1. Key Concepts and Formulas

• Variance ($V$ or $\sigma^2$): A statistical measure that quantifies the average squared deviation of each data point from the mean of the dataset. It describes the spread or dispersion of the data. For a set of $n$ observations $x_1, x_2, \dots, x_n$ with mean $\bar{x}$, the variance $V$ is given by: $V = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$
• Effect of Change of Origin on Variance: If each observation $x_i$ in a dataset is transformed by adding or subtracting a constant value $c$ (i.e., $y_i = x_i + c$), the variance of the new dataset ($V_y$) remains identical to the variance of the original dataset ($V_x$). $V(x_i + c) = V(x_i)$ Reasoning: Adding a constant value to every data point shifts the entire distribution along the number line without changing the internal spread or the relative distances between the data points.
• Effect of Change of Scale on Variance (for contrast): If each observation $x_i$ is transformed by multiplying or dividing by a non-zero constant $k$ (i.e., $y_i = kx_i$), the variance of the new dataset is $k^2$ times the variance of the original dataset. $V(kx_i) = k^2 V(x_i)$ Reasoning: Multiplication by a constant $k$ stretches or compresses the data, thus changing the spread. Since variance involves squared deviations, the scaling factor becomes $k^2$.

2. Step-by-Step Solution

Step 1: Analyze Population A.

• Population A consists of 100 observations: $x_1 = 101, x_2 = 102, \dots, x_{100} = 200$.
• These observations are consecutive integers, forming an arithmetic progression.
• Let $V_A$ denote the variance of Population A. Our goal is to find its relationship with $V_B$.

Step 2: Analyze Population B.

• Population B consists of 100 observations: $y_1 = 151, y_2 = 152, \dots, y_{100} = 250$.
• These observations are also consecutive integers, forming an arithmetic progression. Crucially, Population B also has 100 observations, matching the count in Population A.
• Let $V_B$ denote the variance of Population B.

Step 3: Establish the relationship between Population A and Population B.

• To determine how $V_A$ and $V_B$ are related, we need to find a mathematical transformation that links the observations of Population A to those of Population B.
• Let's compare the corresponding terms:
  • The first observation of A is $x_1 = 101$. The first observation of B is $y_1 = 151$. We notice that $151 = 101 + 50$.
  • The second observation of A is $x_2 = 102$. The second observation of B is $y_2 = 152$. We notice that $152 = 102 + 50$.
  • This pattern holds true for all corresponding observations. In general, for any $i$-th observation, the relationship is: $y_i = x_i + 50$
• This relationship signifies that every observation in Population B is obtained by adding a constant value of 50 to the corresponding observation in Population A. This is precisely what is known as a "change of origin" transformation.

Step 4: Apply the Variance Property.

• According to the key concept regarding the "effect of change of origin on variance," adding a constant to each observation in a dataset does not alter its variance.
• Since Population B's observations ($y_i$) are derived from Population A's observations ($x_i$) by adding a constant (50), their variances must be equal. $V_B = V_A$
• This means that Population B is simply a shifted version of Population A on the number line; the spread of its data points is identical.

Step 5: Calculate the required ratio.

• The problem asks for the ratio $\frac{V_A}{V_B}$.
• Since we have established that $V_A = V_B$, we can substitute $V_A$ for $V_B$ in the ratio expression: $\frac{V_A}{V_B} = \frac{V_A}{V_A} = 1$

3. Common Mistakes & Tips

• Confusing Change of Origin with Change of Scale: A frequent error is to apply the rule for change of scale ($V(kx) = k^2V(x)$) instead of the rule for change of origin. Always differentiate between addition/subtraction (shifts) and multiplication/division (scales).
• Verifying the Transformation: Ensure that the relationship ($y_i = x_i + c$ or $y_i = kx_i$) holds for all corresponding observations, not just the first few.
• Other Statistical Measures: Remember that while variance and standard deviation are invariant under change of origin, measures like mean, median, and mode do change by the constant amount $c$.

4. Summary

This problem is a direct application of a fundamental property in statistics concerning the effect of transformations on variance. By analyzing the observations of Population A and Population B, we found that Population B is obtained by adding a constant value of 50 to each observation of Population A. This "change of origin" transformation does not affect the variance of a dataset. Therefore, the variance of Population A ($V_A$) is equal to the variance of Population B ($V_B$), leading to a ratio of 1.

5. Final Answer

The final answer is $\boxed{1}$, which corresponds to option (A).