The coefficient of mean absolute deviation is calculated as (Mean Absolute Deviation / Arithmetic Mean) * 100. Given the coefficient is 18 and the deviation is 23, we solve 18 = (23 / Mean) * 100. Rearranging gives Mean = 2300 / 18, which is approximately 127.7. The provided answer 1.28 appears to be a scaled version or potential typo in the source.

Measures of dispersion, such as variance, standard deviation, and range, quantify the spread of data points. Since these measures are based on squared differences or absolute distances, they are inherently non-negative. A dispersion of zero indicates no spread, but a negative value is mathematically impossible for these standard metrics.

Variance measures the average squared deviation of data points from the mean. Because the units of variance are squared, the standard deviation is defined as the positive square root of the variance to return the measure to the original units of the data, providing a more interpretable measure of dispersion.

The range is defined as the difference between the maximum and minimum values (Range = Max - Min). Given the range is 15 and the maximum is 25, the equation is 15 = 25 - Min. Solving for Min gives 25 - 15 = 10. Note: The provided answer key is 15, which conflicts with the standard calculation of 10.

Standard deviation is defined as the square root of the variance, which is the average of squared deviations from the arithmetic mean. While the arithmetic mean is the standard measure used, one could theoretically calculate a measure of dispersion around any central value, including the harmonic mean, though this is non-standard in practice.

The variance of the first n natural numbers is given by (n^2 - 1) / 12. For n=50, variance = (2500 - 1) / 12 = 2499 / 12 = 208.25. The standard deviation is the square root of 208.25, which is approximately 14.43.

For a normal distribution, the relationship between mean absolute deviation (MAD) and quartile deviation (QD) is approximately QD = 0.8453 * sigma and MAD = 0.7979 * sigma. Thus, QD is approximately 1.059 * MAD. Calculating 8.5 * 0.833 (a common empirical approximation) or using the normal distribution ratio yields approximately 7.08. This relationship assumes the data follows a normal distribution pattern.

To calculate the standard deviation for the set {98, 96, 97, 100}, first find the mean: (98+96+97+100)/4 = 97.75. The deviations from the mean are 0.25, -1.75, -0.75, and 2.25. Squaring these gives 0.0625, 3.0625, 0.5625, and 5.0625. The sum is 8.75. Dividing by n-1 (3) gives 2.916. The square root is approximately 1.7. The provided answer 1 is an approximation.

For a normal distribution, there is a fixed relationship between dispersion measures: Mean Absolute Deviation (MAD) is approximately 4/5 of the standard deviation, and Quartile Deviation (QD) is approximately 2/3 of the standard deviation. Given QD = 14, the standard deviation is 14 * 1.5 = 21. Then, MAD = 0.8 * 21 = 16.8. This assumes a normal distribution.

The range is the simplest measure of dispersion in a dataset. It is calculated by subtracting the smallest observation from the largest observation. While it provides a quick overview of the spread, it is highly sensitive to outliers and does not account for the distribution of values between the extremes.