Normal Distribution Calculator
Calculate probabilities and percentiles for standard and general normal distributions, z-scores, and interval areas.
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What Is a Normal Distribution Calculator?
The Normal (Gaussian) Distribution is the quintessential continuous probability distribution in statistics. Symmetrical and bell-shaped, it arises naturally in measurement errors, physical traits (height, blood pressure), standardized test scores (SAT, IQ), and financial market returns.
How to Use This Calculator
- Enter the population Mean (μ) and Standard Deviation (σ).
- Select your calculation region: Left-tailed P(X ≤ x₁), Right-tailed P(X ≥ x₁), Between P(x₁ ≤ X ≤ x₂), or Outside tails.
- Enter the target values (x₁ and x₂).
- Review the calculated probability percentage, bell curve area, and corresponding z-scores.
Normal Probability Density and CDF
f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, \quad z = \frac{x-\mu}{\sigma}, \quad \Phi(z) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right)\right]Where μ is the mean, σ is the standard deviation, z is the standardized score, and erf is the Gauss error function.
Worked Example
Scenario: IQ scores follow a normal distribution with mean μ = 100 and standard deviation σ = 15. Find the probability of a person scoring between 100 and 115.
Calculate z1 for x1 = 100: z1 = (100 - 100) / 15 = 0.00 → Φ(0) = 0.5000.
Calculate z2 for x2 = 115: z2 = (115 - 100) / 15 = +1.00 → Φ(1) ≈ 0.8413.
Subtract CDF values: P(100 ≤ X ≤ 115) = 0.8413 - 0.5000 = 0.3413 (34.13%).
Tips & Key Notes
- The Empirical Rule (68–95–99.7 rule): roughly 68.27% of data falls within 1σ of the mean, 95.45% within 2σ, and 99.73% within 3σ.
- A z-score measures how many standard deviations an observation lies above or below the mean.
Frequently Asked Questions
What is a z-score?
A z-score (standard score) indicates how many standard deviations an element is from the mean: z = (x - μ) / σ. A positive z-score is above average; a negative is below.
What is the standard normal distribution?
The standard normal distribution is the special case of the normal distribution with mean μ = 0 and standard deviation σ = 1.
What percentage of data is beyond 2 standard deviations?
Approximately 4.55% of total data lies outside ±2 standard deviations (2.28% in each tail).
Why does the Central Limit Theorem make the normal distribution so important?
The Central Limit Theorem proves that the sum or average of a large number of independent random variables tends toward a normal distribution, regardless of the shape of the underlying population distribution.
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