Empirical Rules and Z-Scores Notes

Empirical Rule

For a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Z-Score

A metric that indicates how many standard deviations a data value is from the mean.

Z-Score Formula

Z=(X−μ)σZ=σ(X−μ)​ where X is the raw score, alpha is the mean, and σ is the standard deviation.`

What does a Z-score tell you?

It tells you how many standard deviations the data value is from the mean.

Unusual Z-Scores

Z-scores greater than 3 are extremely unusual, between 2 and 3 are very unusual, and between 1.5 and 1.75 are maybe unusual.

Population Z-Score Unstandardizing Formula

To find original value X from z-score: X = \mu + Z\sigma.

Sample Z-Score Unstandardizing Formula

To find original value x from z-score: x = \bar{x} + Zs.

Interpretation of Z-scores: greater than 3

Extremely unusual.

Interpretation of Z-scores: between 1 and 1.5

Somewhat low/high but not unusual.

Distribution Shape for Z-Scores

Z-scores are particularly useful when the data distribution is mound-shaped and approximately symmetric.