeric content, use our Number Extractor to extract numbers first.

Embed Median Absolute Deviation Calculator Widget

About Median Absolute Deviation Calculator

Welcome to the Median Absolute Deviation Calculator, a robust statistical tool that calculates MAD with step-by-step formulas, interactive data visualization, and outlier detection insights. MAD is a powerful alternative to standard deviation when your data contains outliers or follows a non-normal distribution.

What is Median Absolute Deviation (MAD)?

Median Absolute Deviation (MAD) is a robust measure of statistical dispersion that describes how spread out the values in a dataset are. Unlike standard deviation, which uses the mean and squared differences, MAD uses the median and absolute differences, making it highly resistant to outliers and extreme values.

Median Absolute Deviation Formula

$$\text{MAD} = \text{median}(|x_i - \text{median}(X)|)$$

In plain terms: MAD is the median of how far each data point is from the overall median of the data.

Why MAD is a "Robust" Measure

A statistic is considered robust if it is not heavily influenced by outliers or violations of assumptions. MAD has a breakdown point of 50%, meaning that up to half of the data can be corrupted before MAD gives an arbitrarily incorrect result. In contrast, the mean and standard deviation have a breakdown point of 0% - even a single outlier can dramatically affect them.

MAD vs Standard Deviation: When to Use Each

Property	MAD	Standard Deviation
Central tendency used	Median	Mean
Deviation type	Absolute values	Squared values
Sensitivity to outliers	Very low (robust)	High (sensitive)
Breakdown point	50%	0%
Best for	Skewed data, outliers	Normal distributions
Efficiency for normal data	~37%	100%

When to Use MAD

Your data may contain outliers or extreme values
The data is skewed or not normally distributed
You need a robust baseline for outlier detection
You want a measure unaffected by a few unusual observations
Working in fields like finance, quality control, or anomaly detection

When to Use Standard Deviation

Your data is confirmed to be normally distributed
You need maximum statistical efficiency
The data is clean with no outliers
You need to use results in parametric tests

The Scale Factor (k = 1.4826)

When comparing MAD to standard deviation, or using MAD as a robust estimate of population standard deviation for normally distributed data, the constant k = 1.4826 is applied:

Scaled MAD (Robust σ Estimate)

$$\hat{\sigma} = k \times \text{MAD} = 1.4826 \times \text{MAD}$$

This constant comes from the relationship:

$$k = \frac{1}{\Phi^{-1}(3/4)} \approx 1.4826$$

Where $\Phi^{-1}$ is the inverse cumulative distribution function of the standard normal distribution. For normally distributed data, scaled MAD will approximately equal the standard deviation.

MAD for Outlier Detection

MAD is excellent for detecting outliers because outliers do not influence the threshold itself. The modified Z-score method uses MAD:

Modified Z-Score

$$M_i = \frac{0.6745 \times (x_i - \text{median})}{\text{MAD}}$$

A data point is typically flagged as an outlier if $|M_i| > 3.5$. This method is more reliable than using standard deviation because:

Outliers do not influence the MAD or median used to calculate the threshold
It works well even when multiple outliers are present (masking effect is avoided)
It is effective for non-normal distributions

How to Use This Calculator

Enter your data: Input numerical values separated by commas, spaces, or line breaks. Use the example buttons for quick testing with different data types.
Select a scale factor: Choose "No scaling" for raw MAD, or k=1.4826 to estimate standard deviation. You can also enter a custom scale factor.
Set decimal precision: Choose from 2 to 15 decimal places.
Calculate and analyze: Click "Calculate MAD" to see comprehensive results including the robustness assessment.
Review step-by-step: Examine the detailed calculation breakdown showing each step of the MAD computation.

Understanding Your Results

Primary Results

MAD: The median absolute deviation - the main result
Scaled MAD: MAD multiplied by your chosen scale factor
Median: The central value of your dataset
Robustness Rating: Assessment comparing MAD to standard deviation

Comparison Statistics

Mean: Arithmetic average for comparison
Standard Deviation: Sample standard deviation for comparison
IQR: Interquartile range (another robust measure)
Q1, Q3: First and third quartiles

Frequently Asked Questions

What is Median Absolute Deviation (MAD)?

Median Absolute Deviation (MAD) is a robust measure of statistical dispersion. It is calculated as the median of the absolute deviations from the data's median: MAD = median(|xᵢ - median(X)|). Unlike standard deviation, MAD is resistant to outliers, making it ideal for datasets with extreme values or non-normal distributions.

How is MAD different from Standard Deviation?

MAD uses the median and absolute values, while standard deviation uses the mean and squared differences. This makes MAD much more robust to outliers - a single extreme value can dramatically increase standard deviation but barely affects MAD. For normally distributed data, MAD multiplied by 1.4826 approximates the standard deviation.

What is the scale factor k=1.4826 for MAD?

The constant 1.4826 is used to make MAD a consistent estimator of the standard deviation for normally distributed data. Mathematically, k = 1/Φ⁻¹(3/4), where Φ⁻¹ is the quantile function of the standard normal distribution. When you multiply MAD by 1.4826, you get a robust estimate of σ.

When should I use MAD instead of Standard Deviation?

Use MAD when your data may contain outliers, is not normally distributed, or when you need a robust measure that will not be skewed by extreme observations. MAD is particularly useful in exploratory data analysis, quality control, finance, and anomaly detection.

How can MAD be used for outlier detection?

MAD is excellent for outlier detection using the modified Z-score: M = 0.6745 × (xᵢ - median) / MAD. Values with |M| > 3.5 are typically considered outliers. This method is more reliable than using standard deviation because outliers do not influence the detection threshold itself.

How many numbers does this MAD Calculator support?

This calculator can handle datasets of virtually any size. We have tested with over 100,000 numbers and the tool provides instant results. Whether you have 3 data points or 100,000, the calculator will efficiently compute the MAD along with all related statistics.

Additional Resources

Reference this content, page, or tool as:

"Median Absolute Deviation Calculator" at https://MiniWebtool.com/median-absolute-deviation-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 19, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Mean Absolute Deviation (MAD) Calculator

Median CalculatorFeatured

Median Absolute Deviation Calculator

Your ad blocker is preventing us from showing ads

About Median Absolute Deviation Calculator

What is Median Absolute Deviation (MAD)?

Why MAD is a "Robust" Measure

MAD vs Standard Deviation: When to Use Each

When to Use MAD

When to Use Standard Deviation

The Scale Factor (k = 1.4826)

MAD for Outlier Detection

How to Use This Calculator

Understanding Your Results

Primary Results

Comparison Statistics

Frequently Asked Questions

What is Median Absolute Deviation (MAD)?

How is MAD different from Standard Deviation?

What is the scale factor k=1.4826 for MAD?

When should I use MAD instead of Standard Deviation?

How can MAD be used for outlier detection?

How many numbers does this MAD Calculator support?

Additional Resources

Related MiniWebtools:

Statistics And Data Analysis:

Top & Updated: