The question in the image asks:

(a) Define the A.M (Arithmetic Mean) for a grouped frequency distribution. Also, state its merits and demerits.

Definition:

The Arithmetic Mean (A.M.) for a grouped frequency distribution is calculated as the sum of the products of the midpoints of the class intervals and their corresponding frequencies, divided by the total frequency. Mathematically, it is expressed as:

$\text{A.M.} = \frac{\sum f_i x_i}{\sum f_i}$

Where:

• $f_i$ = frequency of the $i$-th class
• $x_i$ = midpoint of the $i$-th class
• $\sum f_i$ = total frequency

The midpoints ($x_i$) are obtained by averaging the upper and lower boundaries of each class.

Merits:

1. Simple to calculate: The arithmetic mean is straightforward and can be easily computed.
2. Easy to understand: It provides a clear measure of central tendency.
3. Uses all observations: It takes into account every value in the dataset.
4. Suitable for further statistical analysis: The A.M. is used in other statistical methods, like standard deviation, regression analysis, etc.

Demerits:

1. Sensitive to extreme values: The mean can be heavily affected by outliers, especially if the distribution is skewed.
2. Not always representative: In the case of a highly skewed distribution, the A.M. might not reflect a central value properly.
3. Requires grouped data: Grouping can lead to a loss of individual data detail.
4. Not appropriate for ordinal or categorical data: The A.M. is meaningful only for numerical data.

Would you like further details on any part of this, or have any questions?

Related Questions:

1. What is the difference between A.M. and median in a grouped frequency distribution?
2. How do outliers affect the arithmetic mean in grouped data?
3. Can the A.M. be used for non-numerical data? Why or why not?
4. How do you handle open-ended intervals when calculating the A.M.?
5. What are the different methods to compute the A.M. in a frequency distribution?

Tip:

When calculating the A.M. for grouped data, ensure the class intervals are clearly defined, and be cautious about how you handle any open-ended or unequal intervals.