Median Absolute Deviation Calculator

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:

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:

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

1. Enter your data: Input numerical values separated by commas, spaces, or line breaks. Use the example buttons for quick testing with different data types.
2. 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.
3. Set decimal precision: Choose from 2 to 15 decimal places.
4. Calculate and analyze: Click "Calculate MAD" to see comprehensive results including the robustness assessment.
5. 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

• Median Absolute Deviation - Wikipedia
• Robust Statistics - Wikipedia