Quartile deviation is based on the first and third quartiles, which are positional measures. Because it ignores the extreme values at the tails of a distribution, it is the preferred measure of dispersion when dealing with open-ended class intervals where the exact values of the extremes are unknown.

To calculate the standard deviation, first find the mean: (2+4+6+8+10+12)/6 = 7. Then, calculate the squared deviations from the mean: (-5)^2, (-3)^2, (-1)^2, (1)^2, (3)^2, (5)^2, which are 25, 9, 1, 1, 9, 25. The sum is 70. Dividing by n-1 (5) gives 14. The square root of 14 is approximately 3.74. Note: The provided answer 3.42 may assume population standard deviation or a different divisor.

In a normal distribution, the quartile deviation (Q.D.) is approximately 2/3 of the standard deviation, and the mean absolute deviation (M.A.D.) is approximately 4/5 of the standard deviation. By comparing these ratios, the quartile deviation is approximately 5/6 of the mean absolute deviation. This relationship is a standard approximation used in descriptive statistics to compare measures of dispersion.

To calculate the sample variance, first find the mean: (12+15+18+16+14)/5 = 15. Then, calculate the sum of squared deviations from the mean: (12-15)^2 + (15-15)^2 + (18-15)^2 + (16-15)^2 + (14-15)^2 = 9 + 0 + 9 + 1 + 1 = 20. Finally, divide by (n-1), which is 4: 20/4 = 5. Thus, the sample variance is 5.

Measures of dispersion, such as range, variance, and standard deviation, quantify the spread of data. Because they are based on squared differences or absolute differences, they are always non-negative. Therefore, a measure of dispersion can never be negative.

Variance is defined as the average of the squared deviations of individual observations from their arithmetic mean. It serves as a primary measure of variability in a dataset, quantifying how far the data points are spread out from the average value.

To calculate the sum of squared deviations from x=4, we compute (2-4)^2 + (3-4)^2 + (4-4)^2 + (5-4)^2 + (6-4)^2. This simplifies to (-2)^2 + (-1)^2 + (0)^2 + (1)^2 + (2)^2, which equals 4 + 1 + 0 + 1 + 4 = 10. This measure represents the dispersion of the data points relative to the specified value.

The range is calculated by subtracting the minimum value from the maximum value. In this dataset, the maximum value is 60 and the minimum value is 30. Therefore, 60 - 30 = 30. This range is used to determine the spread for creating frequency distribution intervals.

Absolute measures of dispersion, such as the standard deviation, variance, and range, are expressed in the same units as the data. In contrast, relative measures of dispersion, like the coefficient of variation, are unitless ratios. Absolute measures provide a direct sense of the magnitude of spread within the dataset, whereas relative measures are used to compare variability across different datasets with different units or means.

Standard deviation is defined as the square root of the variance. Since variance is the average of squared deviations, it is always non-negative. Consequently, the standard deviation is defined as the positive square root, ensuring it is always positive or zero.