MAD

Math & Statistics

Mean Absolute Deviation Calculator

Calculate a five-value arithmetic mean, each absolute deviation, mean absolute deviation, maximum deviation, and comparison gap. Distinguish mean absolute deviation from median absolute deviation and review the effect of each entered value.

Value 1Value 2Value 3Value 4Value 5Entered comparison MAD

Mean absolute deviation from mean-

Arithmetic mean-

Absolute deviation 1-

Absolute deviation 2-

Absolute deviation 3-

Absolute deviation 4-

Absolute deviation 5-

MAD minus comparison-

Largest absolute deviation-

Decision view

Absolute residual lollipops around the mean

Absolute residual lollipops around the meanEach observation is connected to the arithmetic mean so its absolute contribution and the final MAD remain visible.

Exact scenario comparisonValue 5 changes while all other entered assumptions remain constant.

Value 5	Mean absolute deviation from mean	Arithmetic mean	Absolute deviation 1	Absolute deviation 2	Absolute deviation 3	Absolute deviation 4	Absolute deviation 5	MAD minus comparison	Largest absolute deviation

How to use Mean Absolute Deviation Calculator

Enter five measurements on the same scale.
Inspect the arithmetic mean before interpreting distances.
Review each residual contribution and the largest deviation.
Compare MAD values only when the underlying units and centers are comparable.

Calculator guide

Understanding Mean Absolute Deviation Calculator

Mean absolute deviation from the mean summarizes the average magnitude of five residuals without squaring them. This page displays every observation-to-mean distance so the final spread measure can be reconstructed.

Center first Every deviation depends on the calculated mean.

No cancellation Absolute values keep opposite residuals from summing to zero.

Original units MAD stays on the same scale as the data.

Not median MAD The reference center and algorithm differ.

Calculation method

How the calculation works

Compute the five-value arithmetic mean, expose each absolute residual, and average those residuals to obtain mean absolute deviation from the mean. In the Mean Absolute Deviation Calculator, the live scenario varies value 5 and tracks mean absolute deviation from mean while the remaining results preserve the reconciliation path. Add the five observations and divide by five to obtain the arithmetic mean. Subtract that mean from each observation, take each absolute value, add the five magnitudes, and divide by five.

Detailed calculation process

Average the five distances from the mean

The default values center exactly at 14, making every signed difference and absolute contribution easy to audit.

General formula: xbar = (1/n) x sum(x_i); MAD_mean = (1/n) x sum(|x_i - xbar|) First calculate the arithmetic center. Then ignore residual direction by taking absolute values and average those five distances.

What each symbol means

x_i Observation i in the original measurement units.

xbar Arithmetic mean in the original measurement units.

n Number of observations; fixed at five on this page.

|x_i - xbar| Absolute distance of observation i from the mean.

MAD_mean Mean absolute deviation from the mean, in original units.

Worked substitution with the default inputs

1. Calculate the mean: (12 + 15 + 14 + 19 + 10) / 5 = 70 / 5 = 14 The same mean becomes the reference point for all residuals.

2. Find signed residuals: -2, +1, 0, +5, -4 Subtract 14 from each observation while retaining direction temporarily.

3. Take absolute distances: 2, 1, 0, 5, 4 Absolute values prevent positive and negative residuals from cancelling.

4. Average the distances: (2 + 1 + 0 + 5 + 4) / 5 = 12 / 5 = 2.4 The average observation lies 2.4 units from the mean under this measure.

5. Compare and locate the maximum: 2.4 - 3 = -0.6; max distance = 5 Default MAD is 0.6 below the entered comparison and the value 19 contributes the largest deviation.

The five defaults have mean 14 and absolute deviations totaling 12, so mean absolute deviation is 2.4.

Residual audit

See which values create the spread

The final average is easier to interpret when every distance remains visible.

At the mean Contributes zero absolute deviation.

Near the mean Adds a small amount to the total.

Far from the mean Adds a larger linear contribution.

Moving center Changing one value shifts every residual through the mean.

Worked situations

Practical examples

The observation 14 contributes zero because it equals the mean.
Values 19 and 10 contribute distances of 5 and 4 respectively.
Replacing an extreme value changes both the mean and every residual, not just one contribution.

Better inputs

Useful tips

State that this is deviation from the arithmetic mean.
Keep original units attached to the MAD result.
Use the lollipop plot to inspect whether one value dominates the average distance.

Before relying on the result

Limitations and common mistakes

This calculator is not the median absolute deviation statistic.
Five observations provide limited information about distribution shape and sampling variability.
Dependence, unequal weights, repeated measurement, and missing-data mechanisms are not represented.

Reference

Key terms

Residual: Observation minus the arithmetic mean.
Absolute residual: Residual magnitude after removing its sign.
Mean absolute deviation: Average absolute residual from the arithmetic mean.
Maximum deviation: Largest absolute residual among the five observations.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why do the residuals sum to zero but MAD is positive?

Signed residuals cancel around the arithmetic mean; absolute residuals remove signs before averaging.

Is MAD less sensitive than standard deviation?

Absolute distances do not square large residuals, but interpretation still depends on the data and purpose.

Can MAD be zero?

Yes, when all five observations are identical.

Can I compare MAD across different units?

Not directly. MAD is measured in the original units unless it is separately standardized.