1. **Problem Statement:** We have a dataset normalized using Z-score: $$z = \frac{x - \mu}{\sigma}$$ (a) Find the differential $$dz$$ in terms of $$dx$$. (b) Given $$\mu = 50$$, $$\sigma = 10$$, and measurement error in $$x$$ is $$\Delta x = 0.5$$, estimate the error in $$z$$. (c) For min-max scaling $$f(x) = \frac{x - x_{min}}{x_{max} - x_{min}}$$, find $$df$$ and analyze its behavior across the input range. 2. **Formulas and Rules:** - The differential of a function $$y = g(x)$$ is $$dy = g'(x) dx$$. - For error propagation, small changes in $$x$$ cause changes in $$z$$ approximated by $$dz = \frac{dz}{dx} dx$$. 3. **Step (a): Find $$dz$$ in terms of $$dx$$** Given $$z = \frac{x - \mu}{\sigma}$$, treat $$\mu$$ and $$\sigma$$ as constants. Calculate derivative: $$\frac{dz}{dx} = \frac{1}{\sigma}$$ Therefore, $$dz = \frac{1}{\sigma} dx$$ 4. **Step (b): Estimate error in $$z$$ given $$\mu = 50$$, $$\sigma = 10$$, and $$\Delta x = 0.5$$** Using the differential approximation for error: $$\Delta z \approx |dz| = \left| \frac{1}{\sigma} \right| \Delta x = \frac{1}{10} \times 0.5 = 0.05$$ So, the estimated error in $$z$$ is $$0.05$$. 5. **Step (c): Find $$df$$ for min-max scaling and analyze behavior** Given: $$f(x) = \frac{x - x_{min}}{x_{max} - x_{min}}$$ Since $$x_{min}$$ and $$x_{max}$$ are constants, derivative is: $$\frac{df}{dx} = \frac{1}{x_{max} - x_{min}}$$ Therefore, $$df = \frac{1}{x_{max} - x_{min}} dx$$ **Analysis:** - The derivative $$\frac{df}{dx}$$ is constant across the input range. - This means the sensitivity of $$f(x)$$ to changes in $$x$$ is uniform regardless of $$x$$. - Unlike Z-score scaling, min-max scaling does not depend on mean or standard deviation but on the range. **Final answers:** (a) $$dz = \frac{1}{\sigma} dx$$ (b) $$\Delta z = 0.05$$ (c) $$df = \frac{1}{x_{max} - x_{min}} dx$$ with constant sensitivity across the input range.