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  4. Z-score Calculator

Z-score Calculator

Compute z from x, μ, σ; convert between z and normal probabilities like a z-table; or area between two z-scores.

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How it works

z counts standard deviations from the mean. Normal areas use Φ(z); any one area determines the rest.

z=(x−μ)/σ · P=Φ(z2)−Φ(z1) · 0→z=|Φ(z)−½|

When to use

Use this tool when you need to: try IQ: x=115, μ=100, σ=15 → z=1. Convert z=1.96 (≈95% central band). Area between −1 and 1 ≈68%.

How to use this tool

  1. Enter your values in the fields above.
  2. Review the formula and any mode options for your problem.
  3. Read the result and the step-by-step solution.

Try IQ: x=115, μ=100, σ=15 → z=1. Convert z=1.96 (≈95% central band). Area between −1 and 1 ≈68%.

Frequently asked questions

What is the Z-score Calculator?
z counts standard deviations from the mean. Normal areas use Φ(z); any one area determines the rest.
How do I use the Z-score Calculator?
Enter the required values, then calculate. The Z-score Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Z-score Calculator use?
z=(x−μ)/σ · P=Φ(z2)−Φ(z1) · 0→z=|Φ(z)−½|

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Z-score calculator

Standardize a raw score, convert either way between z and normal probabilities (like a z-table), or find the area between two z-scores.

Mode

Uses the population formula z = (x − μ) / σ. Results also list left, right, 0-to-z, central, and outer-tail areas.

Try an example

How it works

What is a z-score?

A z-score (standard score) is dimensionless: how many standard deviations a raw observation lies from the mean. z = (x − μ)/σ. Above the mean → positive z; below → negative. Applications include z-tests, comparing scales, process control, and prediction intervals.

Z-tables

A unit-normal table lists standardized areas. Some tables give the area from the mean (0) to z; others list cumulative Φ(z)=P(X≤z). Always check which region a printed table references. This calculator reports both styles so results match either layout.

z ↔ probability converter

Supply any one of z, P(X<Z), P(X>Z), P(0→Z), the central band P(|X|<|z|), or the two outer tails. The remaining quantities follow from Φ and Φ⁻¹. Example: z=1.12 has 0-to-z area ≈0.3686 on a from-mean table.

Area between two z-scores

P(z1 < Z < z2) = Φ(z2) − Φ(z1) is the probability mass between the bounds on the standard normal density — the shaded region in density diagrams.

Related calculators

Probability, counting, sampling design, and normal-based inference tools work together. Descriptive statistics calculators help when you start from raw data.

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