Key Concepts and Formulas

1. Median for Grouped Data: The median is a measure of central tendency that divides a dataset into two equal halves. For a grouped frequency distribution, we estimate the median using a specific formula, as we don't have individual data points.
2. Cumulative Frequency (CF): For a given class interval, its cumulative frequency is the sum of the frequencies of that class and all preceding classes. It helps locate the class interval where the median observation lies.
3. Median Formula for Grouped Data: The formula to calculate the median ($M$) for a grouped frequency distribution is: $M = l + \left( \frac{\frac{N}{2} - C}{f} \right) h$ Where:
   • $l$ = Lower limit of the median class.
   • $N$ = Total frequency (sum of all frequencies).
   • $C$ = Cumulative frequency of the class preceding the median class.
   • $f$ = Frequency of the median class.
   • $h$ = Class size (or class width) of the median class.

Step-by-Step Solution

To determine the median of the given frequency distribution, we will systematically follow these steps:

Step 1: Construct the Cumulative Frequency (CF) Table The first crucial step is to create a cumulative frequency table. This table helps us to quickly identify the class interval within which the median observation falls by showing the running total of observations up to the end of each class.

Explanation: Each cumulative frequency is obtained by adding the current class's frequency to the cumulative frequency of the previous class. For instance, the CF for the $8-12$ class is $10$ (its own frequency) plus $12$ (CF of the $4-8$ class) which totals $22$. This means 22 observations have a value less than or equal to 12.

Step 2: Calculate the Total Frequency ($N$) and Median Position ($N/2$) The total frequency ($N$) represents the total number of observations in the dataset. The median position, $N/2$, indicates the position of the middle observation when all data points are arranged in ascending order.

First, sum all the frequencies to find $N$: $N = \sum f_i = 3 + 9 + 10 + 8 + 6 = 36$ Next, calculate the median position: $\frac{N}{2} = \frac{36}{2} = 18$ Explanation: The median value is associated with the $18^{th}$ observation in this distribution. We now need to find which class interval contains this $18^{th}$ observation.

Step 3: Identify the Median Class The median class is the class interval whose cumulative frequency is just greater than or equal to the median position ($N/2$). We refer to our cumulative frequency table for this.

• The CF for class $0-4$ is $3$.
• The CF for class $4-8$ is $12$.
• The CF for class $8-12$ is $22$.

Since $22$ is the first cumulative frequency that is greater than or equal to $18$, the median class is $8-12$. Explanation: This means the $18^{th}$ observation falls somewhere within the range of $8$ to $12$.

Step 4: Extract Parameters for the Median Formula Now that the median class ($8-12$) has been identified, we can extract all the necessary values to substitute into the median formula:

• $l$ (lower limit of the median class) $= 8$
• $N$ (total frequency) $= 36$ (calculated in Step 2)
• $C$ (cumulative frequency of the class preceding the median class) $= 12$ (This is the CF of the $4-8$ class, which comes immediately before the $8-12$ class.)
• $f$ (frequency of the median class) $= 10$ (This is the frequency of the $8-12$ class itself.)
• $h$ (class size) $= 12 - 8 = 4$ (The difference between the upper and lower limits of the median class. All classes have a uniform width of 4.)

Explanation: Each parameter plays a specific role in adjusting the lower limit of the median class ($l$) to precisely locate the median value. For example, $(N/2 - C)$ tells us how many observations we need to count into the median class from its lower limit.

Step 5: Apply the Median Formula and Calculate M Substitute the extracted values into the median formula: $M = l + \left( \frac{\frac{N}{2} - C}{f} \right) h$ $M = 8 + \left( \frac{18 - 12}{10} \right) 4$ First, calculate the numerator inside the parenthesis: $M = 8 + \left( \frac{6}{10} \right) 4$ Next, perform the division: $M = 8 + (0.6) \times 4$ Then, the multiplication: $M = 8 + 2.4$ Finally, the addition: $M = 10.4$ Thus, the median ($M$) of the frequency distribution is $10.4$. Verification: The calculated median $10.4$ lies within our identified median class $8-12$, which confirms the correctness of our median class selection and calculations up to this point.

Step 6: Calculate the Final Required Value ($20M$) The question asks for the value of $20M$. $20M = 20 \times 10.4$ $20M = 208$

Common Mistakes & Tips

• Class Continuity: Always verify that the class intervals are continuous (upper limit of one class equals lower limit of the next). If they are not (e.g., $0-4, 5-9$), they must be adjusted to continuous classes (e.g., $0-4.5, 4.5-9.5$) before proceeding. In this problem, the classes are already continuous.
• Correct 'C' and 'f': A frequent error is confusing $C$ (cumulative frequency of the preceding class) with the cumulative frequency of the median class itself, or confusing $f$ (frequency of the median class) with the frequency of another class. Pay close attention to these definitions.
• Arithmetic Precision: Be meticulous with calculations, especially the fractions and decimals within the formula, to avoid simple arithmetic errors that can lead to an incorrect final answer.
• Median Verification: After calculating the median $M$, always check if it falls within the identified median class interval. If $M$ is outside this range, it indicates an error in identifying the median class or in the calculation.

Summary

To find the median for grouped data, we first construct a cumulative frequency table to determine the total number of observations ($N$) and the median position ($N/2$). This allows us to accurately identify the median class. Once the median class is known, we extract its specific parameters ($l, N, C, f, h$) and substitute them into the median formula. Following these steps, we calculated the median $M = 10.4$. Finally, we computed the required value $20M = 208$.

The final answer is $\boxed{208}$, which corresponds to option (C).