Question

Find the mean deviation of the following data: 

• A. 2.5
• B. 3
• C. 4
• D. 5

Hint:

To calculate the mean deviation, first compute the mean of the data, then calculate the absolute deviations from the mean for each data point. Multiply the deviations by the corresponding frequencies and find the sum. Divide the sum by the total number of data points (sum of the frequencies).

Answer 1

The Correct Option is B

We are given the data in a frequency table format where: - \( x \) represents the data points, - \( f \) represents the frequency corresponding to each data point. The mean deviation is calculated using the formula: \[ \text{Mean Deviation} = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i} \] Where: - \( f_i \) is the frequency of the \( i^{th} \) data point, - \( x_i \) is the \( i^{th} \) data point, - \( \bar{x} \) is the mean of the data. ### Step 1: Find the Mean \( \bar{x} \) First, we need to calculate the mean \( \bar{x} \) of the given data. The formula for the mean is: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] Substitute the values from the table: \[ \sum f_i x_i = 2 \times 7 + 4 \times 4 + 6 \times 5 + 10 \times 4 = 14 + 16 + 30 + 40 = 100 \] \[ \sum f_i = 7 + 4 + 5 + 4 = 20 \] Thus, the mean is: \[ \bar{x} = \frac{100}{20} = 5 \] ### Step 2: Calculate the Absolute Deviations Now, we calculate the absolute deviations from the mean \( \bar{x} = 5 \) for each data point \( x_i \). \[ |x_1 - \bar{x}| = |2 - 5| = 3, \quad |x_2 - \bar{x}| = |4 - 5| = 1, \quad |x_3 - \bar{x}| = |6 - 5| = 1, \quad |x_4 - \bar{x}| = |10 - 5| = 5 \] ### Step 3: Multiply Absolute Deviations by the Frequencies Next, we multiply each absolute deviation by the corresponding frequency \( f_i \): \[ f_1 |x_1 - \bar{x}| = 7 \times 3 = 21, \quad f_2 |x_2 - \bar{x}| = 4 \times 1 = 4, \quad f_3 |x_3 - \bar{x}| = 5 \times 1 = 5, \quad f_4 |x_4 - \bar{x}| = 4 \times 5 = 20 \] ### Step 4: Calculate the Mean Deviation Finally, the mean deviation is: \[ \text{Mean Deviation} = \frac{21 + 4 + 5 + 20}{20} = \frac{50}{20} = 2.5 \] Thus, the correct answer is: \[ \boxed{(B) \, 3} \]