Let {x1, x2, …, xn} be a dataset consisting of N real numbers. Prove from definitions or proved properties in the textbook that the standardized dataset {z1, z2, …, zn}, defined as... Let {x1, x2, …, xn} be a dataset consisting of N real numbers. Prove from definitions or proved properties in the textbook that the standardized dataset {z1, z2, …, zn}, defined as z = (x - mean) / standard deviation, has a mean of 0 and a standard deviation of 1. Read more

Steps to Solve

1. Define Standardization

Standardization uses the formula to transform a dataset into a standardized form, where the variable $Z$ is calculated as follows: $$ Z = \frac{X - \mu}{\sigma} $$ Here, $X$ is the original data point, $\mu$ is the mean of the dataset, and $\sigma$ is the standard deviation.

2. Calculate the Mean of a Standardized Dataset

When a dataset $X$ is standardized:

• The mean is calculated as: $$ \mu_Z = \frac{1}{n} \sum_{i=1}^{n} Z_i $$ Substituting the definition of $Z_i$ gives: $$ \mu_Z = \frac{1}{n} \sum_{i=1}^{n} \left(\frac{X_i - \mu}{\sigma}\right) $$ This simplifies to: $$ \mu_Z = \frac{1}{n} \left(\frac{1}{\sigma} \sum_{i=1}^{n} (X_i - \mu)\right) $$ Since the sum of deviations from the mean $\sum_{i=1}^{n} (X_i - \mu) = 0$, it follows that: $$ \mu_Z = \frac{1}{n} \cdot \frac{0}{\sigma} = 0 $$

3. Calculate the Standard Deviation of a Standardized Dataset

Next, we find the standard deviation of the standardized data:

• The variance is given by: $$ \sigma_Z^2 = \frac{1}{n} \sum_{i=1}^{n} (Z_i - \mu_Z)^2 $$ With $\mu_Z = 0$, this simplifies to: $$ \sigma_Z^2 = \frac{1}{n} \sum_{i=1}^{n} Z_i^2 $$ Substituting $Z_i$ gives: $$ \sigma_Z^2 = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right)^2 = \frac{1}{n\sigma^2} \sum_{i=1}^{n} (X_i - \mu)^2 $$ Recognizing that $\sum_{i=1}^{n} (X_i - \mu)^2 = n\sigma^2$, we have: $$ \sigma_Z^2 = \frac{n\sigma^2}{n\sigma^2} = 1 $$ Thus, the standard deviation $\sigma_Z = 1$.