📈Normal Distribution Calculator
Calculate Z-scores, probabilities, and percentiles for any normal distribution. Enter mean (μ), standard deviation (σ), and a value X to get P(x < X), P(x > X), and the Z-score. Add a second value X₂ for range probability P(X₁ < x < X₂). Works for any normal distribution, not just standard normal.
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Normal Distribution Calculator: Z-Scores, Probabilities, and the Bell Curve
The normal distribution (bell curve) N(μ, σ²) describes how values cluster symmetrically around a mean μ with spread determined by standard deviation σ. The Z-score standardizes any value: Z = (X − μ) / σ. The probability P(x < X) equals the area under the curve to the left of X, computed via the standard normal CDF Φ(Z).
Key formula: Z = (X − μ) / σ → P(x < X) = Φ(Z)
| Range | Probability | Empirical Rule |
|---|---|---|
| μ ± 1σ | 68.27% | 68% rule |
| μ ± 2σ | 95.45% | 95% rule |
| μ ± 3σ | 99.73% | 99.7% rule |
The normal distribution appears throughout nature and statistics because of the Central Limit Theorem: the sum of many independent random variables converges to a normal distribution regardless of their individual distributions. This makes it the foundation of statistical inference, hypothesis testing, quality control, and predictive modeling.
Understanding the Z-Score
The Z-score measures how many standard deviations a value lies from the mean. Z = (X − μ) / σ. A Z-score of 0 means X equals the mean; Z = 1 means X is one standard deviation above the mean; Z = −2 means X is two standard deviations below the mean. Converting to Z-scores allows comparison across different normal distributions: a test score of 85 in a class with μ=70, σ=10 gives Z=(85−70)/10 = 1.5, while in a class with μ=80, σ=5 gives Z=(85−80)/5 = 1.0 — the first student performed relatively better despite the same absolute score.
Cumulative Probability and the CDF
The cumulative distribution function (CDF) Φ(z) gives the probability that a standard normal variable falls at or below z: Φ(z) = P(Z ≤ z). It equals the area under the bell curve from −∞ to z. Key values: Φ(0) = 0.5 (symmetric around 0); Φ(1) = 0.8413 (84.13% of values below 1σ); Φ(−1) = 0.1587; Φ(1.96) ≈ 0.975 (the basis of 95% confidence intervals). The complementary probability P(Z > z) = 1 − Φ(z). For two-sided problems: P(|Z| > z) = 2(1 − Φ(z)).
Range Probability and the Empirical Rule
The probability that X falls between two values X₁ and X₂ is P(X₁ < x < X₂) = Φ(Z₂) − Φ(Z₁). The "empirical rule" (68-95-99.7 rule) gives three memorable benchmarks: P(μ−σ < x < μ+σ) ≈ 68.27%; P(μ−2σ < x < μ+2σ) ≈ 95.45%; P(μ−3σ < x < μ+3σ) ≈ 99.73%. In practice: if adult male heights are normally distributed with μ=70 inches, σ=3 inches, then 95.45% of men are between 64 and 76 inches tall, and only 0.27% are more than 3σ from the mean (shorter than 61 or taller than 79 inches).
Frequently Asked Questions
What is a Z-score and how is it calculated?
A Z-score (standard score) measures how many standard deviations a data point is from the mean: Z = (X − μ) / σ. Example: if exam scores have μ=75, σ=8 and you scored 91, your Z-score is (91−75)/8 = 2.0 — you scored 2 standard deviations above average. Negative Z-scores mean below average. Z-scores allow comparison across different distributions: Z=1.5 always means the top 6.68% regardless of the original scale. Z-scores are used in standardized testing (SAT, IQ), quality control (Six Sigma), and hypothesis testing.
What is the difference between P(x < X) and P(x > X)?
P(x < X) is the left-tail (cumulative) probability: the probability that a random value drawn from the distribution is less than X. P(x > X) is the right-tail probability: the probability of a value greater than X. They always sum to 1: P(x < X) + P(x > X) = 1. For a symmetric normal distribution, P(x < μ) = P(x > μ) = 0.5. Example: if test scores are N(70, 10²) and X=80: Z=(80−70)/10=1; P(x<80) = Φ(1) ≈ 0.8413 (84.13% of students scored below 80); P(x>80) = 1−0.8413 = 0.1587 (15.87% scored above 80).
How do I find the percentile rank using a normal distribution?
The percentile rank of a value X in a normal distribution equals P(x < X) × 100. Example: if weights are normally distributed with μ=150 lbs, σ=20 lbs, and you weigh 175 lbs: Z=(175−150)/20=1.25; P(x<175) = Φ(1.25) ≈ 0.8944. You are in the 89.44th percentile. Conversely, to find the value at a specific percentile (e.g., 90th percentile): use the inverse normal function (probit) to get Z ≈ 1.2816 for the 90th percentile, then X = μ + Z × σ = 150 + 1.2816 × 20 ≈ 175.6 lbs.
What is the standard normal distribution?
The standard normal distribution is the special case N(0, 1²) — mean = 0 and standard deviation = 1. Any normal distribution can be converted to the standard normal by computing Z = (X − μ) / σ. This standardization is useful because probability tables and software functions typically compute probabilities for N(0,1), then all other normal distributions are handled by first converting to Z-scores. The standard normal PDF is φ(z) = (1/√2π) × e^(−z²/2), and its CDF Φ(z) does not have a closed-form expression — it must be computed numerically.
What is the Central Limit Theorem and why does the normal distribution matter?
The Central Limit Theorem (CLT) states that the sum (or mean) of n independent, identically distributed random variables with finite mean and variance converges to a normal distribution as n → ∞, regardless of the original distribution's shape. This is why the normal distribution is so ubiquitous: heights, measurement errors, test scores, financial returns — any quantity that results from many small independent effects tends to be approximately normal. The CLT also justifies using normal-distribution-based statistical tests (t-tests, ANOVA, confidence intervals) even when the underlying data is not perfectly normal, as long as sample sizes are large enough (typically n ≥ 30).