Mean Absolute Deviation Calculator
Find the average absolute distance from the arithmetic mean. Enter raw observations or an exact value-frequency table, then review every distance, contribution and calculation step.
Last Updated: July 27, 2026
Online MAD Calculator
Enter up to 500 values or value-frequency rows. The calculator uses exact rational arithmetic, so large offsets, tiny distances and equivalent fractions remain accurate.
Your exact MAD, decimal approximation and formula details will appear here.
Exact absolute-deviation table
| Value (x) | Frequency (f) | |x - mean| | f × |x - mean| |
|---|---|---|---|
| — | — | — | Calculate a data set |
Duplicate exact values are combined before calculation.
Calculation steps
- Enter numerical data and select Calculate MAD.
Recent calculations
- Your last six results will appear here.
How to Use This Mean Absolute Deviation Calculator
- Choose Raw observations when every measurement is listed. Choose Values and frequencies when values repeat known whole-number counts.
- Enter the numerical data. You may use signed integers, decimals, fractions, mixed numbers or scientific notation.
- In frequency mode, enter one count for every value. Keep the two lists aligned.
- Select the number of decimal places for displayed approximations.
- Select Calculate MAD. Review the exact mean, total absolute deviation, MAD, range, contribution table and steps.
A comma separates observations, so enter 1200 rather than 1,200. Blank separators are ignored. Exact equivalents such as 0.5, 1/2, 2/4 and 5e-1 are grouped as the same value.
What Is Mean Absolute Deviation?
Mean absolute deviation measures the average distance between each observation and the arithmetic mean. Distance is nonnegative, so the formula uses absolute values. A small result means the data stays close to its mean. A larger result means observations sit farther from the mean on average.
Absolute distance = |x - mean|
Mean absolute deviation = Σ|x - mean| / N
The result uses the same units as the original data. If delivery times use minutes, MAD also uses minutes. Unlike variance, the distances are not squared. This keeps the numerical scale easy to interpret.
Mean Absolute Deviation Formula
For observations x1 through xN, first calculate the arithmetic mean. Then subtract the mean from every observation, take each absolute value, add the distances and divide by N.
MAD = [|x1 - x̄| + |x2 - x̄| + ... + |xN - x̄|] / N
This descriptive formula divides by N for both a listed population and a listed sample. There is no n - 1 option on this page. The familiar n - 1 correction belongs to the usual unbiased sample-variance estimator, not to the standard mean absolute deviation summary.
Worked Mean Absolute Deviation Example
Use the data 2, 4, 4, 4, 5, 5, 7 and 9. The total is 40 and N is 8, so the arithmetic mean is 5. The absolute distances from 5 are 3, 1, 1, 1, 0, 0, 2 and 4.
MAD = 12 / 8 = 3/2 = 1.5
The answer means an observation is 1.5 data units away from the arithmetic mean on average. It does not say every observation is exactly 1.5 units away. The table shows the individual distances behind the average.
Using Values and Frequencies
Frequency mode represents repeated observations without making you type every copy. Enter values 1, 2 and 3 with frequencies 2, 1 and 1 to represent 1, 1, 2 and 3. Total frequency is 4, not the three visible rows.
Mean = Σ(f × x) / N
MAD = Σ[f × |x - mean|] / N
The calculator merges duplicate exact values and ignores zero-frequency rows. It never expands a large frequency into millions of browser entries. Frequencies must be nonnegative whole-number observation counts. Survey weights, probability weights, reliability weights and negative weights do not have the same interpretation and are outside this page.
Mean Absolute Deviation Versus Median Absolute Deviation
The abbreviation MAD is ambiguous. In elementary descriptive statistics, MAD often means mean absolute deviation about the arithmetic mean. In robust statistics, MAD often means median absolute deviation about the median. They use different centers and different final summaries.
| Measure | Center | Final summary | Main property |
|---|---|---|---|
| Mean absolute deviation | Arithmetic mean | Mean of absolute distances | Uses every distance directly |
| Median absolute deviation | Median | Median of absolute distances | More resistant to extreme observations |
This calculator reports the first definition only. It does not multiply a median-based statistic by 1.4826 or any other consistency factor. State the full measure name when software, a course or a report uses the abbreviation MAD.
MAD Compared with Variance and Standard Deviation
All three measures describe spread around a center, but they treat distance differently. Mean absolute deviation keeps absolute distances. Variance squares deviations. Standard deviation takes the square root of variance.
| Measure | Distance treatment | Units | Effect of extremes |
|---|---|---|---|
| Mean absolute deviation | Absolute values | Original data units | Linear increase with distance |
| Variance | Squared deviations | Squared data units | Quadratic increase with distance |
| Standard deviation | Square root of average squared deviation | Original data units | Inherits squared-distance emphasis |
| Range | Maximum minus minimum | Original data units | Uses only two endpoints |
MAD is often easier to explain because it answers a direct question: how far is an observation from the mean on average? Variance and standard deviation remain central in probability models, regression, standard errors and many inferential procedures.
How to Interpret the Result
A MAD of zero means every positive-frequency observation equals the same value. Any variation produces a positive result. There is no universal good or bad MAD. Interpretation depends on the measurement scale, units, context and acceptable variation.
Adding the same constant to every observation leaves MAD unchanged. Multiplying every observation by a number a multiplies MAD by |a|. Changing metres to centimetres therefore multiplies the displayed MAD by 100, even though the physical spread has not changed.
- Compare groups only when their units and measurement definitions match.
- Keep the arithmetic mean unrounded until every distance is calculated.
- Inspect the underlying values because one summary cannot show clusters or gaps.
- Report whether MAD means mean absolute deviation or median absolute deviation.
Exact Fractions, Large Offsets and Tiny Distances
Ordinary browser numbers store many decimal values approximately. They may also lose small differences beside a large shared baseline. The values 10000000000000001, 10000000000000002 and 10000000000000003 have mean 10000000000000002 and MAD 2/3. A floating-point conversion performed too early may erase those one-unit differences.
This tool parses supported inputs as reduced fractions backed by arbitrary-size integers. The sum, mean, each absolute distance, total deviation and MAD remain exact. The decimal selector changes only the displayed approximation. A small positive result switches to scientific notation instead of appearing as zero.
Exact arithmetic has a 4,000-digit combined numerator-and-denominator safety limit for intermediate fractions. A large collection of unrelated denominators may reach it. The calculator reports a controlled error instead of silently switching to lower-precision arithmetic.
Mean Absolute Deviation Versus Mean Absolute Error
Mean absolute deviation and mean absolute error use the same absolute-distance idea but answer different questions. This page calculates distances from the arithmetic mean of the entered observations. Mean absolute error usually compares observed values with predictions, forecasts or another fixed reference. The result is an error measure only when a prediction or target supplies the center.
For example, if actual sales are 90, 110 and 100, their arithmetic mean is 100 and their mean absolute deviation is 20/3. If three forecasts are 95, 105 and 120, their absolute forecast errors are 5, 5 and 20, so mean absolute error is 10. Do not substitute one statistic for the other without checking which reference values the task defines.
Input Rules and Limits
- Enter from 1 through 500 raw observations or paired frequency rows.
- Numerical parts may contain up to 50 digits. Scientific exponents may range from -100 through 100.
- Use a period as the decimal mark. Commas separate observations and cannot serve as thousands separators.
- Fractions require a nonzero denominator. Mixed numbers use a space, such as -2 1/3.
- Frequency counts must be whole numbers from 0 through 1,000,000,000,000.
- The value and frequency lists must contain matching entry counts.
- Units, currency symbols, percentages, expressions, class intervals, NaN and infinity are rejected.
- Missing observations are not estimated, and suspected outliers are not removed.
Common Mean Absolute Deviation Mistakes
- Dropping the absolute value: signed deviations around the arithmetic mean add to zero.
- Using the median as the center: that creates a different statistic from the one calculated here.
- Rounding the mean early: calculate distances from the exact mean before rounding the final MAD.
- Dividing by distinct rows: frequency data divides by total frequency N.
- Applying n - 1: the standard descriptive MAD formula divides by the number of observations.
- Squaring deviations: squared distances belong to variance, not mean absolute deviation.
- Using arbitrary weights as counts: frequency mode represents repeated observations only.
Practical Uses and Scope
Mean absolute deviation helps summarize consistency in delivery times, manufacturing measurements, prices, temperatures and classroom data. A related measure, mean absolute error, averages paired distances between actual values and their own predictions. This calculator does not accept paired predictions or calculate mean absolute error. If you enter a list of forecast errors here, it measures their spread around the error list's arithmetic mean, not their average distance from zero.
This page describes only the entered data around its arithmetic mean. It does not infer missing values, test a population hypothesis, construct a confidence interval or identify the cause of variation. Check data collection, units, rounding requirements and the requested MAD definition before using the result in formal work.
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Frequently Asked Questions
How do I calculate mean absolute deviation?
Find the arithmetic mean, calculate the absolute distance from each observation to that mean, add the distances and divide by the number of observations.
What formula does this MAD calculator use?
It uses MAD equals the sum of absolute values of x minus the arithmetic mean, divided by N. Frequency mode multiplies each distance by its whole-number count.
Does mean absolute deviation divide by n or n minus 1?
The standard descriptive mean absolute deviation formula divides by the number of observations, N. The n minus 1 correction belongs to the usual sample-variance formula.
Can mean absolute deviation be negative?
No. Every absolute distance is nonnegative. MAD equals zero only when all positive-frequency observations have the same value.
What is the MAD of one value?
It is zero. The arithmetic mean equals the only observation, so its absolute distance from the mean is zero.
Does MAD use the same units as the data?
Yes. Absolute distances and their mean stay in the original data units. Variance differs because it uses squared units.
Is mean absolute deviation the same as median absolute deviation?
No. Mean absolute deviation averages distances from the arithmetic mean. Median absolute deviation takes the median of distances from the median and is more resistant to extreme values.
Can I enter fractions and scientific notation?
Yes. The calculator accepts integers, decimals, fractions, mixed numbers and scientific notation, then reduces supported inputs to exact rational values.
Are 0.5 and 1/2 treated as the same value?
Yes. Both reduce to the exact fraction 1/2. Equivalent forms such as 2/4 and 5e-1 are grouped together.
How does frequency mode calculate MAD?
Each frequency repeats its matching value that many times. The calculator uses total frequency as N, merges duplicate exact values and does not expand large counts.
Can I use decimal or negative weights?
No. Frequency mode accepts nonnegative whole-number observation counts only. Arbitrary, probability, reliability and negative weights have different meanings.
Why might my MAD answer differ from another calculator?
The other tool may use the median instead of the mean, apply a scaling factor, round the center early, divide by a different count or interpret weights differently.
This calculator is an educational descriptive-statistics tool. Confirm the requested MAD definition, units, data quality, frequency meaning and rounding rules before relying on a result.