Does adding a constant or reducing all the numbers by 5% change the z score?

Question

I have a set of numbers.

I have two cases:

1. If I add a constant to them, does the mean, standard deviation and z score change?

What I think: Mean changes, std deviation doesn't change and z score doesn't change.

2. If I reduce all the numbers by 5% does the mean, standard deviation and z score change?

What I think: Mean changes by 5%, standard deviation changes but the percentage difference remains the same, and the z score changes by 2.5%

Am I thinking correctly?

Answer 1

Suppose that you have $x_1, x_2, \cdots, x_n$. Using the definitions of those statistics, you have $$\bar{x} =\frac{\sum x_i}{n}$$ $$s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$$ $$z-score \; of\; x_i=\frac{x_i-\bar{x}}{s}$$ Now if you add a constant to all observations, it is easy to see that $\bar{x}$ changes by the same amount, but the other two statistics remain the same.

If you reduce all numbers by 5%, each $x_i$ becomes $0.95x_i$. Thus, both $\bar{x}$ and $s$ decrease by 5% as well, but the z-score remains the same.