Descriptive Statistics
Mean Absolute Deviation Calculator
Mean absolute deviation averages the distances from the arithmetic mean, and it does not substitute the median for that center.
Worksheet
Press Calculate to see the result.
Mean absolute deviation averages the distances from the arithmetic mean, and it does not substitute the median for that center.
What this calculator answers
Scores 72, 80, 85, 90, and 93 have a mean of 84. The absolute distances from 84 are 12, 4, 1, 6, and 9 points. Their average is 6.4 points, and that average is the mean absolute deviation. The center is the mean on purpose. The median of these scores is 85, and the median absolute deviation from 85 is a different page and a different number. Squaring is also avoided, so a gap of 12 points counts as 12 points, not as 144. The result stays in the unit of the scores.
The headline is the mean of the absolute deviations from the arithmetic mean. The mean itself is shown so the center is not confused with the median. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Mean absolute deviation from the mean uses this relationship.
mean absolute deviation = (sum of |x - mean|) / n
- mean. Arithmetic mean, the center of every distance.
- |x - mean|. Absolute distance of one score from that mean.
- n. How many distances are averaged.
- MAD. On this page, the mean of those absolute distances, not the median of them.
Absolute value stops distances above and below the mean from cancelling. Dividing by n, not by n minus 1, treats the result as an average distance in the batch you typed. No square and no median enters the formula. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The values are 72, 80, 85, 90, 93. The arithmetic mean is 84. Distances are measured from this mean, not from the median. The absolute distances are 12, 4, 1, 6, 9.
Their sum is 32. Divide by the count 5: 32 / 5 = 6.4. The headline is that mean absolute deviation, 6.4.
The median of these values is 85, and it was not used as the center. The gaps were not squared. A sample standard deviation would be a different number, 8.3367.
| Mean absolute deviation | 6.4 |
|---|---|
| Mean | 84 |
| Count | 5 |
A second situation
93 becomes 140, which pulls the mean and the absolute distances
The values are 72, 80, 85, 90, 140. The arithmetic mean is 93.4. Distances are measured from this mean, not from the median. The absolute distances are 21.4, 13.4, 8.4, 3.4, 46.6.
Their sum is 93.2. Divide by the count 5: 93.2 / 5 = 18.64. The headline is that mean absolute deviation, 18.64.
The median of these values is 85, and it was not used as the center. The gaps were not squared. A sample standard deviation would be a different number, 26.8849.
| Mean absolute deviation | 18.64 |
|---|---|
| Mean | 93.4 |
| Count | 5 |
Replacing 93 with 140 moves the mean from 84 to 93.4 and raises the mean absolute deviation from 6.4 to 18.64. The new point pulls the center and then still sits far from that new center. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
93 was replaced by 140. The headline result went from 6.4 to 18.64.
The nudged inputs
The values are 72, 80, 85, 90, 140. The arithmetic mean is 93.4. Distances are measured from this mean, not from the median. The absolute distances are 21.4, 13.4, 8.4, 3.4, 46.6.
Their sum is 93.2. Divide by the count 5: 93.2 / 5 = 18.64. The headline is that mean absolute deviation, 18.64.
The median of these values is 85, and it was not used as the center. The gaps were not squared. A sample standard deviation would be a different number, 26.8849.
| Mean absolute deviation | 18.64 |
|---|---|
| Mean | 93.4 |
| Count | 5 |
How to read the result
A mean absolute deviation of 6.4 points says the average score sits 6.4 points away from 84, ignoring sign. It is larger than a typical description might suggest if you only looked at the middle three scores, because 72 contributes a 12-point distance. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The center is the arithmetic mean of this list.
- Every typed score should contribute its absolute distance once.
- The divisor is n, the count of distances.
- You do not want the median absolute deviation.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Measuring the distances from the median and still using this name.
- Squaring the distances and calling the root a mean absolute deviation.
- Dividing by n minus 1 as if this were a sample standard deviation.
- Dropping the largest distance before averaging, which is a trim this page does not do.
Where a different calculator is the honest choice
Use the median absolute deviation calculator when the center and the summary should both be medians. Use the mean calculator for the center alone. Use the standard deviation calculator when the gaps should be squared. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Median Absolute Deviation Calculator, Mean Calculator, Standard Deviation Calculator.
Check yourself
Scores are 72, 80, 85, 90, and 93. What is the mean absolute deviation from the mean?
The mean absolute deviation is 6.4 points.
Using the result in a sentence
Forecast errors, score spreads you want in the original unit, and a check against a squared deviation all use an average absolute distance from the mean. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Because the mean itself moves when a tail score moves, the distances are not independent of that score. A robust scale belongs on the median absolute deviation page. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Mean Absolute Deviation Calculator commit to?
This page uses Mean absolute deviation from the mean: mean absolute deviation = (sum of |x - mean|) / n. Absolute value stops distances above and below the mean from cancelling. Dividing by n, not by n minus 1, treats the result as an average distance in the batch you typed. No square and no median enters the formula. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numbers, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the mean absolute deviation is 6.4. A mean absolute deviation of 6.4 points says the average score sits 6.4 points away from 84, ignoring sign. It is larger than a typical description might suggest if you only looked at the middle three scores, because 72 contributes a 12-point distance. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Measuring the distances from the median and still using this name. Because the mean itself moves when a tail score moves, the distances are not independent of that score. A robust scale belongs on the median absolute deviation page. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the median absolute deviation calculator when the center and the summary should both be medians. Use the mean calculator for the center alone. Use the standard deviation calculator when the gaps should be squared. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The mean absolute deviation is 6.4 points. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.