Normal Distribution Calculator

X ↔ Probability Bell Curve Custom μ σ Free · No Signup

Free online normal distribution calculator for any μ and σ: compute P(X ≤ x) probabilities, find X from percentile (inverse normal), or calculate P(a ≤ X ≤ b) range probability. Interactive Plotly bell curve, step-by-step KaTeX formulas, and Python scipy export.

Normal Distribution Calculator
Find the probability for this value

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Enter values and click Calculate

Find probabilities, percentiles, and Z-scores for your normal distribution.

Normal Distribution Curve

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Calculate to see the bell curve.

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What Is the Normal Distribution?

The normal distribution (Gaussian distribution) is a symmetric, bell-shaped probability distribution defined by two parameters: the mean (μ) and standard deviation (σ). It is the most important distribution in statistics because many natural phenomena follow it.

Probability Density Function:  f(x) = (1 / σ√(2π)) × e−(x−μ)² / (2σ²)
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Symmetric

Perfectly symmetric about the mean. Mean = median = mode, all at the center.

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Defined by μ and σ

μ sets the center, σ controls the spread. Larger σ means a flatter, wider bell.

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Asymptotic

Tails extend infinitely but approach zero. Total area under the curve equals 1.

The 68-95-99.7 Rule (Empirical Rule)

68% 95% μ−2σ μ−1σ μ μ+1σ μ+2σ
RangeProbabilityExample (IQ: μ=100, σ=15)
μ ± 1σ68.27%IQ 85 – 115
μ ± 2σ95.45%IQ 70 – 130
μ ± 3σ99.73%IQ 55 – 145

Standardization & Z-Scores

Any normal distribution N(μ, σ) can be converted to the standard normal N(0, 1) by computing the Z-score:

Z = (X − μ) / σ
Worked Example: IQ scores follow N(100, 15). What Z-score corresponds to IQ = 130?
Z = (130 − 100) / 15 = 30 / 15 = 2.0
Φ(2.0) = 0.9772 → IQ 130 is at the 97.7th percentile
Z-ScoreLeft Tail P(Z ≤ z)Percentile
−2.3260.01001st
−1.6450.05005th
0.0000.500050th
+1.6450.950095th
+1.9600.975097.5th
+2.3260.990099th

Real-World Applications

IQ & Standardized Testing

IQ scores follow N(100, 15). SAT scores approximate N(1060, 195). Z-scores enable comparison across different tests.

Quality Control

Manufacturing uses normal distribution to set tolerance limits. Six Sigma targets Z = ±6 for defect rates below 3.4 per million.

Biological Measurements

Heights, weights, blood pressure, and many biological variables are approximately normally distributed in populations.

Central Limit Theorem

Sample means approach a normal distribution as n increases, regardless of the population shape. This is the foundation of inferential statistics.

Frequently Asked Questions

A normal distribution (Gaussian distribution) is a symmetric, bell-shaped probability distribution defined by its mean μ and standard deviation σ. It is the most common distribution in statistics because many natural phenomena follow it due to the Central Limit Theorem.
First standardize by computing Z = (x − μ) / σ. Then look up the cumulative distribution function Φ(Z). This gives the left-tail probability. For the right tail, subtract from 1: P(X ≥ x) = 1 − Φ(Z).
The inverse normal function Φ−1(p) returns the X value such that P(X ≤ x) = p. It converts a probability or percentile back to a value on the distribution. For example, the 95th percentile of N(100, 15) is about 124.67.
Use P(a ≤ X ≤ b) = Φ(Zb) − Φ(Za), where Za and Zb are the standardized values. This calculator computes it automatically in Range mode.
Z-scores standardize any normal distribution to the standard normal N(0, 1). The formula Z = (X − μ) / σ transforms values so they can be compared across different distributions. A Z-score of 2 always means 2 standard deviations above the mean, regardless of the original scale.
Data often follows a normal distribution when it results from many small, independent effects. Common examples include heights, measurement errors, and test scores. Use normality tests like Shapiro-Wilk or Q-Q plots to verify. The Central Limit Theorem guarantees sample means are approximately normal for large samples.

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