The Sample Median Absolute Deviation

In the box below, please enter the sample median absolute deviation of the data given above, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

You got the correct answer of 29.6516. Congratulations!

Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.

Show Formula

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The formula for the median absolute deviation is

In this formula, there are a couple symbols to know:

The formula is nice, as it compactly gives the directions for calculating the sample median absolute deviation. Here are the steps it describes:

1. Calculate the sample median of the data, \( \tilde{x} \)
2. Calculate the difference between each data value and the sample median, \( x_i - \tilde{x} \)
3. Take the absolute value of those differences, \( \left| x_i - \tilde{x} \right| \)
4. Calculate the median of those absolute values, \( \text{median} \left( \left| x_i - \tilde{x} \right| \right) \)
5. Finally, divide that median by 0.6745, \( \text{median} \left( \left| x_i - \tilde{x} \right| \right)/0.6745 \)

And here is an interesting question: Why 0.6745?

The divisor was selected to ensure that the median absolute deviation is an unbiased estimator of the standard deviation when the data are generated from a Normal distribution.

Show Solution

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The sample median absolute deviation is the median absolute deviation from the sample median.

1. Calculate the sample median of the data
2. Calculate the difference between each data value and the sample median
3. Take the absolute value of those differences
4. Calculate the median of those absolute values
5. Finally, divide that median by 0.6745

The table below shows the schema:

The table shows steps 2 and 3. Step 4 is to calculate the median of the final column: 20. The final step is to divide this median by 0.6745. Doing so gives the sample median absolute deviation as MAD = 29.6516.

Show the R Code

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Copy and paste the following code into your R script window, then run it from there.

In the R output, the sample median absolute deviation is the number output by the script.

Show the Excel Code

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Copy and paste the following code into your Excel spreadsheet window, making sure your cursor is in A1 when you paste.

Copy and paste the following code into your Excel spreadsheet window, making sure the value sample1 ends up in A1 after pasting.

Note that there is no MAD function in Excel. The only way to calculate the median absolute deviation is to perform the calculations described in the formula.