Descriptive Statistics
Median Absolute Deviation Calculator
Median absolute deviation is the median of the distances from the median, and the page multiplies it by 1.4826 for a normal-consistent scale.
Worksheet
Press Calculate to see the result.
Median absolute deviation is the median of the distances from the median, and the page multiplies it by 1.4826 for a normal-consistent scale.
What this calculator answers
Scores 72, 80, 85, 90, and 93 have a median of 85. The absolute distances from 85 are 13, 5, 0, 5, and 8, and the median of those distances is 5. That 5 is the median absolute deviation. It does not square the gaps, and it does not start from the mean, so one distant score has to cross the middle of the distances before it moves the result. Multiplying by 1.4826 produces a scale that matches the standard deviation when the data really are normal. The mean absolute deviation page is a different formula and will not match this number.
The headline is the median absolute deviation from the median. The second figure is 1.4826 times that deviation, a consistent scale for a normal curve. 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
Median absolute deviation uses this relationship.
MAD = median of |x - median|. Consistent scale = 1.4826 * MAD.
- median. Center the distances are measured from.
- |x - median|. Absolute distance of one observation from that center.
- MAD. Median of those absolute distances.
- 1.4826. Factor that lines MAD up with a normal standard deviation.
Using the median twice, once as the center and once as the summary of the distances, keeps a single extreme from owning the scale. The factor 1.4826 is 1 over the inverse normal of 0.75, so that a normal population's MAD estimates its standard deviation. 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 median, which is the center for these distances, is 85. The absolute distances from 85 are 13, 5, 0, 5, 8.
The median of those distances is 5. That median of absolute deviations is the headline. The mean was not used as the center. Multiply by 1.4826 to put the scale on a normal-curve footing: 1.4826 * 5 = 7.413.
The sample standard deviation of this list is a different number, 8.3367, because it squares gaps from the mean. A single extreme moves this MAD only if it changes the median or the middle absolute distance.
| Median absolute deviation | 5 |
|---|---|
| Consistent scale 1.4826 times MAD | 7.413 |
| Median | 85 |
A second situation
The scores spread out to 60, 70, 85, 100, and 110
The values are 60, 70, 85, 100, 110. The median, which is the center for these distances, is 85. The absolute distances from 85 are 25, 15, 0, 15, 25.
The median of those distances is 15. That median of absolute deviations is the headline. The mean was not used as the center. Multiply by 1.4826 to put the scale on a normal-curve footing: 1.4826 * 15 = 22.239.
The sample standard deviation of this list is a different number, 20.6155, because it squares gaps from the mean. A single extreme moves this MAD only if it changes the median or the middle absolute distance.
| Median absolute deviation | 15 |
|---|---|
| Consistent scale 1.4826 times MAD | 22.239 |
| Median | 85 |
The wider scores 60, 70, 85, 100, and 110 still have median 85, but the median of the absolute distances rises from 5 to 15. The consistent scale rises from 7.413 to 22.239. 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
The batch was replaced by 60, 70, 85, 100, and 110. The headline result went from 5 to 15.
The nudged inputs
The values are 60, 70, 85, 100, 110. The median, which is the center for these distances, is 85. The absolute distances from 85 are 25, 15, 0, 15, 25.
The median of those distances is 15. That median of absolute deviations is the headline. The mean was not used as the center. Multiply by 1.4826 to put the scale on a normal-curve footing: 1.4826 * 15 = 22.239.
The sample standard deviation of this list is a different number, 20.6155, because it squares gaps from the mean. A single extreme moves this MAD only if it changes the median or the middle absolute distance.
| Median absolute deviation | 15 |
|---|---|
| Consistent scale 1.4826 times MAD | 22.239 |
| Median | 85 |
How to read the result
A MAD of 5 points says the middle distance from the median is 5 points. The scaled figure 7.413 is the one to compare with a normal-theory standard deviation, and it is still not the sample standard deviation of these five scores. 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
- You want resistance to a long tail.
- The center of the distances is the median, not the mean.
- The 1.4826 factor is relevant only as a normal-theory scale.
- Ties in the distances are allowed.
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
- Averaging the absolute distances from the mean and calling that a MAD.
- Comparing raw MAD with a standard deviation without the 1.4826 factor.
- Using the mean as the center because it is more familiar.
- Expecting MAD to notice a single extreme that does not cross the middle distance.
Where a different calculator is the honest choice
Use the mean absolute deviation calculator when the center must be the mean. Use the median calculator when you only need the center. Use the standard deviation calculator when squared gaps are the definition you want. 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: Mean Absolute Deviation Calculator, Median Calculator, Standard Deviation Calculator.
Check yourself
Scores are 72, 80, 85, 90, and 93. What is the median absolute deviation?
The median absolute deviation is 5 and the consistent scale is 7.413.
Using the result in a sentence
Robust summaries of scores, assay replicates with an occasional spike, and any batch where one bad point should not set the scale. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
MAD ignores how far the farthest points are, once they sit beyond the middle distance. That resistance is the feature, and it is also why a quality problem in the tail can hide. 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 Median Absolute Deviation Calculator commit to?
This page uses Median absolute deviation: MAD = median of |x - median|. Consistent scale = 1.4826 * MAD.. Using the median twice, once as the center and once as the summary of the distances, keeps a single extreme from owning the scale. The factor 1.4826 is 1 over the inverse normal of 0.75, so that a normal population's MAD estimates its standard deviation. 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 median absolute deviation is 5. A MAD of 5 points says the middle distance from the median is 5 points. The scaled figure 7.413 is the one to compare with a normal-theory standard deviation, and it is still not the sample standard deviation of these five scores. 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?
Averaging the absolute distances from the mean and calling that a MAD. MAD ignores how far the farthest points are, once they sit beyond the middle distance. That resistance is the feature, and it is also why a quality problem in the tail can hide. 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 mean absolute deviation calculator when the center must be the mean. Use the median calculator when you only need the center. Use the standard deviation calculator when squared gaps are the definition you want. 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 median absolute deviation is 5 and the consistent scale is 7.413. 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.