Mean Absolute Deviation
Print PageA Mean Absolute Deviation Calculator (also known as a MAD Calculator, Average Absolute Error Utility, Mean Absolute Forecast Error Generator, or Linear Variability Analyzer) computes sample arithmetic means (X̄ = ∑ xi ÷ n), individual absolute deviations (|xi - X̄|), total sum of absolute deviations (∑ |xi - X̄|), Mean Absolute Deviation (MAD = ∑ |xi - X̄| ÷ n), Mean Absolute Deviation about the Median (MADmedian = ∑ |xi - Median| ÷ n), Median Absolute Deviation (MADmed), and Coefficient of Mean Deviation (CMD = MAD ÷ X̄).
In supply chain inventory demand forecasting, financial portfolio risk analysis, manufacturing machine calibration, and educational scoring, **Mean Absolute Deviation (MAD)** measures the average distance between each data point and the dataset mean. Because MAD takes absolute values rather than squaring differences (like standard deviation), it provides a linear, highly intuitive measure of error variability that avoids over-penalizing occasional large outliers.
Our free online Mean Absolute Deviation Calculator provides instant calculations across all linear variability parameters:
- Mean Absolute Deviation Formula (MAD about Mean):
MAD = ∑ | xi - X̄ | ÷ n. - Mean Absolute Deviation about Median:
MADmedian = ∑ | xi - Median | ÷ n. - Median Absolute Deviation (MADmed):
MADmed = Median( | x1 - Median |, | x2 - Median |, … ). - Coefficient of Mean Absolute Deviation (CMD Percentage):
CMD = ( MAD ÷ X̄ ) · 100%. - Mean Absolute Error (MAE): Equivalent to MAD when measuring forecast error relative to actual target values (
MAE = ∑ | Actuali - Forecasti | ÷ n).
Master MAD Reference Table (Retail Supply Chain Demand Forecasting: n = 6 Weekly Sales Errors)
The table below displays weekly sales demand forecast errors (in units), mean deviations, absolute deviations, and comparative variance metrics for 6 retail sales weeks (n = 6 Weeks: 12, 15, 18, 20, 22, 27 units):
| Weekly Observation Number | Weekly Sales Units (xi) | Raw Deviation (xi – X̄) | Absolute Deviation |xi – X̄| | Squared Deviation (xi – X̄)2 |
|---|---|---|---|---|
| Week 1 | 12 units | 12 – 19 = -7.000 | 7.000 units | 49.00 units2 |
| Week 2 | 15 units | 15 – 19 = -4.000 | 4.000 units | 16.00 units2 |
| Week 3 | 18 units | 18 – 19 = -1.000 | 1.000 units | 1.00 units2 |
| Week 4 | 20 units | 20 – 19 = +1.000 | 1.000 units | 1.00 units2 |
| Week 5 | 22 units | 22 – 19 = +3.000 | 3.000 units | 9.00 units2 |
| Week 6 | 27 units | 27 – 19 = +8.000 | 8.000 units | 64.00 units2 |
| SUMMATIONS (∑) | ∑ x = 114 units | ∑ (x – X̄) = 0.000 | ∑ |x – X̄| = 24.000 units | ∑ SS = 140.00 units2 |
| FINAL METRIC EVALUATION | Mean X̄ = 19.00 units | Zero Sum Proof | MAD = 4.000 units (24 ÷ 6) | Sample SD s = 5.292 units |
Step-by-Step Retail Demand MAD Calculation
To calculate Mean Absolute Deviation (MAD) for 6 weekly sales figures (12, 15, 18, 20, 22, 27):
Step 1 (Calculate Sample Mean X̄): X̄ = (12 + 15 + 18 + 20 + 22 + 27) ÷ 6 = 114 ÷ 6 = 19.000 units
Step 2 (Calculate Absolute Deviations |x_i - 19|):
- |12 - 19| = |-7| = 7.000
- |15 - 19| = |-4| = 4.000
- |18 - 19| = |-1| = 1.000
- |20 - 19| = |+1| = 1.000
- |22 - 19| = |+3| = 3.000
- |27 - 19| = |+8| = 8.000
Step 3 (Calculate Sum of Absolute Deviations): ∑ |x_i - X̄| = 7 + 4 + 1 + 1 + 3 + 8 = 24.000 units
Step 4 (Calculate MAD): MAD = 24.000 ÷ 6 = 4.0000 units
Step 5 (Calculate Coefficient of MAD CMD): CMD = (4.000 ÷ 19.000) · 100% = 21.0526% ≈ 21.05%
Thus, sales demand fluctuates by an average of 4.000 units per week (MAD = 4.00) around the 19.00-unit weekly mean, yielding a 21.05% relative forecast volatility.
Dispersion Measures Comparison: MAD vs. Standard Deviation (SD) vs. Median Absolute Deviation (MADmed)
Below is a comparative reference chart detailing when to use MAD versus alternative measures of variability:
| Variability Metric | Mathematical Distance Weighting | Sensitivity to Outliers | Primary Practical Application |
|---|---|---|---|
| Mean Absolute Deviation (MAD) | LINEAR (|xi – X̄|) | MODERATE (Treats errors proportionally) | Demand forecasting (MAE), inventory buffer stock, quality control. |
| Standard Deviation (SD / s) | QUADRATIC / SQUARED ((xi – X̄)2) | HIGH (Square term heavily penalizes large errors) | Normal distributions, parametric t-tests, ANOVA. |
| Median Absolute Deviation (MADmed) | MEDIAN ABSOLUTE DISTANCE | EXTREMELY ROBUST (Breakdown point = 50%) | Heavy-tailed distributions, machine learning robust scaling. |
History & Mathematics: 1756 Thomas Simpson to 1816 Carl Friedrich Gauss
1756 Thomas Simpson & Observational Errors
In 1756, English mathematician Thomas Simpson first utilized average absolute deviations in astronomical error analysis, demonstrating that averaging multiple observations reduces total deviation.
1816 Carl Friedrich Gauss & Efficiency Proofs
In 1816, German mathematician Carl Friedrich Gauss evaluated MAD versus Standard Deviation, proving that while standard deviation is mathematically optimal for perfect normal distributions, MAD is superior and more robust when handling real-world contaminated data with outliers.
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Frequently Asked Questions (FAQ)
What is the formula for Mean Absolute Deviation (MAD)?
The formula is MAD = ∑ | xi - X̄ | ÷ n, where xi represents each score, X̄ is the mean, and n is sample size.
Why is MAD useful in inventory demand forecasting?
In forecasting, MAD measures average unit error. A MAD of 4.00 units means your inventory forecast is off by an average of 4 units per week, making safety stock calculations simple.
What is the difference between MAD and Standard Deviation?
MAD uses linear absolute distances (|x - X̄|), treating errors proportionally. Standard Deviation squares deviations ((x - X̄)2), heavily penalizing large outliers.