When standardizing a dataset to z-scores using $z_i = \frac{x_i - \bar{x}}{s}$, which of the following MUST be true about the standardized data?

(1) The mean of the z-scores is 0 (2) The standard deviation of the z-scores is 1 (3) The median of the z-scores equals 0 (4) The minimum z-score is $\frac{x_{min} - \bar{x}}{s}$ and the maximum is $\frac{x_{max} - \bar{x}}{s}$ (5) The distribution shape is unchanged (same skewness, kurtosis, etc.)