1. Key Concepts and Formulas

• Mean ($\bar{x}$): The sum of all observations divided by the number of observations. $\bar{x} = \frac{\sum x_i}{N}$
• Variance ($\sigma^2$): A measure of the spread of data. It's the average of the squared differences from the mean. A computationally convenient formula is: $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$ The standard deviation ($\sigma$) is the square root of the variance.
• Mode: The observation that appears most frequently in a data set.
• Mean Deviation about the Mode (MD$_{\text{Mode}}$): The average of the absolute differences between each observation and the mode. $\text{MD}_{\text{Mode}} = \frac{\sum |x_i - \text{Mode}|}{N}$

2. Step-by-Step Solution

Let the given observations be $x_i$: $2, 3, 3, 4, 5, 7, a, b$. The total number of observations, $N = 8$.

Step 2.1: Use the Mean to find a relationship between $a$ and $b$.

• What we are doing: We use the given mean to form an equation involving the unknown values $a$ and $b$.
• Why: This is the first step in determining the complete set of observations.
• Given: Mean ($\bar{x}$) = 4.
• Formula: $\bar{x} = \frac{\sum x_i}{N}$

Substitute the values: $4 = \frac{2+3+3+4+5+7+a+b}{8}$ Sum the known observations: $2+3+3+4+5+7 = 24$. $4 = \frac{24+a+b}{8}$ Multiply both sides by 8: $32 = 24+a+b$ $a+b = 32 - 24$ $\mathbf{a+b = 8 \quad \text{(Equation 1)}}$

Step 2.2: Use the Standard Deviation (Variance) to find a second relationship between $a$ and $b$.

• What we are doing: We use the given standard deviation to form another equation, this time involving $a^2$ and $b^2$.
• Why: We need two independent equations to solve for two unknowns ($a$ and $b$). Working with variance ($\sigma^2$) is often easier as it removes the square root.
• Given: Standard Deviation ($\sigma$) = $\sqrt{2}$.
• Calculation: Variance ($\sigma^2$) = $(\sqrt{2})^2 = 2$.
• Formula: $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$

We know $\bar{x}=4$, so $(\bar{x})^2 = 4^2 = 16$. First, calculate the sum of the squares of the known observations: $2^2+3^2+3^2+4^2+5^2+7^2 = 4+9+9+16+25+49 = 112$.

Now, substitute all known values into the variance formula: $2 = \frac{112+a^2+b^2}{8} - 16$ Add 16 to both sides: $18 = \frac{112+a^2+b^2}{8}$ Multiply both sides by 8: $144 = 112+a^2+b^2$ $a^2+b^2 = 144 - 112$ $\mathbf{a^2+b^2 = 32 \quad \text{(Equation 2)}}$

Step 2.3: Solve the system of equations for $a$ and $b$.

• What we are doing: We solve the two equations from Step 2.1 and Step 2.2 to find the specific values of $a$ and $b$.
• Why: Knowing $a$ and $b$ completes our data set, which is necessary for calculating the mode and mean deviation.
• Equations:
  1. $a+b = 8$
  2. $a^2+b^2 = 32$

From Equation 1, we can express $b$ in terms of $a$: $b = 8-a$ Substitute this into Equation 2: $a^2 + (8-a)^2 = 32$ Expand the term $(8-a)^2$: $a^2 + (64 - 16a + a^2) = 32$ Combine like terms: $2a^2 - 16a + 64 = 32$ Subtract 32 from both sides to form a quadratic equation: $2a^2 - 16a + 32 = 0$ Divide by 2 to simplify: $a^2 - 8a + 16 = 0$ This is a perfect square trinomial, which factors as $(a-4)^2$: $(a-4)^2 = 0$ Taking the square root: $a-4 = 0 \implies a = 4$ Now, substitute $a=4$ back into $b=8-a$: $b = 8-4 \implies b = 4$ So, the unknown observations are $a=4$ and $b=4$.

Step 2.4: Find the Mode of the complete data set.

• What we are doing: We identify the most frequent observation in the complete data set.
• Why: The question asks for the mean deviation about the mode.

The complete set of observations is $2, 3, 3, 4, 5, 7, 4, 4$. Let's list them in ascending order to easily count frequencies: $2, 3, 3, 4, 4, 4, 5, 7$.

• 2 appears once.
• 3 appears twice.
• 4 appears three times.
• 5 appears once.
• 7 appears once.

The value that appears most frequently is 4. Therefore, the Mode = 4.

Step 2.5: Calculate the Mean Deviation about the Mode.

• What we are doing: We calculate the average of the absolute differences between each observation and the mode.
• Why: This is the final step to answer the question.
• Formula: $\text{MD}_{\text{Mode}} = \frac{\sum |x_i - \text{Mode}|}{N}$

Calculate $|x_i - \text{Mode}|$ for each observation (with Mode = 4):

• $|2-4| = |-2| = 2$
• $|3-4| = |-1| = 1$
• $|3-4| = |-1| = 1$
• $|4-4| = |0| = 0$
• $|4-4| = |0| = 0$
• $|4-4| = |0| = 0$
• $|5-4| = |1| = 1$
• $|7-4| = |3| = 3$

Sum these absolute differences: $\sum |x_i - \text{Mode}| = 2+1+1+0+0+0+1+3 = 8$

Finally, calculate the Mean Deviation about the Mode: $\text{MD}_{\text{Mode}} = \frac{8}{8}$ $\mathbf{\text{MD}_{\text{Mode}} = 1}$

3. Common Mistakes & Tips

• Equation Setup: Carefully sum the known observations and their squares. A small arithmetic error here will propagate throughout the entire solution.
• Solving Quadratics: Be mindful of the signs when expanding $(x-y)^2$ and solving the quadratic equation. A perfect square trinomial indicates a unique solution for the unknown values.
• Mode Identification: Ensure you count frequencies correctly, especially in larger datasets or when there might be multiple modes (though not the case here).
• Mean Deviation Calculation: Remember to take the absolute difference $|x_i - \text{Mode}|$ before summing.
• Variance Formula: The formula $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$ is generally more robust and less prone to rounding errors than $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}$, especially if $\bar{x}$ is not an integer.

4. Summary

This problem required a systematic application of several statistical concepts. First, we used the given mean and standard deviation to form a system of two equations, which we solved to find the unknown observations $a=4$ and $b=4$. With the complete dataset, we identified the mode as 4. Finally, we calculated the mean deviation about this mode by summing the absolute differences of each observation from the mode and dividing by the total number of observations, resulting in a value of 1.

5. Final Answer

The mean deviation about the mode of these observations is 1, which corresponds to option (C).

The final answer is $\boxed{1}$.