Informational
How to calculate a z-score
Subtract the mean from the value, then divide by the standard deviation: z = (x − μ) / σ. The result says how many standard deviations x sits above or below the mean. Positive is above; negative is below.
When a z-score is the right tool
Use it to compare values from different scales (exam vs exam) or to find a tail probability on a normal curve. For a sample mean, the standardized statistic is (x̄ − μ) / (σ / √n) — same idea, extra √n. If σ is unknown and n is small, you likely want a t procedure instead.
Formula
For a sample mean: z = (x̄ − μ) / (σ / √n). You still need a known population SD for a true z.
Worked example
Exam scores are approximately normal with μ = 72 and σ = 8. A student scores 84.
- x − μ = 84 − 72 = 12.
- 12 / 8 = 1.5.
- The score is 1.5 standard deviations above the mean.
z = 1.5. On a standard normal curve, that is above average but not an extreme outlier.
From z to probability
A z-table or a normal CDF gives the area to the left of z. For z = 1.5, P(Z ≤ 1.5) ≈ 0.933. The right-tail probability is 1 − 0.933 = 0.067. Statistics Problem Solver Pro includes a Normal CDF calculator and a distribution visualizer with shaded regions if you want the area without a paper table.
Useful tips
- |z| around 2 is often called “unusual”; |z| around 3 is rare under a normal model. These are rules of thumb, not theorems.
- Keep the sign. Dropping the minus turns a below-average score into an above-average one.
- If you must compute σ from data first, see how to calculate standard deviation.
Common mistakes
- Dividing by n or √n when the question asked for a single-value z = (x − μ) / σ.
- Using s in a z formula and still calling it a z-test with a small sample.
- Looking up a two-tail area when the homework asked for one tail (or the reverse).
How Statistics Problem Solver Pro helps
The Z-Score calculator is free and offline. For a full word problem, scan or type it and read the substitutions. The formula reference lists the exact expression, and quizzes in the app include a Distributions & Z-Scores set if you want practice items.
Check a z-score on iPhone
Use the free calculator, or scan the full problem for every step.
Related guides
FAQ
What three numbers do I need for a z-score?
The raw value x, the mean μ (or x̄, depending on the formula), and the standard deviation σ. For a mean, you also need n.
Open the iPhone calculator overview →Is a z-score the same as a percentile?
No. A z-score is a standardized distance. A percentile is an area (probability) under the curve. You convert with a normal table or CDF.
See related formulas →