A z-score tells you how many standard deviations a value lies from the mean: z = (x − μ) / σ. The Toolenza calculator computes z from raw values, plus the corresponding percentile (probability of seeing a smaller value) via the normal CDF.

The intuition

• z = 0 — exactly average.
• z = +1 — 1 standard deviation above average; 84th percentile.
• z = +2 — 2 σ above; 97.7th percentile.
• z = +3 — 3 σ above; 99.87th percentile ("three-sigma event").
• z = −2 — 2 σ below; 2.3rd percentile.

The famous 68–95–99.7 rule: about 68% of data falls within ±1 σ, 95% within ±2 σ, 99.7% within ±3 σ.

Where z-scores actually get used

• Standardised tests — SAT and IQ scores are z-scores rescaled to mean 500 (SAT) or 100 (IQ).
• Paediatrics — child growth charts use z-scores ("BMI z-score of +2 means 97.7th percentile for that age").
• Finance — Altman Z-score for bankruptcy prediction (a different formula but same idea).
• Outlier detection — z > 3 or z < −3 is often flagged as anomalous.
• Quality control — six-sigma processes target a z = 6 defect rate (3.4 per million).

Pitfalls

The normal-distribution percentile only applies if your data is roughly normal. Income distribution is heavily right-skewed; a z-score of 3 in income doesn't mean 99.87th percentile — it means top fraction of a percent. For skewed distributions, rank-based percentiles are more honest.