Z-score Calculator

Calculated using standard mathematical and industry formulas.

Free online Z Score Calculator calculator and unit conversion tool for everyday, engineering, and home projects.

How to use this calculator?

1. Enter the mean and standard deviation of your distribution.
2. In 'Raw score → Z-score' mode: enter the raw score x to get z and its percentile.
3. In 'Z-score → Raw score' mode: enter z to find the corresponding raw value.

Formula & calculation method

Z-score formula and normal CDF

z = (x − μ) / σ · x = μ + z × σ · P(Z < z) computed from normal CDF

CDF computed via polynomial approximation (Abramowitz & Stegun). Accurate to ±5 × 10⁻⁴.

Step-by-step calculation example

x = 85, μ = 70, σ = 10: z = (85 − 70) / 10 = 1.5. P(Z < 1.5) ≈ 0.9332 → 93.3rd percentile.

Key insights to remember

• Z = 0 is exactly at the mean; Z = ±1 covers the middle 68.3% of a normal distribution.
• Z = ±2 covers 95.4%; Z = ±3 covers 99.7% (the '68-95-99.7 rule').
• Z-scores enable comparison across different scales: a test score and a height can be placed on the same standard scale.
• The empirical rule: Z > 1.96 places you in the top 2.5% for a two-tailed test at α = 0.05.

How to interpret your results?

Percentile

P(Z < z) × 100 gives the percentile: z = 1.5 means your value exceeds 93.3% of the distribution.

Standardisation

Z-scores remove units and scale differences, allowing direct comparison across datasets with different means and SDs.

Frequently asked questions

How do I use the Z Score Calculator calculator?

Enter your numerical input and select desired units to get instant accurate results.

How accurate are the conversions?

Conversions use official ISO conversion factors for maximum precision.

Does it convert metric and imperial units?

Yes, seamless conversions between metric (meters, kg) and imperial (feet, lbs) are supported.

Is my data stored?

No, all calculations execute 100% locally in your browser with zero data logging.

Can I use this for DIY or engineering projects?

Yes, it provides reliable estimates for construction, crafting, science, and everyday math.

What if I enter invalid inputs?

The calculator will display a friendly error prompt if non-numeric values are entered.

Related calculators

• Standard Deviation Calculator — Calculate sample and population standard deviation, variance, standard error, and coefficient of variation for any dataset.
• Confidence Interval Calculator — Calculate confidence intervals for a sample mean or proportion at 80%, 90%, 95%, 98%, or 99% confidence levels.
• P-value Calculator — Calculate the P-value from a Z-statistic for one-tailed or two-tailed hypothesis tests and determine statistical significance.

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