Question 1

What does a z-score tell you?

Accepted Answer

A z-score tells you how far a value is from the mean, measured in standard deviation units. A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean; −2 means two standard deviations below. Under a normal distribution: ~68% of values have z-scores between −1 and +1; ~95% between −2 and +2; ~99.7% between −3 and +3. This is the empirical rule (68–95–99.7 rule).

Question 2

How do you interpret a z-score in practice?

Accepted Answer

Example: a student scores 82 on an exam with mean 70 and standard deviation 10. z = (82 − 70) / 10 = 1.2. This means the student scored 1.2 standard deviations above the class average — approximately the 88th percentile. Z-scores allow comparisons across different exams, scales, or populations without needing to know the original units.

Question 3

What is the difference between z-score and t-score?

Accepted Answer

Both standardise a value relative to a distribution. Use a z-score when the population standard deviation (σ) is known, or when the sample size is large (n ≥ 30). Use a t-score when the population standard deviation is unknown and must be estimated from the sample — typically with small samples. For large samples, z and t values converge to nearly the same result.

Question 4

How do you convert a z-score to a percentile?

Accepted Answer

A percentile represents the percentage of values at or below a given point. For a normal distribution, a z-score of 0 corresponds to the 50th percentile. A z-score of +1 corresponds to approximately the 84.1st percentile. A z-score of −1.645 corresponds to the 5th percentile (often used as the critical value for one-tailed 5% significance tests). This calculator uses the normal CDF to convert z-scores to percentiles automatically.

Question 5

Can z-scores be used with non-normal distributions?

Accepted Answer

Yes — the z-score formula itself works for any distribution. However, the percentile and probability interpretations (using the normal CDF) are only accurate for data that is approximately normally distributed. For heavily skewed, bimodal, or non-normal distributions, z-scores still measure distance from the mean in SD units, but the corresponding percentile estimates will be inaccurate.

Question 6

What are typical z-score critical values?

Accepted Answer

Common critical values: ±1.645 for 90% confidence interval (5% in each tail); ±1.960 for 95% CI (2.5% each tail); ±2.326 for 98% CI; ±2.576 for 99% CI. In one-tailed hypothesis tests: 1.282 (10%), 1.645 (5%), 2.326 (1%). These values come from the standard normal table and are used extensively in statistical inference.

Z-Score	P(X ≤ z)	Percentile	Use Case
−2.576	0.50%	0.5th	99% CI lower tail
−1.960	2.50%	2.5th	95% CI lower tail
−1.645	5.00%	5th	90% CI lower tail / one-tail 5%
0.000	50.00%	50th	Mean — median for symmetric dist.
+1.282	90.00%	90th	One-tail 10% significance
+1.645	95.00%	95th	One-tail 5% / 90% CI upper tail
+1.960	97.50%	97.5th	95% CI upper tail
+2.326	99.00%	99th	98% CI upper tail / one-tail 1%
+2.576	99.50%	99.5th	99% CI upper tail
+3.000	99.87%	99.87th	Six-sigma quality baseline

Condition	Use Z	Use T
Population σ known	✓
Population σ unknown		✓
Large sample (n ≥ 30)	✓	Acceptable
Small sample (n < 30)	Not ideal	✓
Data is normal	✓	✓
Comparing means across populations	✓ (known σ)	✓ (unknown σ)

Z-Score Calculator

Formula

The 68–95–99.7 Rule (Empirical Rule)

Common Critical Z-Values

Z-Score vs T-Score — When to Use Which

Related Calculators

Disclaimer

Frequently Asked Questions