By definition, the standard deviation is calculated as the square root of the variance, which is the average of squared deviations from the arithmetic mean. Using the Harmonic Mean as the reference point for calculating deviations would not result in the standard deviation, as the arithmetic mean is the unique value that minimizes the sum of squared deviations.

Measures of dispersion, such as variance, standard deviation, and range, are calculated using squared differences or absolute differences between data points. Because these operations yield non-negative results, measures of dispersion are mathematically constrained to be non-negative. In practice, they are typically positive unless all data points are identical, in which case the dispersion is zero.

In statistics, the variance is defined as the square of the standard deviation. Given that the standard deviation (σ) is 9, the variance (σ²) is calculated by squaring 9, which equals 81. This measure quantifies the spread of data points around the mean.

The coefficient of range is (H-L)/(H+L). Given H-L = 20 and the coefficient = 0.077, we have 20 / (H+L) = 0.077. Solving for (H+L) gives 20 / 0.077, which is approximately 259.74, rounding to 260.

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

The sample variance is defined as the sum of squared deviations from the mean divided by (n-1). Assuming the total sum of squares refers to the sum of squared deviations from the mean, we have 5 = 20 / (n-1). Solving for n-1 gives 4, which implies n = 5. However, if the question implies variance as the mean squared deviation (sum of squares / n), then 5 = 20 / n, resulting in n = 4.

Quartile deviation measures the spread of the middle 50% of the data. A larger quartile deviation indicates that the middle 50% of the observations are more spread out, which signifies less uniformity or higher dispersion around the median.

The variance is defined as the average of the squared differences from the mean. It measures the spread of a set of data points around their mean value. By squaring the deviations, we ensure that negative and positive differences do not cancel each other out, providing a non-negative measure of dispersion.

When all values in a dataset are identical, there is no dispersion or spread. Consequently, the mean deviation, variance, and standard deviation are all equal to zero, as every data point is exactly equal to the mean of the set.

For grouped data, the range is determined by subtracting the lower class boundary of the first class from the upper class boundary of the last class. Using boundaries instead of limits ensures that the entire span of the data is accounted for, including the gaps between classes.