Math
Z-Score Calculator
Calculate z-scores and probabilities for normal distributions
z = (x − μ) / σ = (75 − 70) / 10 = 0.5000
Z-score
0.5000
within 1 standard deviation of the mean (68% of data)
P(X < x)
69.15%
cumulative / below
P(X > x)
30.85%
above / right tail
P(±|z|)
38.29%
two-tailed interval
Normal distribution reference
±1σ: 68.27%
±2σ: 95.45%
±3σ: 99.73%
## Z-Score Calculator — Standard Scores and Normal Distribution Probabilities
The z-score is one of the most fundamental tools in statistics, used in hypothesis testing, quality control, medical diagnostics, educational testing, and financial risk analysis. Our calculator handles three common scenarios: calculating a z-score from a raw value, finding the probability associated with a z-score, and reversing the process to find a value at a given percentile.
### The Z-Score Formula
$$z = \frac{x - \mu}{\sigma}$$
Where:
- **x** = observed value
- **μ** = population mean
- **σ** = population standard deviation
The result tells you how many standard deviations the value sits above (positive) or below (negative) the mean.
### Understanding the Normal Distribution
In a standard normal distribution (μ = 0, σ = 1):
- **68.27%** of values fall within ±1 SD (z between −1 and 1)
- **95.45%** of values fall within ±2 SD
- **99.73%** of values fall within ±3 SD (the "3-sigma rule")
### Probability Interpretation
For any z-score, our calculator gives three probabilities:
| Probability | Meaning |
|------------|---------|
| P(X < x) | Area to the left — percentile rank |
| P(X > x) | Area to the right — probability of exceeding this value |
| P(−|z| < Z < |z|) | Area between ±z — central probability |
### Common Z-Score Critical Values
| z-score | % below | Use in practice |
|---------|---------|-----------------|
| 1.282 | 90% | 90% confidence interval (one-tailed) |
| 1.645 | 95% | 95% CI (one-tailed) / 5% significance level |
| 1.960 | 97.5% | 95% CI (two-tailed) — most common |
| 2.576 | 99.5% | 99% CI (two-tailed) |
| 3.000 | 99.87% | Three-sigma quality control |
### Practical Applications
**Education**: Compare test scores across different exams. A score of 80 on one test and 70 on another may be equivalent or different depending on the mean and SD of each.
**Medicine**: Growth charts, blood test reference ranges, and BMI percentiles all use z-scores to compare an individual to population norms.
**Finance**: Value at Risk (VaR) calculations use z-scores to estimate the probability of portfolio losses exceeding a threshold.
**Manufacturing**: Six Sigma processes target ±6 standard deviations from the mean, corresponding to 3.4 defects per million opportunities.
Frequently Asked Questions
What is a z-score?
A z-score (or standard score) tells you how many standard deviations a data point is from the mean of its distribution. The formula is z = (x − μ) / σ. Positive z-scores are above the mean, negative ones are below. A z-score of 0 means the value equals the mean exactly.
What is a "good" or "bad" z-score?
Z-scores themselves have no inherent good/bad interpretation — it depends on context. For test scores, a high z-score is good. For medical markers like blood pressure, a high z-score may indicate a problem. The absolute value tells you how unusual the data point is relative to the rest of the distribution.
What does a z-score of 1.96 represent?
Z = ±1.96 corresponds to the 95% confidence level. This means 95% of all values in a standard normal distribution fall within 1.96 standard deviations of the mean. This value is widely used in hypothesis testing and confidence interval construction.
What is the area under the normal curve?
The area under the entire normal distribution curve equals 1 (or 100%). The area to the left of a given z-score gives P(X < x) — the probability of randomly drawing a value less than x. Our calculator shows both the area below and above any z-score.
Can I convert a probability back to a z-score?
Yes — this is called the inverse normal (or probit) function. If you know that a value is at the 90th percentile (probability = 0.90), the reverse calculation gives you z ≈ 1.282. Our calculator supports this "reverse" mode.
When should I use a z-score vs a t-score?
Use z-scores when the population standard deviation is known and the sample is large (n > 30). Use t-scores when the population SD is unknown and you are using a sample standard deviation, especially for small samples.