A z-score tells you how far a value is from the mean in units of standard deviations. Once you know the formula, z-score questions become quick plug-in-and-check problems.
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The definition of a z-score is:
- x = the value you’re comparing
- μ (mu) = the mean
- σ (sigma) = the standard deviation
Interpretation is simple: z = 2 means “2 standard deviations above the mean,” and z = −1 means “1 standard deviation below the mean.”
Problem: Test scores are normally distributed with mean μ = 75 and standard deviation σ = 8. If you scored x = 91, what is your z-score?
z = (x − μ) / σ
z = (91 − 75) / 8
z = 16 / 8
z = 2
Interpretation: A z-score of 2 means you scored 2 standard deviations above the mean.
Problem: Heights have mean μ = 66 inches and standard deviation σ = 3 inches. What height corresponds to z = −1.5?
x = μ + zσ
x = 66 + (−1.5)(3)
x = 66 − 4.5
x = 61.5
Interpretation: 61.5 inches is 1.5 standard deviations below the mean.
Some problems ask for “the probability of scoring above 91” or “what percentile is a z-score of 1.2?” That’s an area under the normal curve question. The typical flow is:
If you need a conceptual refresher on the standard normal curve and what those areas mean, the OpenStax discussion of the standard normal distribution connects z-scores to probabilities with clear visuals.
The mean (μ) is the center. The standard deviation (σ) is the spread. Don’t subtract σ from x.
If x is below the mean, z must be negative. Use parentheses when x − μ is negative.
“Above” means right tail; “below” means left tail. For “between,” subtract two cumulative areas.
A z-score of 10 is extremely unlikely in most real datasets. Re-check arithmetic if you get huge values.
Paste μ, σ, and x, and get the full work plus a quick reasonableness check.
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