Which one of the following statements is incorrect?

The correct answer is The sum of absolute deviations taken from Median is the largest.

Key PointsThe algebraic 

Algebraic sum of deviations taken from Arithmetic Mean is zero:
This statement is true. The algebraic sum of deviations from the arithmetic mean is always zero. This means that the sum of the positive deviations is equal to the sum of the negative deviations, resulting in a net sum of zero.

The sum of absolute deviations taken from Median is the largest:
This statement is false. The sum of absolute deviations taken from the median is not necessarily the largest. The sum of absolute deviations from the mean is always smaller or equal to the sum of absolute deviations from the median. The median minimizes the sum of absolute deviations, making it smaller or equal to the sum of absolute deviations from any other point.

Standard deviation is always less than the range:
This statement is true. The range is the difference between the maximum and minimum values in a dataset. The standard deviation measures the dispersion of the data around the mean. Depending on the distribution of the data, the standard deviation can be smaller or larger than the range. For example, in a dataset with a large range and relatively small variability around the mean, the standard deviation can be smaller than the range.

The sum of the squares of deviations of a given set of observations is minimum when taken about the arithmetic mean:
This statement is true. The sum of the squares of deviations is minimized when taken about the arithmetic mean. This is the basis for calculating the variance, which is the average of the squared deviations from the mean. The arithmetic mean provides the best estimate for minimizing the sum of the squares of deviations in a dataset.

In summary:

• The algebraic sum of deviations from the arithmetic mean is zero.
• The sum of absolute deviations from the median is not necessarily the largest.
• The standard deviation can be either smaller or larger than the range.
• The sum of the squares of deviations is minimized when taken about the arithmetic mean.