z-scores

What is z-scores

z_{(s a m p l e)} = \frac{x_{i} - \bar{x}}{s_{x}}, z_{(p o p u l a t i o n)} = \frac{x_{i} - μ}{σ}

Z-score interpretation: The value we get is the number of standard deviations the value is away from the mean

positive z-scores are larger than the mean
negative z-scores are smaller than the mean

Properties

If we standardize a variable, what we are doing is scaling each value so that the mean becomes 0 and the standard deviation becomes 1.

puts values on the same scale
mean (center) = 0, standard deviation (spread) = 1
no units

Transformation

We can standardize all the values of a variable. The resulting standardized Values are either squeezed or stretched, but their relative positions stay the same

68-95-99 Rule

Useful property of the mean and standard deviation when we have a bell-shaped distribution (unimodal and symmetric).