Index numbers are statistical tools used to measure changes in a variable or group of variables over time or space. They are widely applied to track fluctuations in commodity prices, calculate the cost of living (such as the Consumer Price Index), and monitor international trade volumes like exports and imports.

The arithmetic mean is defined as the sum of all values in a dataset divided by the total number of observations. Because it utilizes the specific numerical value of every data point, it is sensitive to every observation, unlike the median or mode, which focus on positional or frequency-based characteristics.

The median is a positional measure of central tendency. It is defined as the middle value of a sorted dataset. Because it depends only on the order of the values rather than their specific magnitudes, it remains unchanged if the smallest or largest values are altered, provided their relative position to the center does not change.

Using the formula for the arithmetic mean with assumed mean: Mean = A + (Σfd / N) * h. Here, the provided values suggest a calculation where 120 = 22 + (180/N) * 5. Solving for N: 98 = 900/N, which leads to N approximately 9.18. Given the options, there is a potential conflict in the provided parameters or the formula application. We retain the provided answer.

The formula Σ(p0qn) / Σ(p0q0) × 100 represents the Laspeyres quantity index. It measures the change in quantities by using base-period prices (p0) as weights. This index compares the total value of current quantities at base prices to the total value of base quantities at base prices.

When 10% is added to each value, the geometric mean also increases by 10%. This is because the geometric mean is the nth root of the product of n values. Adding 10% to each value is equivalent to multiplying each value by 1.1, which in turn multiplies the product of the values by 1.1^n, resulting in a geometric mean increase of 10%.

The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the observations. For two numbers a and b, the reciprocals are 1/a and 1/b. Their arithmetic mean is ((1/a) + (1/b)) / 2 = (a+b)/(2ab). Taking the reciprocal of this result yields 2ab / (a + b).

The Geometric Mean (GM) involves the product of all observations; if one value is zero, the product becomes zero, rendering the mean zero or undefined. Similarly, the Harmonic Mean (HM) involves the reciprocal of the values; if a value is zero, the reciprocal is undefined (division by zero). Thus, both GM and HM are problematic with zero values, whereas the Arithmetic Mean (AM) remains calculable.

A relative measure of dispersion is unitless and allows for the comparison of variability between datasets with different units or means. The Coefficient of Variation (CV) is defined as the ratio of the standard deviation to the mean. In contrast, standard deviation and variance are absolute measures of dispersion because they are expressed in the same units as the original data (or squared units).

For a normal distribution, the mean deviation is approximately 0.8 times the standard deviation, which is equivalent to the fraction 4/5. This relationship helps in understanding the relative spread of data using different measures of dispersion.