Calculate modified Z scores from a list of data or from the median and MAD, flag outliers with the Iglewicz–Hoaglin 3.5 rule, and compare with IQR fences.

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Modified Z Score Formula

The following formula is used to calculate the modified Z score for a given observation.

Z = 0.6745 × (X - M) / MAD

Variables:

  • Z is the modified Z score
  • X is the observation
  • M is the median
  • MAD is the median absolute deviation

To calculate the modified Z score, subtract the median from the observation, divide the result by the median absolute deviation (MAD), and then multiply by 0.6745.

What is a Modified Z Score?

A modified Z score is a statistical measure that equals 0.6745 times the observation’s distance from the median divided by the median absolute deviation (MAD). Unlike the traditional Z score, which uses the mean and standard deviation, the modified Z score uses the median and MAD, making it more robust to outliers. This measure is particularly useful in identifying outliers in a dataset, because median and MAD are less sensitive to extreme observations. Potential-outlier flags require investigation, not automatic removal.

How to Calculate Modified Z Score?

The following steps outline how to calculate the Modified Z Score.

  1. First, determine the observation (X).
  2. Next, determine the median (M) of the dataset.
  3. Next, calculate the median absolute deviation (MAD).
  4. Finally, calculate the Modified Z Score using the formula Z = 0.6745 * (X – M) / MAD.
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Observation (X) = 10

Median (M) = 5

Median Absolute Deviation (MAD) = 2