Key Concepts and Formulas

• Mean ($\mu$ or $\bar{x}$): For a set of $N$ observations $x_1, x_2, \ldots, x_N$, the mean is the arithmetic average: $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$
• Standard Deviation ($\sigma$): A measure of the spread of data around the mean. Its square is the variance ($\sigma^2$). The definitional formula for variance is $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$.
• Computational Variance Formula: For practical calculations, especially when dealing with sums of squares, an alternative formula for variance is extremely useful: $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - \mu^2$ This formula directly relates the variance, the mean, and the mean of the squares of the observations. It will be the cornerstone of our solution.

Step-by-Step Solution

We are given the following information for 50 observations, $x_1, x_2, \ldots, x_{50}$:

• Number of observations, $N = 50$.
• Mean of these observations, $\mu = 16$.
• Standard deviation of these observations, $\sigma = 16$.

Our objective is to find the mean of the new set of observations: $(x_1 – 4)^2, (x_2 – 4)^2, \ldots, (x_{50} – 4)^2$. Let's denote this required mean as $\mu'$. $\mu' = \frac{\sum_{i=1}^{50} (x_i - 4)^2}{50}$

Step 1: Calculate the Sum of Observations ($\sum x_i$)

• What: We will use the given mean to find the sum of the original observations, $\sum x_i$.
• Why: The expression for $\mu'$ involves terms with $x_i$ after algebraic expansion. Calculating $\sum x_i$ (or rather, $\frac{\sum x_i}{N}$ which is the mean itself) upfront will simplify the later substitution.
• Math: The definition of the mean is: $\mu = \frac{\sum x_i}{N}$ Substitute the given values $\mu = 16$ and $N = 50$: $16 = \frac{\sum x_i}{50}$ Now, solve for $\sum x_i$: $\sum x_i = 16 \times 50$ $\sum x_i = 800$

Step 2: Calculate the Mean of Squares ($\frac{\sum x_i^2}{N}$)

• What: We will use the computational formula for variance to find the mean of the squares of the original observations, $\frac{\sum x_i^2}{N}$.
• Why: The expression for $\mu'$ also involves terms with $x_i^2$ after expansion. This value is directly obtainable from the variance formula and will be crucial for the final calculation.
• Math: The computational formula for variance is: $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$ We are given $\sigma = 16$, so $\sigma^2 = 16^2 = 256$. We are also given $\mu = 16$, so $\mu^2 = 16^2 = 256$. Substitute these values into the formula: $256 = \frac{\sum x_i^2}{50} - 256$ To isolate $\frac{\sum x_i^2}{50}$, add 256 to both sides: $\frac{\sum x_i^2}{50} = 256 + 256$ $\frac{\sum x_i^2}{50} = 512$

Step 3: Evaluate the Required Mean of $(x_i - 4)^2$

• What: Now we will use the results from Step 1 and Step 2 to calculate the desired mean, $\mu'$.
• Why: By algebraically expanding the term $(x_i - 4)^2$ and applying the properties of summation, we can express $\mu'$ in terms of $\frac{\sum x_i^2}{N}$ and $\frac{\sum x_i}{N}$, which we have already determined.
• Math: The required mean is: $\mu' = \frac{\sum_{i=1}^{50} (x_i - 4)^2}{50}$ First, expand the term $(x_i - 4)^2$ using the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$: $(x_i - 4)^2 = x_i^2 - 2(x_i)(4) + 4^2$ $(x_i - 4)^2 = x_i^2 - 8x_i + 16$ Substitute this expanded form back into the expression for $\mu'$: $\mu' = \frac{\sum_{i=1}^{50} (x_i^2 - 8x_i + 16)}{50}$ Now, distribute the summation and the division by $N=50$ over each term: $\mu' = \frac{\sum x_i^2}{50} - \frac{\sum 8x_i}{50} + \frac{\sum 16}{50}$ Using the property $\sum c \cdot a_i = c \cdot \sum a_i$ and $\sum_{i=1}^N C = N \times C$: $\mu' = \left(\frac{\sum x_i^2}{50}\right) - 8 \left(\frac{\sum x_i}{50}\right) + \left(\frac{50 \times 16}{50}\right)$ Now, substitute the values we have:
  • From Step 2, $\frac{\sum x_i^2}{50} = 512$.
  • The term $\frac{\sum x_i}{50}$ is the original mean, $\mu = 16$.
  • The term $\frac{50 \times 16}{50}$ simplifies to 16. Substitute these values into the equation for $\mu'$: $\mu' = 512 - 8(16) + 16$ Perform the arithmetic: $\mu' = 512 - 128 + 16$ $\mu' = 384 + 16$ $\mu' = 400$

Common Mistakes & Tips

• Master the Computational Variance Formula: The formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$ is incredibly powerful and efficient. Ensure you remember and understand how to apply it.
• Algebraic Precision: Always be careful when expanding algebraic expressions like $(a-b)^2$. A common error is forgetting the middle term, e.g., writing $(x_i - 4)^2 = x_i^2 + 16$ instead of $x_i^2 - 8x_i + 16$.
• Properties of Summation: Remember that summation distributes over addition and subtraction ($\sum (A \pm B) = \sum A \pm \sum B$), and constants can be pulled out ($\sum cA = c \sum A$). Also, the sum of a constant $C$ over $N$ terms is $N \times C$ ($\sum_{i=1}^N C = N C$). These properties are vital for simplifying expressions.

Summary

This problem effectively demonstrates the interconnections between fundamental statistical measures: mean and standard deviation. The key to solving it was leveraging the computational formula for variance, $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$, to determine the mean of the squares of the observations. Once we had this value, along with the original mean, we could algebraically expand the expression for the target mean, $(x_i - 4)^2$, and substitute the pre-calculated components to arrive at the final answer.

The final answer is $\boxed{400}$, which corresponds to option (A).