This mean absolute deviation calculator finds the average absolute distance from the mean. Enter observations to see the center, distances and result, or explicitly choose unscaled median absolute deviation.
Measure average distance from the mean
Mean absolute deviation describes average distance from the arithmetic mean. Find that mean, take the absolute distance of every observation from it, and average those distances. Absolute values keep opposite-side distances from cancelling. The result uses the original measurement unit. The abbreviation MAD can also mean median absolute deviation, which is a different statistic. This page defaults to mean absolute deviation. Its alternate option explicitly calculates unscaled median absolute deviation: find the data median, then take the median of the absolute distances from that median. Enter observations in one common unit. Every valid value contributes; the tool does not infer a population model, remove extreme values, or divide the mean absolute distance by a sample-size-minus-one denominator. The NIST measures-of-scale reference defines both absolute-spread measures and distinguishes them from standard deviation.
Mean absolute deviation uses the mean and averages distances. Median absolute deviation uses the median twice and is unscaled here.
The absolute-spread answer has the same measurement unit as the observations.
Every valid observation contributes. Nonnumeric tokens are rejected rather than silently discarded.
Check the center and absolute distances
The mean preset contains 5 observations: 2 4 6 8 10. Its arithmetic mean is 6. In input order, the absolute distances from that mean are 4, 2, 0, 2, 4. Their sum is 12. Divide by the full count 5 to obtain mean absolute deviation 2.4.
For each value, absolute distance means the nonnegative size of its difference from the center. Signed deviations could cancel, so they are not averaged as a spread measure. Squared deviations answer another question. The NIST absolute-spread definitions specify the full-count mean absolute distance.
The engine computes distances before display rounding. Its centered base-and-offset working retains small representable gaps next to large values; recombining those parts into a displayed center can round the midpoint. The individual observation details retain the entered numeric values and show approximate distances.
Distinguish mean and median absolute deviation
For the median preset 1 2 3 4 100, the data median is 3. Distances from that median, in input order, are 2, 1, 0, 1, 97. The middle sorted distance is 1, so unscaled median absolute deviation is 1. This is a median of distances, not their sum divided by the count.
Using the same observations with the mean option gives center 22 and mean absolute deviation 31.2. The two results differ because both the center and the aggregation change. Median absolute deviation is less sensitive to extreme tails; this does not mean that a particular result proves an outlier or licenses automatic deletion. NIST discusses the different influence of tail values on these measures.
Even-length medians use the average of the two middle sorted values. The additional median check 1 2 8 10 gives data median 5, with absolute distances 4, 3, 3, 5. Its two middle sorted distances are 3 and 4, producing median absolute deviation 3.5. The OpenStax center definitions explain this midpoint convention.
The median option is unscaled. No normal-consistency multiplier, modified z score or inferred probability model is applied. The detail-only sum of absolute distances is audit information in median mode; it is not used as that mode’s numerator.
Keep the units and every valid observation
Mean absolute deviation keeps original-unit distances. Standard deviation squares deviations, applies its selected variance denominator, then takes a square root. Both can describe spread, but their aggregation and sensitivity differ. Use the Standard Deviation Calculator when you need that squared-deviation measure or its sample/population convention.
The Mean Median Mode Calculator compares summaries of center. The Interquartile Range Calculator describes ordered middle spread. The Coefficient of Variation Calculator measures spread relative to a nonzero mean and explains measurement-scale limits. This absolute-deviation tool does not establish a universal acceptable-spread threshold.
A single observation or a constant dataset has absolute deviation zero under either supported measure. No sample SD is being estimated here. Repeated observations remain repeated entries; weighted/frequency inputs and grouped midpoint approximations are outside this page.
Local limits are 5,000 observations, 100,000 characters and 128 characters per number. Nonzero entered magnitudes range from 1e-100 to 1e+100. Simple fractions, decimals and scientific notation are accepted. Omit thousands separators. Details show the first 100 observation distances, while all valid observations contribute.
The engine uses IEEE754 numbers, compensated sums and centered distances. Display values round to 10 significant digits after calculation, and approximation signs distinguish rounded working. Values or gaps already lost when an input is represented cannot be recovered. The result supplies descriptive spread, without automatic deletion, normal scaling, a confidence interval or a quality verdict.