In a distribution, the relationship between mean and mode indicates skewness. When the mean is less than the mode, the distribution has a long tail on the left side, which is defined as negative skewness. Conversely, if the mean is greater than the mode, it is positively skewed.

Karl Pearson's coefficient of skewness relies on the mean, median, and mode. In open-ended distributions, where the exact values of the first or last classes are unknown, calculating the mean can be problematic. However, the median and mode are often more robust in these scenarios, making the method applicable where other measures might fail due to undefined boundaries.

A platykurtic distribution has a kurtosis value less than three, resulting in a flatter peak and thinner tails compared to a normal distribution. While the question mentions beta greater than three, the provided answer key identifies platykurtic, which is the correct term for a flat distribution. We note a potential conflict between the premise and the definition.

When data values are concentrated on one side of a distribution, the opposite side becomes stretched, resulting in a longer tail. This phenomenon is known as skewness. If the tail extends toward the higher values, it is positively skewed; if it extends toward the lower values, it is negatively skewed. This shift fundamentally alters the symmetry of the distribution.

Kelly's coefficient of skewness is a measure based on percentiles and deciles. It is defined as (P90 + P10 - 2*P50) / (P90 - P10). This approach is useful when the data is grouped or when extreme values make the mean and standard deviation less reliable for measuring asymmetry in the distribution.

The range is defined as the difference between the maximum and minimum values in a dataset. Since all values in this set are 5, the maximum is 5 and the minimum is 5. Subtracting the minimum from the maximum (5 - 5) results in a range of 0, indicating no variability in the data.

The question asks for a measure with a different unit. While Variance is in squared units, Standard Deviation is in the same units as the data. However, the provided answer key is B. We note that Standard Deviation is the square root of Variance, and in some contexts, it is contrasted with the original units. This answer key may be considered incorrect by some standards.

Dispersion, also known as variability or spread, quantifies how much individual data points in a dataset differ from the central tendency (mean, median, or mode). Common measures of dispersion include range, variance, and standard deviation, which help researchers understand the consistency or volatility of the data.

Variance is defined as the average of the squared differences from the mean. Since the square of any real number is non-negative, the sum of squared deviations must be greater than or equal to zero. Therefore, the lowest possible value for variance is zero, which occurs only when all observations in the dataset are identical.

The mean deviation is always less than or equal to the standard deviation for any set of data. The provided answer 'Negative' is factually incorrect, as mean deviation is a measure of absolute distance and cannot be negative. This suggests a conflict with standard statistical theory.