Standard deviation vs mean absolute deviation
The standard deviation and the mean absolute deviation answer the same question — how far do the values typically sit from the centre? — and get two different numbers from it. The SD squares every deviation before averaging, which is what lets the rest of statistics work; the MAD keeps the absolute value, which is what makes it the more honest summary for skewed or contaminated data.
The two formulas on one dataset
SD: s = √[ Σ(x − x̄)² / (n − 1)] MAD: MAD = Σ|x − x̄| / n
Seven measurements, in centimetres: 3, 1, 5, 2, 1, 4, 2. The mean is
18 ÷ 7 = 2.5714. Both measures start from the same seven deviations; they differ in what
happens next:
| x | x − x̄ | |x − x̄| | (x − x̄)² |
|---|---|---|---|
| 3 | 0.4286 | 0.4286 | 0.1837 |
| 1 | −1.5714 | 1.5714 | 2.4694 |
| 5 | 2.4286 | 2.4286 | 5.8979 |
| 2 | −0.5714 | 0.5714 | 0.3265 |
| 1 | −1.5714 | 1.5714 | 2.4694 |
| 4 | 1.4286 | 1.4286 | 2.0408 |
| 2 | −0.5714 | 0.5714 | 0.3265 |
| Σ | 0 | 8.5714 | 13.7143 |
The MAD is the top row's total divided by n: 8.5714 ÷ 7 = 1.2245 cm. The SD divides the squared total by n − 1 and takes the root: √(13.7143 ÷ 6) = s = 1.5119 cm; treating the seven measurements as the whole group, the population value is √(13.7143 ÷ 7) = σ = 1.3997 cm. The SD is the larger number, as it almost always is: squaring hands the far-out values 5 and 1 an outsized vote.
Why the standard deviation won anyway
Three properties of the square made the SD the backbone of statistics while the MAD fell out of mainstream use:
- Variances add. For independent quantities, Var(X + Y) = Var(X) + Var(Y). That one rule is what makes ANOVA, portfolio mathematics and error propagation possible. No equivalent rule exists for sums of absolute deviations.
- The square is differentiable. Least-squares estimation — fitting a line, finding a maximum-likelihood normal mean — falls out of calculus because the objective, Σ(x − x̄)², is smooth. The absolute value has a corner at zero and gives up the trick.
- Efficiency on normal data. When the data really is normal, the SD extracts more information from the same sample than the MAD, so it estimates the population spread with smaller error.
For that reason, when the data is roughly symmetric and uncontaminated, report the SD. The trade-off runs the other way as soon as the assumptions break, which is the point below.
When the MAD is the better number
The MAD is the honest summary when one of three things is true:
- The data is skewed. Reaction times, incomes and error magnitudes are long-tailed. A skew inflates the SD toward the tail while the MAD stays near what a typical value actually deviates.
- Genuine outliers are present. One 100-minute delivery among nine near 16 cannot be averaged away; the SD's squared term makes it dominate the answer, the MAD keeps it proportionate. The outlier example shows the two diverging on the same data.
- The measurement is coarse or ordinal. Likert-scale data and rounded readings do not support the arithmetic the SD leans on. The SD of Likert data is a related case of the same caution.
When you do report the MAD, report it next to the median: M = 16.5, MAD = 1.5, not a mean ± MAD, because the absolute deviations are measured from whatever centre you chose. The MAD calculator returns both the mean-based and median-based versions on the same data.
Converting between the two
For normal data the two are pinned together: the MAD about the mean is
√(2/π) ≈ 0.7979 times the SD, so the SD is larger by the reciprocal
√(π/2) ≈ 1.2533. On the seven values above the ratio is 1.3997 ÷ 1.2245 =
1.14, smaller than 1.25 because a seven-value sample is far too small for the normal
ratio to have settled in, and because the data is not normal. Use the 1.2533 factor only
when the distribution is close to normal; otherwise compute both and look at the gap —
a gap well above 1.25 is itself a sign of heavy tails.
Which to put in the report
| Situation | Report |
|---|---|
| Roughly symmetric, clean data (most lab work, exam scores) | SD, next to the mean |
| Skewed data (times, amounts, magnitudes) | MAD next to the median, and say why the SD was not used |
| Data containing genuine outliers you are keeping | Both: the SD as the conventional figure, the MAD as the resistant check |
| Inferential statistics (t-tests, CIs, ANOVA) | SD only; the formulas are built on the variance |
| Roughly normal data where one number must be quoted | SD: it is the more precise estimator |
Related calculators
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Standard deviation calculator
The default measure of spread, worked from your data.
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Mean absolute deviation calculator
The un-squared counterpart, about the mean or the median.
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SD vs variance
The squared-deviation measure one step removed.
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What is standard deviation?
The meaning, the units, and when a high or low SD is good or bad.
Common questions
When should I report the mean absolute deviation instead of the standard deviation?
When the data is far from normal. The SD assumes the far-out values are what should dominate the measure; for skewed data, data with genuine outliers, or data measured on an ordinal scale, the MAD is more representative of a typical deviation. In those cases pair it with the median, not the mean: report median ± MAD, not mean ± MAD.
Which is affected more by outliers, the SD or the MAD?
The SD. A value's influence is squared, so one observation at ten standard deviations contributes a hundred times as much as one at one. The MAD grows in direct proportion to the value, and if it is built about the median (the usual definition) a single outlier cannot even move the centre it is measured from. That is the whole reason MAD survived in robust statistics and forecasting.
What is the ratio between the standard deviation and the MAD for normal data?
For normally distributed data, MAD about the mean equals √(2/π) ≈ 0.7979 times the standard deviation, so the SD is larger by the reciprocal, √(π/2) ≈ 1.2533. That fixed ratio is what makes the two directly convertible for normal data: multiply the MAD by 1.2533 and you recover the SD.
Can I use the MAD in place of the SD in a t-test or confidence interval?
No. The formulas behind t-tests, confidence intervals and ANOVA are built on the variance, not on the absolute deviation, because variances of independent quantities add while absolute deviations do not. Use the MAD to describe the data, then compute the SD for the inferential step, and say which you are reporting and why.