Question

Let the mean and the standard deviation of the observations $ 2, 3, 4, 5, 7, a, b $ be $ 4 $ and $ \sqrt{2} $ respectively. Then the mean deviation about the mode of these observations is:

• A. 1
• B. \( \frac{3}{4} \)
• C. 2
• D. \( \frac{1}{2} \)

Hint:

To calculate the mean deviation about the mode, use the absolute differences of each observation from the mode and take the average.

Answer 1

The Correct Option is A

We are given the observations \( 2, 3, 4, 5, 7, a, b \), and the mean \( \mu = 4 \) and standard deviation \( \sigma = \sqrt{2} \).
Step 1: Calculate the sum of the observations
The mean of the observations is given by: \[ \frac{2 + 3 + 4 + 5 + 7 + a + b}{7} = 4 \] This simplifies to: \[ 2 + 3 + 4 + 5 + 7 + a + b = 28 \] So, we have: \[ 21 + a + b = 28 \quad \Rightarrow \quad a + b = 7 \]
Step 2: Calculate the sum of squared deviations
The formula for the standard deviation is: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \] Substitute the known values: \[ 2^2 + 3^2 + 4^2 + 5^2 + 7^2 + a^2 + b^2 = 7 \cdot 2 \] Simplifying: \[ 4 + 9 + 16 + 25 + 49 + a^2 + b^2 = 14 \] \[ 103 + a^2 + b^2 = 14 \quad \Rightarrow \quad a^2 + b^2 = 14 - 103 = -89 \] Now, substitute \( a + b = 7 \) into this equation: \[ a^2 + b^2 = (a + b)^2 - 2ab \] \[ -89 = 49 - 2ab \quad \Rightarrow \quad ab = 64 \]
Step 3: Mode Calculation
The mode of the observations is 4 (since it appears most frequently).
Step 4: Calculate the Mean Deviation
The mean deviation about the mode is given by: \[ \frac{|2 - 4| + |3 - 4| + |4 - 4| + |5 - 4| + |7 - 4| + |a - 4| + |b - 4|}{7} \] Substituting the values, we find: \[ \frac{2 + 1 + 0 + 1 + 3 + |a - 4| + |b - 4|}{7} \] Since \( a = 4 \) and \( b = 4 \), the mean deviation simplifies to: \[ \frac{2 + 1 + 0 + 1 + 3 + 0 + 0}{7} = \frac{7}{7} = 1 \]
Thus, the mean deviation about the mode is \( 1 \).

Answer 2

Step 1: The equation is given as: 

\[ \frac{24 + a + b}{8} = 4 \]

Step 2: Simplifying the equation:

\[ a + b = 8 \]

Step 3: The next equation is:

\[ 2 = \frac{4 + 1 + 1 + 0 + 1 + 9 + (a - 4)^2 + (b - 4)^2}{8} \] Simplifying: \[ 16 = 48 + a^2 + b^2 - 8a - 8b \]

Step 4: This leads to:

\[ a^2 + b^2 = 32 \]

Step 5: We also know:

\[ 32 = 2ab \] Solving for \( ab \): \[ ab = 16 \]

Step 6: From \( a + b = 8 \) and \( ab = 16 \), we find:

\[ a = 4, \quad b = 4 \]

Step 7: The mode is given as:

\[ \text{Mode} = 4 \]

Step 8: The mean deviation is calculated as:

\[ \text{Mean Deviation} = \frac{2 + 1 + 1 + 0 + 1 + 1 + 3 + 0 + 0}{8} = 1 \]