Topic 5: z-Scores (Standardized Scores)

summary

Raw scores (X) often lack context. A z-score transforms a raw score to describe its exact location within a distribution, measured in units of standard deviation. This allows us to compare scores from completely different distributions (e.g., comparing a Biology grade to a Psychology grade).

Purpose of z-score: To identify a precise location

Tells us if a score is above average, below average, or extreme.

A raw score is a lonely number. It has no friends and no context. A z-score gives that number a home, a neighborhood, and a rank. It tells us not just what you got, but where you stand.

Purpose of z-score: To standardize a distribution

Allows different distributions to be made equivalent so they can be directly compared

Property of z-score distribution

When an entire distribution of X values is transformed into z-scores, the new distribution will always have these three properties: Shape, The Mean of a z-distribution (μz), The Standard Deviation of a z-distribution (σz)

Property of z-score distribution: Shape

Exactly the same as the original distribution.

Property of z-score distribution: Mean

Always 0. The mean of the original distribution becomes the center point (0).

Property of z-score distribution: Standard Deviation

Always 1. One point in z-score units is exactly one standard deviation.

Population

z = (X – μ) / σ

Sample

z = (X – M) / s

Standardizing to a New Distribution

Steps:

1. Convert the original raw score into a z-score.

2. Use that z-score to find a new X value in a distribution with a "prettier" mean and standard deviation. [ eg: Moving a score from a distribution of (μ =57, σ =14) to one of (μ =50, σ =10). ]