Understanding Mean Absolute Deviation: A Beginner’s Guide

Introduction

When I first analyzed stock market returns, I needed a way to measure how much investments typically deviate from their average performance. Standard deviation seemed complex, so I discovered a simpler alternative: mean absolute deviation (MAD). This straightforward measure tells me how spread out data points are from their average value, without squaring differences like standard deviation does.

What Is Mean Absolute Deviation?

Mean absolute deviation calculates the average distance between each data point and the dataset’s mean. Unlike variance or standard deviation, MAD uses absolute values, making it easier to interpret for beginners.

The formula appears simple:

MAD = \frac{1}{n}\sum_{i=1}^{n}|x_i - \mu|

Where:

• n = number of observations
• x_i = individual data point
• \mu = mean of all data points

Why MAD Matters in Finance

1. Easier Interpretation

MAD measures average deviation in the original units (dollars, percentages), unlike variance which uses squared units.

2. Robustness to Outliers

Extreme values affect MAD less than standard deviation. A single outlier won’t distort the measurement as severely.

3. Portfolio Risk Assessment

I use MAD to compare volatility across assets when building diversified portfolios.

Step-by-Step Calculation

Let’s compute MAD for Apple’s daily returns over five trading days:

Day	Return (%)
1	1.5
2	-0.8
3	2.1
4	-1.2
5	0.6

Step 1: Calculate Mean (\mu)

\mu = \frac{1.5 + (-0.8) + 2.1 + (-1.2) + 0.6}{5} = 0.44\%

Step 2: Find Absolute Deviations

Day	Deviation from Mean	Absolute Value
1	1.5 – 0.44 = 1.06	1.06
2	-0.8 – 0.44 = -1.24	1.24
3	2.1 – 0.44 = 1.66	1.66
4	-1.2 – 0.44 = -1.64	1.64
5	0.6 – 0.44 = 0.16	0.16

Step 3: Average Absolute Deviations

MAD = \frac{1.06 + 1.24 + 1.66 + 1.64 + 0.16}{5} = 1.15\%

This means Apple’s returns typically deviate by 1.15% from their average during this period.

Comparing MAD to Standard Deviation

While both measure spread, key differences emerge:

Feature	MAD	Standard Deviation
Calculation	Uses absolute values	Squares deviations
Sensitivity	Less affected by outliers	More affected by outliers
Units	Same as original data	Squared units (variance)
Complexity	Easier to compute manually	Requires more steps

For our Apple example, the standard deviation calculates to 1.38% versus MAD’s 1.15%. The 20% difference shows how outlier treatment varies.

Practical Applications

1. Stock Volatility Analysis

I compare MAD values across stocks to identify which exhibit more consistent returns. A lower MAD suggests steadier performance.

2. Economic Forecasting

Economists use MAD to evaluate prediction model accuracy by measuring average error magnitudes.

3. Quality Control

Manufacturers apply MAD to monitor product consistency, like tablet weights in pharmaceuticals.

Limitations to Consider

1. Less Common in Finance

While standard deviation dominates risk metrics (e.g., Sharpe ratio), MAD sees limited use in formal models.

2. Mathematical Properties

The absolute value function isn’t differentiable at zero, complicating calculus-based optimizations.

3. Sample Size Effects

Small datasets may underrepresent true variability, affecting both MAD and standard deviation similarly.

Advanced Insights

Relationship to Normal Distribution

For normally distributed data, MAD relates to standard deviation (\sigma):

MAD \approx 0.7979\sigma

This conversion helps when comparing studies using different dispersion measures.

Robust Statistics Connection

MAD serves as a robust estimator of scale, forming the basis for:

• Modified Z-scores
• Median absolute deviation (a variation using medians)

Historical Context

The concept dates to 1755 when Roger Boscovich used absolute deviations for geodesic measurements. Its simplicity made it popular before computational tools eased standard deviation calculations.

Implementing MAD in Excel

I calculate MAD quickly using:

=AVERAGE(ABS(range - AVERAGE(range)))  

For our Apple example:

=AVERAGE(ABS(B2:B6 - AVERAGE(B2:B6)))  

Conclusion

Mean absolute deviation offers a transparent way to quantify variability, especially when introducing statistical concepts. While finance favors standard deviation for theoretical reasons, MAD provides intuitive, outlier-resistant insights. I use both measures depending on whether I prioritize mathematical convenience (standard deviation) or interpretability (MAD).