All three statistics the acronym names, and what their ratio to the SD tells you.
Three statistics share this acronym
A ratio of 0.7979 — the normal value to four decimals.
n = 10, mean 133.8000, median 134.0000
Mean absolute deviation = 8.400000
About the median it is 8.400000, and the MEDIAN absolute deviation is 7.500000. All three are called MAD.
Mean abs. dev. (about mean)
8.400000
the school textbook one
Mean abs. dev. (about median)
8.400000
never larger
Median absolute deviation
7.500000
the robust one
Scaled by 1.4826
11.119517
estimates σ for normal data
Standard deviation
10.528269
sample, ÷ 9
MAD ÷ SD
0.797852
0.7979 for a normal
Gap from normal
-0.000033
Closest reference shape
Normal
ratio 0.7979
All three of those numbers are called “MAD”, and they are not interchangeable. The first is what a school textbook and most spreadsheets mean. The second is smaller — the median minimises the sum of absolute deviations, so no other centre can beat it, and the tool’s suite checks that on a swept grid rather than taking it on trust. The third is a different statistic entirely: the median of the absolute deviations, used in robust statistics because a single wild value cannot move it.
The MAD-to-SD ratio is 0.797852, against 0.797885 for a normal distribution. That is close enough to be unremarkable: nothing about the spread of this data argues against normality. The ratio is √(2/π) for a normal exactly, so it is a free shape check that costs one division and needs no test.
The 1.4826 is not arbitrary. It is 1/Φ⁻¹(0.75) — the reciprocal of the normal distribution’s third quartile, 0.674490. Multiplying by it makes the median absolute deviation estimate the same σ the standard deviation does, so the two are comparable. Without it the raw figure is about two thirds of the SD for normal data, and reporting it as a spread invites a comparison it will lose.
What this tool shows
MAD names three different statistics. Mean absolute deviation about the mean, about the median, and the median absolute deviation — all computed here. And the ratio of the first to the standard deviation is exactly √(2/π) = 0.7979 for a normal distribution, which turns a homework quantity into a shape diagnostic that costs one division.
Mean absolute deviation about the mean — the school textbook definition
About the median, which is never larger and is minimal over all centres
The median absolute deviation, and the 1.4826 that makes it comparable to a SD
The MAD-to-SD ratio, with the exact value for four reference distributions
Sample standard deviation alongside, so the two can be compared directly
A flag when the SD and the robust estimate disagree by more than a factor of five
Three definitions 0.7979 for a normal Robust estimate too Exact constants
Three statistics, one acronym, and a ratio that reads as a shape check.
Updated 12 September 2026 · Works in any browser, no installation
Mean absolute deviation is the average distance from the centre, without squaring. That makes it directly interpretable in the data’s own units — unlike variance, whose units are squared — and much less sensitive to extreme values than the standard deviation, which squares them.
At a glance
Formula shown
Mean absolute deviation about the mean is (1/n)·Σ|x − x̄|; about the median it is (1/n)·Σ|x − median|, and it is never larger because the median minimises Σ|x − c| over every centre c. The median absolute deviation is median(|x − median|), a different statistic, and multiplying it by 1.4826 = 1/Φ⁻¹(0.75) makes it estimate the same σ the standard deviation does. For a normal distribution the mean absolute deviation is σ·√(2/π).
Scenario support
Reporting spread in the data’s own units, school and GCSE statistics coursework, forecast accuracy as mean absolute error, quality control charts, and any dataset where a few extreme values would dominate a standard deviation.
Educational estimate
Planning support from the values you enter — not professional advice.
Three statistics, one acronym
Ask for “the MAD” and you may get any of three numbers. They are not close on real data, and nothing in the name distinguishes them.
Mean absolute deviation about the mean. (1/n)·Σ|x − x̄|. This is what a school textbook means, what spreadsheets compute, and what most people searching for this are after.
Mean absolute deviation about the median. Same average, different centre — and it is never larger, because the median is the minimiser of the sum of absolute deviations. That is a theorem rather than a tendency, and the suite checks it by sweeping every candidate centre on a grid.
The median absolute deviation. median(|x − median|), which is a different statistic entirely: the middle of the distances rather than their average. It has a 50% breakdown point — half the data can be arbitrarily corrupted before it moves — and it is what robust statistics means by MAD.
On the outlier preset they are 22.68, 13.40 and 1.00. A factor of twenty-two between the first and the third, from the same ten numbers. Picking one without saying which is how a spread gets reported that is right for a question nobody asked.
Which is why this tool prints all three. The disambiguation is the useful part, and deciding between them is a decision about whether extreme values are signal or noise.
The ratio to SD is a shape check
Dividing the mean absolute deviation by the standard deviation gives a number that depends only on the shape of the distribution, not on its scale — and for several standard shapes the value is exact.
For a normal distribution it is √(2/π) = 0.797885. Exactly. No approximation, no simulation.
The other reference values are exact too. Uniform √12/4 = 0.866025, exponential 2/e = 0.735759, Laplace 1/√2 = 0.707107. Heavier tails give a lower ratio, because the squaring in the standard deviation reacts to them far more than the absolute value does.
So the ratio reads directly as a shape statement. The daily-sales preset lands on 0.7979 — the normal value to four decimals. The one with a single outlier drops to 0.5690. The skewed response times give 0.7037.
It is not a test, and that is a feature. A formal normality test at small n cannot detect a deviation that matters, and at large n flags one that does not. A ratio you can read in one division tells you the direction and the size of the departure, which is usually the question.
A low ratio points at the same data a residual plot would. Both are saying a few observations are carrying the spread. The outlier calculator identifies which, and the residual calculator shows whether they are also influencing a fit.
Where 1.4826 comes from
The constant appears in robust statistics code everywhere and is almost never explained, which makes it look like a fudge factor. It is not.
It is 1/Φ⁻¹(0.75). The third quartile of a standard normal is 0.6744898, and 1/0.6744898 = 1.4826022.
The reasoning is one line. For normal data, half the observations fall within 0.6745σ of the median — that is what the quartile means — so the median absolute deviation is about 0.6745σ. Dividing by 0.6745, or multiplying by 1.4826, recovers σ.
It makes the robust estimate comparable with the standard deviation. Without it the raw median absolute deviation is about two thirds of the SD on normal data, and anyone comparing the two concludes the robust estimate is smaller when it is simply on a different scale.
The constant is calibrated to the normal, and only to the normal. On heavy-tailed data the scaled figure no longer estimates the SD — it estimates what the SD would be if the bulk of the data were normal, which is usually the more useful quantity but is a different one.
Which is exactly why the pair is diagnostic. When the scaled median absolute deviation and the standard deviation agree, the data is behaving normally. When they diverge by a factor of five, as on the outlier preset, the SD is reporting on a handful of points and the robust figure on the rest.
Why the standard deviation won anyway
Absolute deviation is easier to explain, easier to compute by hand, and less sensitive to outliers. It still lost, and the reasons are worth knowing because they are mathematical rather than historical accident.
Squares are differentiable and absolute values are not. Every closed form in statistics — least squares, the normal equations, the variance decomposition — comes from setting a derivative to zero. |x| has a corner at zero, and the corner is where the minimum sits.
Variances add and absolute deviations do not. The variance of a sum of independent variables is the sum of their variances. There is no such rule for absolute deviation, which removes almost every useful decomposition — including the sum-of-squares split that ANOVA is built on.
The standard deviation is more efficient for normal data. Its sampling variance is smaller, so it estimates the spread of a normal population more precisely from the same n. Fisher made this argument against Eddington in the 1920s and it settled the matter for most of a century.
The efficiency argument reverses under contamination. Tukey showed that with even a small fraction of outliers the absolute deviation becomes the more efficient estimator — which is the founding observation of robust statistics, and the reason the median absolute deviation exists at all.
For reporting, absolute deviation is often the better number. “Sales varied by about 8 units a day on average” is a sentence anyone can act on. “The standard deviation was 10.5” needs a further explanation of what a standard deviation is, and usually does not get one.
The same statistic under other names
Mean absolute deviation turns up across several fields wearing different labels, and recognising them saves working out the same thing twice.
Mean absolute error, in forecasting. The average of |actual − forecast|. It is the same computation with the forecast playing the role of the centre, and it is preferred to root mean squared error whenever large misses should not be punished quadratically.
Mean absolute percentage error, when scale varies. The same quantity divided by the actual value, which makes series of different magnitudes comparable — and breaks completely wherever an actual value is zero.
The L1 norm, in optimisation. Minimising the sum of absolute deviations is L1 regression, or quantile regression at the median. It fits the median rather than the mean, which is precisely the property that makes it robust.
Average deviation, in older quality-control texts. Same statistic, and it is what control charts used before computing a standard deviation became free.
They share the minimisation property. Every one of them is minimised at the median rather than the mean — which is why an L1 forecast targets the median outcome, and why a model trained on mean absolute error behaves differently from one trained on squared error even on identical data.
Reporting it without ambiguity
Because the acronym is overloaded, a figure reported as “MAD = 7.4” is not a complete statement. Four things make it one.
Say which centre. “Mean absolute deviation about the mean” or “about the median”. The difference is systematic, not incidental — the second is always smaller.
Say whether it is the mean or the median of the deviations. These are the two statistics people most often confuse, and on skewed data they differ by a lot.
Say whether the 1.4826 was applied. Scaled and unscaled median absolute deviations differ by 48%, and software differs on the default: R’s mad() scales, many other implementations do not.
Report the standard deviation alongside. The ratio is the informative part, and it costs one extra number. A reader who knows 0.7979 is the normal value can then read your shape for free.
And say n. Every one of these is a sample statistic, and the median absolute deviation in particular is noisy at small n — it is the median of n numbers, so it moves in steps. The unbiased-for-normal correction some software applies at small n is another thing to state.
Sources and methodology
References for the statistics and for the argument about which spread to use.
Method. All three statistics are computed from the deviations directly rather than through a computational shortcut, and the consistency constant is carried as 1/Φ⁻¹(0.75) to full double precision rather than as the rounded 1.4826. The suite asserts that deviation about the median never exceeds deviation about the mean across 1,500 random datasets, and separately verifies the minimisation property by sweeping every candidate centre on a grid — a stronger check than comparing two formulas. It confirms the MAD-to-SD ratio lands on √(2/π) over 200,000 simulated normal draws, that the scaled median absolute deviation recovers σ = 1 there, and that a single wild value multiplies the SD by more than a thousand while moving the median absolute deviation by under 20%. That engine is verified on every change against 75 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
VarianceSample and population variance from your data, with a live simulation that shows exactly how much the wrong divisor costs — 20% low at n = 5, closing as the sample grows.
OutlierRuns four outlier rules together and says when each is failing — the 3-sigma rule cannot flag anything below n=11, two outliers hide each other, and the quartile convention alone can decide the answer.
Mean, Median and ModeAll three centres marked on your own data, every mode rather than just the first, and the mean-median gap read as a direct measure of skew.
Coefficient of VariationRelative standard deviation with the scale trap made visible — the same five temperatures give a CV of 22.6%, 41.6% or 3.0% depending on the unit, because only a true zero makes the ratio mean anything.
IQRInterquartile range, the 1.5 × IQR outlier fences, and exactly which values fall outside — with a slider that corrupts one value so you can watch the range move while the IQR does not.
An educational tool. “MAD” names three distinct statistics and software defaults differ on whether the 1.4826 scaling is applied; the MAD-to-SD ratio is a descriptive diagnostic rather than a test of normality.
Published a calculator that computes all three statistics the acronym MAD names, because the search term does not distinguish them: mean absolute deviation about the mean, about the median, and the MEDIAN absolute deviation. On the built-in outlier preset they are 22.68, 13.40 and 1.00 — a factor of twenty-two from the same ten numbers.
Prints the ratio of the mean absolute deviation to the standard deviation, which for a normal distribution is exactly sqrt(2/pi) = 0.797885. That turns a homework quantity into a shape diagnostic costing one division: the daily-sales preset lands on 0.7979 to four decimals, the outlier preset drops to 0.5690, and the skewed one gives 0.7037. The exact reference values for uniform, exponential and Laplace — 0.8660, 0.7358, 0.7071 — are printed for comparison.
Carries the consistency constant as 1/Phi-inverse(0.75) to full double precision rather than as the rounded 1.4826, and explains where it comes from: half the observations of a normal fall within 0.6745 sigma of the median, so dividing by that recovers sigma and makes the robust estimate comparable with a standard deviation.
The suite verifies the minimisation property by SWEEPING every candidate centre on a grid rather than by comparing two formulas, confirms the ratio lands on sqrt(2/pi) over 200,000 simulated normal draws, and checks that one wild value multiplies the SD by more than a thousand while moving the median absolute deviation by under 20%.
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