# Median Absolute Deviation Calculator

> Compute the median absolute deviation, its normal-consistent scaled version (×1.4826) and modified z-score outliers from a list of numbers.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/median-absolute-deviation-calculator

MAD is the median distance of the data from their median. It measures spread like a standard deviation but is barely moved by outliers.

Also called: MAD calculator.

## Inputs
- Numbers (default 2, 4, 4, 4, 5, 5, 7, 9)

## Results
- Median absolute deviation (raw)
- Scaled MAD (estimates σ for normal data)
- Median
- Mean absolute deviation from the mean
- Outliers (|modified z| > 3.5)
- Count

## Formula

```
MAD = median(|xᵢ − median(x)|);  scaled MAD = 1.4826 × MAD;  modified z = 0.6745 × (xᵢ − median) ÷ MAD
```

Where:
- xᵢ: each value
- 1.4826: 1 ÷ Φ⁻¹(0.75), which makes MAD estimate σ for normal data
- 3.5: Iglewicz–Hoaglin outlier cut-off for modified z

## Worked example
1. Median of the sorted data = (4 + 5) ÷ 2 = 4.5.
2. Absolute deviations: 2.5, 0.5, 0.5, 0.5, 0.5, 0.5, 2.5, 4.5; their median is 0.5.
3. Scaled MAD = 1.4826 × 0.5 ≈ 0.7413 (R’s mad() default).

**Result:** The MAD is 0.5, or 0.7413 scaled to the standard-deviation scale.

## What the result means

Compare the scaled MAD with the standard deviation: if the SD is much larger, a few extreme values are inflating it. Values with a modified z-score above 3.5 deserve a closer look.

## Assumptions and limits
- The scale factor 1.4826 assumes the bulk of the data is roughly normal.
- Outliers are flagged with the Iglewicz–Hoaglin modified z-score rule.
- If more than half the values are identical, MAD is 0 and modified z-scores cannot be computed.

## Questions

### Is MAD the same as mean absolute deviation?

No. Median absolute deviation uses medians throughout and resists outliers; mean absolute deviation averages distances from the mean. Both are shown.

### Which software matches this?

R’s mad(x) returns the scaled value; mad(x, constant = 1) returns the raw MAD. Excel has no built-in MAD.

## Related calculators
- [Standard Deviation Calculator](https://calculators.geekmidst.com/calculators/standard-deviation-calculator)
- [Interquartile Range Calculator](https://calculators.geekmidst.com/calculators/interquartile-range-calculator)
- [Tukey Fence Outlier Calculator](https://calculators.geekmidst.com/calculators/outlier-tukey-fence-calculator)
- [Statistics Calculator](https://calculators.geekmidst.com/calculators/statistics-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/median-absolute-deviation-calculator