Enter a value, mean, and standard deviation to compute the z-score, cumulative probability, and percentile rank under the normal (Gaussian) distribution curve.
The normal distribution (also called Gaussian distribution or "bell curve") is the most important probability distribution in statistics. Many natural phenomena follow approximately normal distributions: heights, blood pressure, test scores, measurement errors, IQ scores, and many physical quantities. The Central Limit Theorem says that sample means from any distribution approach normality as sample size grows — making normal distribution the foundation of inferential statistics.
This calculator returns z-score, cumulative probability, and percentile rank for a value from a normal distribution with specified mean and standard deviation. The cumulative probability P(X ≤ x) gives the area under the curve to the left of x; percentile is the same value as a percentage. For example, z = 1.0 corresponds to probability 0.8413 = 84.13th percentile.
The normal distribution is defined by two parameters: mean (μ, center) and standard deviation (σ, spread). The standard normal distribution has μ = 0 and σ = 1. Any normal distribution can be standardized by computing z-scores. The empirical rule (68-95-99.7) is fundamental: 68% of data within ±1σ, 95% within ±2σ, 99.7% within ±3σ of the mean.
**Scenario:** Student scored 85 on test. Class: mean 75, SD 10. **Calculation:** z = (85-75)/10 = 1.0. P(Z ≤ 1.0) = 0.8413. Percentile: 84.13. **Result:** Student scored at the 84th percentile (better than 84% of classmates). 1 standard deviation above mean. In the top ~16% of class.
**Scenario:** Adult male heights: mean 175 cm, SD 7 cm. Probability someone is over 190 cm? **Calculation:** z = (190-175)/7 = 2.14. P(Z > 2.14) = 1 - 0.984 = 0.016. **Result:** Only ~1.6% of adult males are over 190 cm tall. This matches expectation that very tall men are rare in the general population.
**Scenario:** Spec: 100 ± 3. Process distribution: mean 100, SD 1. What proportion of parts are within spec? **Calculation:** P(97 ≤ X ≤ 103) = P((97-100)/1 ≤ Z ≤ (103-100)/1) = P(-3 ≤ Z ≤ 3) = 0.9987 - 0.0013 = 0.9973. **Result:** 99.73% of parts within spec. This matches the empirical rule (within ±3 SD = ~99.7%). Process is well-centered and capable.
**Use normal distribution for:**
- **Continuous variables**: heights, weights, scores, measurements. - **Random sampling theory**: sample means follow normal (CLT). - **Hypothesis testing**: z-tests, t-tests assume normality. - **Confidence intervals**: based on normal critical values. - **Quality control**: process capability uses normal assumption. - **Risk modeling**: Value at Risk, portfolio theory. - **Test scoring**: standardizing assessment results.
**Identifying when data is normal:**
- Roughly symmetric histogram. - Q-Q plot points close to diagonal line. - Shapiro-Wilk test p > 0.05 (fail to reject normality). - Mean ≈ median. - Empirical rule (68-95-99.7) approximately holds.
**Distributions related to normal:**
- **Chi-square**: sum of squared standard normals. - **t-distribution**: standard normal divided by sample SD. - **F-distribution**: ratio of chi-squares. - **Log-normal**: log of normal variable.
**Standardization process:**
For any normal variable X ~ N(μ, σ²): 1. Subtract mean: X - μ. 2. Divide by SD: (X - μ)/σ. 3. Result: Z ~ N(0, 1) (standard normal).
This enables comparing values across different distributions.
**Computing tail probabilities:**
- **Right tail**: P(X > a) = 1 - Φ((a-μ)/σ) - **Left tail**: P(X < a) = Φ((a-μ)/σ) - **Both tails (symmetric)**: 2 × (1 - Φ(|z|)) - **Between values**: Φ((b-μ)/σ) - Φ((a-μ)/σ)
**Why normal distribution is special:**
1. **Central Limit Theorem**: sample means from any distribution approach normal. 2. **Maximum entropy**: most "spread out" distribution given mean and variance. 3. **Sum of independent normals**: still normal. 4. **Stable distribution**: many natural processes converge to normal. 5. **Easy mathematics**: linear transformations preserve normality.
**Common misuses:**
- Applying to skewed data without transformation. - Assuming small samples from any distribution are normal. - Using empirical rule for non-normal data. - Computing CI/p-values without checking normality assumption.
**Normality testing decision:**
Sample size: - **n < 30**: critical to check normality. - **n = 30-100**: still important to check. - **n > 100**: CLT often makes mean approximately normal regardless.
Tests for normality may detect minor non-normality in large samples. Practical importance depends on subsequent analysis.
**Common standardized test scores:**
| Test | Mean | SD | Notes | |---|---|---|---| | WISC IQ | 100 | 15 | Most common IQ test | | Stanford-Binet IQ | 100 | 16 | Older IQ scale | | SAT (each section) | 500 | 100 | Pre-2016 | | SAT (sections, 2016+) | 500 | varies | Modern SAT | | ACT (composite) | 21 | varies | American College Test | | GRE (Q+V) | 305 | 30 | Graduate Record Exam | | LSAT | 150 | varies | Law School Admission | | GMAT | 550 | varies | Graduate Management |
**Bayesian perspective:**
Normal distribution is the conjugate prior for the mean of a normal likelihood when SD is known. This makes Bayesian analysis with normal models particularly elegant.
**Real-world variations:**
Few real distributions are exactly normal: - **Heights**: approximately normal in adults. - **Test scores**: designed to be normal. - **Manufacturing**: approximately normal (with quality control). - **Stock returns**: approximately normal short-term, fatter tails long-term. - **Income**: log-normal usually (highly right-skewed). - **Earthquakes**: log-normal for magnitude.
Calculate the z-score from a value, population mean, and standard deviation.
Calculate P(X=k) and cumulative probabilities for a binomial distribution.
Calculate population and sample standard deviation from a data set.
Calculate P(X=k) and cumulative probabilities for a Poisson distribution.
Calculate the interquartile range (IQR) from a data set of up to 10 values.
Find the percentile rank of a value in a data set.
Z-Score
1.0000
P(X <= x)
0.841345
Percentile
84.13%
P(X > x)
0.158655
PDF f(x)
0.024197
68% Range (1 SD)
65.00 to 85.00
95% Range (2 SD)
55.00 to 95.00
99.7% Range (3 SD)
45.00 to 105.00