The Normal Model & z-Scores

You’ll be able to

Standardize a value into a z-score and interpret it in context
Apply the 68–95–99.7 (empirical) rule to a Normal distribution
Use z-scores to compare values from different distributions

The Normal curve

Many quantitative variables are approximately Normal: a symmetric, single-peaked, bell-shaped distribution described completely by its mean μ (the center) and standard deviation σ (the spread). The curve is a model, not the data themselves — it lets us convert "how far from average?" into a probability or percentile.

z-scores: distance in standard deviations

A z-score rescales a value to say how many standard deviations it sits above or below the mean. A z of +2 means "two standard deviations above average"; a z of −1.5 means "one and a half below." Standardizing strips away the original units, which is what lets you compare a test score to a height or one exam to another.

z-score (standardizing)

z = (x − μ) / σ

Subtract the mean, divide by the standard deviation. Positive z is above the mean, negative z is below. z has no units.

Empirical (68–95–99.7) rule

μ ± 1σ ≈ 68% · μ ± 2σ ≈ 95% · μ ± 3σ ≈ 99.7%

For a Normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

Worked example

Adult resting heart rate is approximately Normal with μ = 70 bpm and σ = 8 bpm. What proportion of adults have a resting heart rate above 86 bpm?

1.Standardize the boundary: z = (86 − 70) / 8 = 16 / 8 = 2. So 86 bpm is exactly 2 standard deviations above the mean.
2.By the empirical rule, about 95% of values lie within 2 standard deviations, leaving about 5% in the two tails combined.
3.By symmetry, half of that 5% is in the upper tail: about 2.5% above z = 2.

Answer: About 2.5% of adults have a resting heart rate above 86 bpm, since 86 is 2 standard deviations above the mean and roughly 2.5% of a Normal distribution lies beyond z = +2.

Checkpoint

On a test with mean 500 and standard deviation 100, Maria scored 640. What is her z-score, and what does it mean?

Watch out

A z-score is a count of standard deviations, not a raw difference and not a percentile. "z = 2" means "two standard deviations above the mean," which corresponds to about the 97.5th percentile in a Normal model — but the z-value itself is 2, not 97.5.

Checkpoint

IQ scores are approximately Normal with μ = 100 and σ = 15. Approximately what percentage of people have IQ scores between 85 and 115?

Answer the 2 checkpoints as you read.