For the calculation of Arithmetic Mean the Class Intervals should be :
(i) Exclusive
(ii) Unequal
(iii) Equal

Question

For the calculation of Arithmetic Mean the Class Intervals should be : 
(i) Exclusive 
(ii) Unequal 
(iii) Equal

Accepted Answer

The correct option is 4:
The calculation of the Arithmetic Mean (Mean) for grouped data relies on finding the midpoint (class mark) of each class interval. Mathematically, the calculation method is robust and does not depend on the specific structural properties of the class intervals—whether they are exclusive, unequal, or equal.1.  **Exclusive Classes (i)**: Exclusive intervals (e.g., 0-10, 10-20) are common because they prevent overlapping and ambiguity regarding which class a boundary value belongs to. However, they are not strictly necessary for calculating the mean, as inclusive series can be used after adjustment.
2.  **Unequal Classes (ii)**: If the class intervals are unequal in width (e.g., 0-10, 10-30, 30-35), the arithmetic mean formula remains fully applicable. The midpoints of these unequal classes are calculated and used in the same weighted average process as equal classes.
3.  **Equal Classes (iii)**: Equal classes (e.g., all classes having a width of 10) simplify the process and are preferred for ease of data representation and other statistical measures like the mode, but they are not a prerequisite for mean calculation.Therefore, since the calculation of the mean only requires determining the class mark for each interval, all three characteristics—exclusive, unequal, or equal—are mathematically possible and valid, making option 4 correct.

The incorrect option is 1:
Both (i) and (ii) are possible, but this option incorrectly excludes the possibility of using equal class intervals. Equal class intervals are the most typical and desirable form of grouping for data analysis, and the mean can certainly be calculated with them.

The incorrect option is 2:
Both (i) and (iii) are possible, but this option incorrectly excludes the possibility of using unequal class intervals. While unequal intervals can make other analyses challenging, they do not impede the calculation of the mean, as the midpoints can always be accurately determined regardless of the class width.

The incorrect option is 3:
Only (i) is possible suggests that only exclusive classes are valid. This is false. While exclusive classes are preferred for clarity, the mean can also be calculated from inclusive class intervals after necessary adjustments. More importantly, this option ignores the possibility of both equal and unequal class widths, which are mathematically permissible.