Calculating the Mean Gain in Weight

The question asks us to find the mean value of the gain in weight for 25 children over a year, based on the provided frequency table. The mean for a frequency distribution is calculated using the formula:

\[ \text{Mean} (\bar{x}) = \frac{\sum (f_i \times x_i)}{\sum f_i} \]

Where:

• \(x_i\) represents the gain in weight for each category.
• \(f_i\) represents the number of children (frequency) for each gain in weight category.
• \(\sum (f_i \times x_i)\) is the sum of the products of each gain in weight value and its corresponding frequency.
• \(\sum f_i\) is the sum of all frequencies, which represents the total number of children.

Understanding the Gain in Weight Data

The provided table data needs careful interpretation based on the layout and the total number of children mentioned (25). The values appear to be listed sequentially as (gain in weight, number of children):

• 1.5 kg gain for 4 children
• 2 kg gain for 5 children (from '25')
• 2.4 kg gain for 8 children
• 3 kg gain for 5 children (from '35')
• 3.2 kg gain for 2 children
• 3.4 kg gain for 1 child

Let's verify the total number of children by summing the frequencies: \(4 + 5 + 8 + 5 + 2 + 1 = 25\). This matches the total number of children mentioned in the question.

Organizing Data for Mean Calculation

We can create a table to list the gain in weight (\(x_i\)), the number of children (\(f_i\)), and the product (\(f_i \times x_i\)).

Calculating the Mean Gain in Weight

Now we can substitute the totals into the mean formula:

\[ \text{Mean} (\bar{x}) = \frac{\sum (f_i \times x_i)}{\sum f_i} = \frac{60.0}{25} \]

Performing the division:

\[ \bar{x} = \frac{60}{25} = \frac{12 \times 5}{5 \times 5} = \frac{12}{5} = 2.4 \]

The mean value of the gain in weight is 2.4 kg.

Final Answer Confirmation

The calculated mean gain in weight is 2.4 kg, which corresponds to one of the given options.

Revision Table: Mean Calculation for Frequency Data

Additional Information: Applications of Mean

The mean is a widely used measure of central tendency in statistics. It provides a single value that represents the typical or average value of a dataset. In the context of this problem, the mean gain in weight of 2.4 kg gives us a single figure representing the average weight gain across the group of 25 children.

Understanding how to calculate the mean from a frequency distribution is important because data is often presented in this summarized form, especially when dealing with large datasets. This method is applicable in various fields, including:

• Education (average test scores)
• Economics (average income)
• Biology (average height, weight, etc., of a population)
• Market Research (average customer spending)

While the mean is useful, it can be affected by extreme values (outliers). Other measures of central tendency, such as the median and mode, might be considered depending on the nature of the data and the required analysis.