The mean absolute deviation is zero when: 

Key Points

• The mean absolute deviation (MAD) is a measure of dispersion that calculates the average absolute difference between each data point and the mean of the dataset.
• Among the alternatives provided, the mean absolute deviation (MAD) is zero when all observations are equal.
• In a dataset where all observations are equal, the absolute difference between each data point and the mean will be zero for every observation. When calculating the average of these zero differences, the result will also be zero.

Additional Information

• When all observations are different from each other:
  • In this scenario, the MAD will not be zero. Since all observations are different, there will be non-zero absolute differences between each data point and the mean. When averaged, these non-zero differences will result in a non-zero MAD.
• When few observations are equal to one and few are equal to two:
  • In this scenario, the MAD will not be zero either. As there are different values present in the dataset (one and two), the absolute differences between each data point and the mean will be non-zero. Averaging these non-zero differences will lead to a non-zero MAD.
• When few observations are zero and few are non-zero:
  • Similar to the previous scenarios, the MAD will not be zero in this case. Since there are both zero and non-zero values in the dataset, the absolute differences between each data point and the mean will be non-zero. Computing the average of these non-zero differences will yield a non-zero MAD.
• When all observations are equal:
  • When all observations in a dataset are equal, the absolute difference between each data point and the mean will be zero for every observation. Consequently, when averaging these zero differences, the result will be zero. Thus, in this scenario, the MAD will be zero.

Hence, when all observations in a dataset are equal, the mean absolute deviation (MAD) will be zero.