Consider a finite sequence of random values $X = [x_1, x_2, \dots, x_n]$. Let $\mu_x$ be the mean and $\sigma_x$ be the standard deviation of $X$. Let another finite sequence $Y$ of equal length be derived from this as $y_i = a * x_i + b$, where $a$ and $b$ are positive constants. Let $\mu_y$ be the mean and $\sigma_y$ be the standard deviation of this sequence. Which one of the following statements is **INCORRECT**?

Given the linear transformation $y_i = a x_i + b$, where $a > 0$ and $b > 0$:

1. Mode and Median positions: Since $a > 0$, the linear transformation is strictly monotonically increasing. Hence, the ordering and relative frequencies of elements are preserved. The index position of the mode and median in $X$ will be identical to their positions in $Y$. Therefore, statements (A) and (B) are correct.

2. Mean: By linearity of expectation,

   $$\mu_y = \frac{1}{n} \sum_{i=1}^n (a x_i + b) = a \mu_x + b$$

   Thus, statement (C) is correct.

3. Standard Deviation: Standard deviation measures dispersion and is invariant under shift (addition of a constant $b$). For $a > 0$:

   $$\sigma_y = a \sigma_x$$

   Statement (D) claims $\sigma_y = a\sigma_x + b$, which incorrectly adds the shift $b$.

Hence, statement (D) is INCORRECT.