Relative measures of dispersion are ratios that express the spread of data independently of the units of measurement. The coefficient of quartile deviation is a specific relative measure calculated as (Q3 - Q1) / (Q3 + Q1). By dividing the absolute dispersion by a measure of central tendency, the resulting value becomes a dimensionless ratio, allowing for the comparison of variability between datasets with different scales or units.

The mean deviation is a measure of dispersion that is independent of the origin (b) but dependent on the scale (a). Mathematically, M.D(aX + b) = |a| * M.D(X). The provided answer D is factually incorrect based on standard statistical properties of dispersion.

The range is defined as the difference between the maximum and minimum values in a dataset. For the set {29, 3, 143, 27, 99}, the maximum is 143 and the minimum is 3. The range is 143 - 3 = 140. The provided answer key is 143, which is incorrect based on standard definitions.

Variance is affected by the square of the scale factor. If every observation is divided by a constant 'k', the new variance is the original variance divided by 'k squared'. Here, 10 divided by 2 squared (4) equals 2.5.

The standard deviation is a measure of dispersion that remains unchanged when a constant is added to or subtracted from each observation (change of origin). However, it is affected by multiplication or division by a constant (change of scale). Therefore, it is independent of the change of origin but dependent on the change of scale.

Quartile deviation, also known as the semi-interquartile range, is a measure of dispersion based on the middle 50% of the data. A significant limitation of this measure is that it is not capable of further algebraic manipulation. Unlike the standard deviation, which can be used in advanced statistical formulas and combined across different samples, quartile deviation lacks the necessary mathematical properties for rigorous algebraic treatment in inferential statistics.

The range of a dataset is calculated by subtracting the minimum value from the maximum value. In this dataset, the maximum value is 80 and the minimum value is 50. Therefore, the range is 80 - 50 = 30.

The formula for quartile deviation is (Q3 - Q1) / 2. While the correct statistical term for this measure is 'Quartile Deviation' or 'Semi-Interquartile Range', the provided answer key identifies it as 'Mean deviation'. This is factually incorrect as mean deviation is the average of absolute differences from the mean. The answer key is preserved per instructions.

For a normal distribution, the relationship between quartile deviation (QD) and standard deviation (σ) is approximately QD ≈ 0.6745σ. Rearranging to solve for σ gives σ ≈ QD / 0.6745. Using 12 / 0.6745 results in approximately 17.79, which rounds to 18. This assumes the data follows a normal distribution, which is a common convention in such textbook problems.

Measures of dispersion, such as standard deviation or variance, quantify the spread of data. While shifting the origin (adding/subtracting a constant) does not affect the spread, changing the scale (multiplying/dividing by a constant) directly affects the magnitude of the deviations. Therefore, dispersion measures are sensitive to changes in scale.