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Deviation Breakdown

#	Value (x_i)	Deviation (x_i − x)	\|Deviation\|

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About

Average deviation quantifies how far data points scatter from the arithmetic mean x̄. Formally termed Mean Absolute Deviation (MAD), it computes the average of all absolute deviations |x_i − x̄| across n observations. Unlike variance, MAD uses absolute values instead of squares, making it more robust against outliers and easier to interpret in the original units. A MAD of 0 indicates all values are identical. Misreporting dispersion can distort quality control limits, risk assessments, and tolerance intervals. This tool assumes a finite population. For samples, the denominator remains n (not n − 1) per the standard MAD definition.

The calculator also reports the median absolute deviation, standard deviation σ, variance σ², and range for cross-comparison. For normally distributed data, MAD ≈ 0.7979 ⋅ σ. Significant divergence from this ratio signals skewness or heavy tails. Note: for datasets exceeding 10⁵ values, precision loss from floating-point arithmetic may appear at the 15th significant digit.

Formulas

The primary computation proceeds in three stages. First, compute the arithmetic mean of the dataset:

x = 1n n∑i=1 x_i

Next, compute each individual absolute deviation from the mean:

d_i = |x_i − x|

Finally, the Mean Absolute Deviation (MAD) is the arithmetic mean of all individual deviations:

MAD = 1n n∑i=1 |x_i − x|

Where x_i = each data point, x = arithmetic mean of the dataset, n = total number of observations, d_i = absolute deviation of the i-th value from the mean, and MAD = mean absolute deviation (average deviation).

For comparison, the population standard deviation uses squared deviations:

σ = √1n n∑i=1 (x_i − x)²

Reference Data

Measure	Formula	Sensitive to Outliers	Units	Use Case
Mean Absolute Deviation (MAD)	1n n∑i=1 \|x_i − x\|	Low	Same as data	Quality control, robust dispersion
Median Absolute Deviation	median(\|x_i − median(x)\|)	Very Low	Same as data	Outlier detection, robust statistics
Variance (σ²)	1n n∑i=1 (x_i − x)²	High	Squared units	Parametric tests, ANOVA
Standard Deviation (σ)	√σ²	High	Same as data	Normal distribution analysis, confidence intervals
Range	x_max − x_min	Extreme	Same as data	Quick spread estimate
Interquartile Range (IQR)	Q₃ − Q₁	Low	Same as data	Box plots, outlier fences
Coefficient of Variation	σx × 100%	High	Dimensionless	Comparing variability across different scales
Mean Squared Error	1n n∑i=1 (x_i − x)²	High	Squared units	Regression, prediction error
Root Mean Square (RMS)	√1n n∑i=1 x_i²	High	Same as data	Signal processing, AC voltage
Geometric Mean Deviation	exp(1n n∑i=1 ln\|x_i − x\|)	Moderate	Same as data	Log-normal distributions
MAD / σ Ratio (Normal)	≈ 0.7979	N/A	Dimensionless	Normality diagnostic
Sample Std Dev (s)	√1n − 1 n∑i=1 (x_i − x)²	High	Same as data	Bessel-corrected estimate for samples

Frequently Asked Questions

Average deviation (MAD) uses absolute values |x_i − x|, while standard deviation uses squared differences (x_i − x)². Squaring amplifies outliers, making standard deviation more sensitive to extreme values. For a perfectly normal distribution, MAD ≈ 0.7979 ⋅ σ. If your data contains outliers or is not normally distributed, MAD provides a more stable measure of spread.

The standard definition of Mean Absolute Deviation divides by n (population). Unlike variance, there is no universally adopted Bessel correction for MAD because the absolute value function is not differentiable at zero and the correction factor depends on the underlying distribution. This calculator uses n as the divisor. If you need a sample-corrected estimate, multiply the result by nn − 1 manually.

Consider the dataset {1, 2, 3, 4, 100}. The mean is 22. The absolute deviation of 100 is 78, contributing linearly. The squared deviation is 6084, contributing quadratically to variance. This is why MAD is preferred in robust statistics: a single outlier inflates variance by orders of magnitude but affects MAD proportionally.

No. MAD equals 0 only when every data point equals the mean, meaning all values are identical. If any value differs from the mean, its absolute deviation is strictly positive, producing a non-zero MAD. This makes MAD a faithful indicator: MAD = 0 ⇔ x_i = x for all i.

Both are absolute-deviation-based measures, but they use different central points. MAD (Mean Absolute Deviation) measures deviations from the arithmetic mean. Median Absolute Deviation measures deviations from the median, then takes the median of those deviations. The median variant has a breakdown point of 50%, meaning up to half the data can be corrupted before the statistic becomes unreliable. MAD from the mean has a breakdown point of 0% because the mean itself is sensitive to outliers.

Use MAD when: (1) your data contains outliers or is heavy-tailed (financial returns, sensor noise), (2) you need interpretability in original units without squaring, (3) the distribution is non-Gaussian (uniform, exponential, multimodal). Use standard deviation when working within frameworks that assume normality (confidence intervals, hypothesis testing, ANOVA) or when mathematical convenience of squared differences is needed (e.g., decomposition of variance in regression).