What is the normal distribution?

The normal (Gaussian) distribution is a symmetric, bell-shaped curve defined by its mean (center) and standard deviation (spread). About 68% of data falls within 1 SD of mean, 95% within 2 SD, and 99.7% within 3 SD. It's the most important distribution in statistics due to the Central Limit Theorem.

What is the difference between probability and percentile?

The cumulative probability P(X ≤ x) gives the proportion of distribution falling at or below x. Percentile is the same value expressed as a percentage. P(Z ≤ 1.0) = 0.8413 = 84.13th percentile. They're different ways to express the same idea.

Why is normal distribution so important?

Three reasons: (1) Central Limit Theorem says sample means approach normal for large samples regardless of underlying distribution, (2) many natural phenomena are approximately normal (heights, scores, errors), (3) the math works out nicely — linear combinations of normals are normal, easing analysis.

How can I tell if my data is normally distributed?

Visual: plot a histogram and look for bell shape; create a Q-Q plot and check for straight diagonal. Tests: Shapiro-Wilk (powerful), Kolmogorov-Smirnov, Anderson-Darling. p > 0.05 = fail to reject normality (data could be normal). Note: with very large samples, even minor deviations trigger rejection.

What if my data isn't normal?

Options: (1) transform data (log for right-skew, square-root for moderate skew), (2) use non-parametric tests (Mann-Whitney, Wilcoxon, Kruskal-Wallis), (3) increase sample size and rely on CLT for sample means, (4) use Bayesian methods with appropriate priors. Many tests are robust to mild non-normality.

What does the empirical rule (68-95-99.7) mean?

For approximately normal distributions: 68% of data falls within ±1 SD of mean. 95% within ±2 SD. 99.7% within ±3 SD. This is the "empirical rule" or "68-95-99.7 rule." It helps quickly assess how unusual a value is. ONLY holds for normal distributions.

Normal Distribution Calculator

Q: What is the standard normal distribution?

The standard normal distribution N(0, 1) has mean 0 and standard deviation 1. Any normal distribution can be standardized by computing z-scores: z = (x - μ) / σ. The standard normal is foundational because tables and software are based on it.

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.

Inputs

Value (x)

Mean (mu)

Standard Deviation (sigma)

Results

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

Last updated: May 29, 2026

Formula

**Normal distribution PDF (probability density function):** f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²)) **Standard normal distribution (μ=0, σ=1):** φ(z) = (1/√(2π)) × e^(-z²/2) **Cumulative distribution function (CDF):** Φ(z) = P(Z ≤ z) Returns probability the random variable falls at or below z. **Z-score calculation:** z = (x - μ) / σ **Z and probability conversion (key values):** | z | P(Z ≤ z) | Percentile | |---|---|---| | -3 | 0.0013 | 0.13 | | -2 | 0.0228 | 2.28 | | -1 | 0.1587 | 15.87 | | -0.5 | 0.3085 | 30.85 | | 0 | 0.5000 | 50.00 | | 0.5 | 0.6915 | 69.15 | | 1 | 0.8413 | 84.13 | | 1.5 | 0.9332 | 93.32 | | 1.645 | 0.9500 | 95.00 | | 1.96 | 0.9750 | 97.50 | | 2 | 0.9772 | 97.72 | | 2.576 | 0.9950 | 99.50 | | 3 | 0.9987 | 99.87 | **Empirical Rule (68-95-99.7):** For approximately normal distributions: - ~68% within ±1 SD - ~95% within ±2 SD - ~99.7% within ±3 SD **Properties of normal distribution:** - **Symmetric** about mean: same shape on both sides. - **Bell-shaped**: highest probability at center. - **Continuous**: defined for all real numbers. - **Inflection points** at μ ± σ. - **Mean = Median = Mode**: all at center. - **Skewness = 0, Kurtosis = 3** (mesokurtic). **Worked example: Value 85, mean 75, SD 10** z = (85 - 75) / 10 = 1.0 P(X ≤ 85) = P(Z ≤ 1.0) = 0.8413 Percentile: 84.13% So 84% of values fall below 85 in this distribution. **Common applications:** | Field | Variable | Mean | SD | |---|---|---|---| | Education | IQ scores | 100 | 15 | | Education | SAT total | 1000 | 200 | | Anthropometry | Male height (cm) | 175 | 7 | | Anthropometry | Female height (cm) | 162 | 7 | | Manufacturing | Part dimension | spec | tolerance/3 | | Finance | Stock returns | mean return | volatility | | Healthcare | Blood pressure systolic | 120 | 12 | **Probability calculations:** For any normal distribution: - **P(X < a)** = Φ((a-μ)/σ) - **P(X > a)** = 1 - Φ((a-μ)/σ) - **P(a < X < b)** = Φ((b-μ)/σ) - Φ((a-μ)/σ) - **P(|X-μ| < kσ)** = Φ(k) - Φ(-k) = 2Φ(k) - 1 **Central Limit Theorem:** If X₁, X₂, ..., Xₙ are independent observations from any distribution (not necessarily normal) with mean μ and finite variance σ², then: Sample mean X̄ → Normal(μ, σ²/n) as n → ∞ This is why so much statistics depends on normal distribution. **When to use normal distribution:** ✓ Continuous quantitative variables. ✓ Many small independent contributions (CLT). ✓ Approximately symmetric, bell-shaped data. ✓ Most statistical methods (t-test, ANOVA, regression) assume normality. **When NOT to use normal:** ✗ Discrete counts (use Poisson or binomial). ✗ Strongly skewed data (use log-normal or other). ✗ Bounded variables (e.g., 0-100%) without transformation. ✗ Small samples of non-normal data. **Testing for normality:** - **Visual**: histogram, Q-Q plot. - **Shapiro-Wilk test**: powerful for small samples. - **Kolmogorov-Smirnov test**: another test. - **Anderson-Darling test**: emphasizes tails. **Transformations for non-normal data:** - **Log transformation**: for right-skewed data. - **Square root**: moderate right skew. - **Inverse**: extreme right skew. - **Box-Cox**: optimizes lambda parameter. **Critical values for hypothesis testing:** | Confidence | α | Z critical (two-tail) | |---|---|---| | 90% | 0.10 | ±1.645 | | 95% | 0.05 | ±1.960 | | 99% | 0.01 | ±2.576 | | 99.9% | 0.001 | ±3.291 | **Standardized scores derived from normal:** - **IQ**: z × 15 + 100 - **SAT section**: z × 100 + 500 - **T-scores**: z × 10 + 50 - **Stanines**: 9-point scale based on z **Probability software:** - **Excel**: NORM.DIST, NORM.S.DIST for CDF; NORM.INV for inverse. - **R**: pnorm(), qnorm(), dnorm(). - **Python (scipy.stats)**: norm.cdf, norm.ppf, norm.pdf. - **SPSS**: Compute Variable with PDF.NORMAL or CDF.NORMAL. **Real-world example notation:** X ~ N(μ, σ²) means X follows normal distribution with mean μ and variance σ². Example: Test scores X ~ N(75, 100) means mean = 75, variance = 100, SD = 10.

How to use this calculator

Enter a value (x).
Enter the distribution mean (μ).
Enter the standard deviation (σ).
Calculator returns z-score, P(X ≤ x), and percentile.
Use percentile to communicate relative position in distribution.
Use probability for hypothesis testing and confidence intervals.

Worked examples

Test score percentile

**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.

Adult male height

**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.

Manufacturing process check

**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.

When to use this calculator

**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.

Common mistakes to avoid

Assuming all continuous data is normal. Check with Q-Q plots or Shapiro-Wilk.
Applying empirical rule to non-normal data. 68-95-99.7 only holds for normal.
Forgetting Central Limit Theorem applies to sample means, not raw data.
Using normal for bounded variables (0-100%) without transformation.
Treating right-tailed probability as percentile. They're related but different.
Skipping normality check before t-tests, ANOVA, regression.
Using normal for discrete counts. Use Poisson, binomial, or normal approximation appropriately.

Frequently Asked Questions

Sources & further reading

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