Questions: Compute the Mean, Median, and Mode of the following grouped frequency table. Classes 10 - 20 20 - 30 30 - 40 Mean Frequency 8 11 9 Median Mode

Solution Steps

To compute the Mean, Median, and Mode from a grouped frequency table, follow these steps:

1. Mean: Calculate the midpoint for each class interval, multiply each midpoint by its corresponding frequency, sum these products, and divide by the total frequency.
2. Median: Identify the median class, which is the class where the cumulative frequency reaches or exceeds half of the total frequency. Use the median formula for grouped data to find the median.
3. Mode: Identify the modal class, which is the class with the highest frequency. Use the mode formula for grouped data to find the mode.

To find the mean of the grouped data, we use the formula:

\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

where \( f_i \) is the frequency and \( x_i \) is the midpoint of each class interval. The midpoints are calculated as follows:

• For \( 10 - 20 \): \( x_1 = \frac{10 + 20}{2} = 15 \)
• For \( 20 - 30 \): \( x_2 = \frac{20 + 30}{2} = 25 \)
• For \( 30 - 40 \): \( x_3 = \frac{30 + 40}{2} = 35 \)

The mean is then calculated as:

\[ \text{Mean} = \frac{(8 \cdot 15) + (11 \cdot 25) + (9 \cdot 35)}{8 + 11 + 9} = \frac{203}{28} \approx 25.36 \]

The median is found using the formula for grouped data:

\[ \text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \cdot h \]

where:

• \( L \) is the lower boundary of the median class,
• \( N \) is the total frequency,
• \( F \) is the cumulative frequency before the median class,
• \( f \) is the frequency of the median class,
• \( h \) is the class width.

The cumulative frequencies are \( [8, 19, 28] \), and \( \frac{N}{2} = 14 \). The median class is \( 20 - 30 \) because the cumulative frequency first exceeds 14 in this class. Thus:

• \( L = 20 \)
• \( F = 8 \)
• \( f = 11 \)
• \( h = 10 \)

The median is:

\[ \text{Median} = 20 + \left(\frac{14 - 8}{11}\right) \cdot 10 \approx 25.45 \]

The mode is found using the formula for grouped data:

\[ \text{Mode} = L + \left(\frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)}\right) \cdot h \]

where:

• \( L \) is the lower boundary of the modal class,
• \( f_1 \) is the frequency of the modal class,
• \( f_0 \) is the frequency of the class before the modal class,
• \( f_2 \) is the frequency of the class after the modal class,
• \( h \) is the class width.

The modal class is \( 20 - 30 \) because it has the highest frequency. Thus:

• \( L = 20 \)
• \( f_1 = 11 \)
• \( f_0 = 8 \)
• \( f_2 = 9 \)
• \( h = 10 \)

The mode is:

\[ \text{Mode} = 20 + \left(\frac{11 - 8}{(11 - 8) + (11 - 9)}\right) \cdot 10 = 26.0 \]

Final Answer