Calculate PDF (Probability Density Function) and CDF (Cumulative Distribution Function)
X Value
Mean (μ)
Std Dev (σ)
Results
Step-by-Step Calculation
Normal Distribution Formulas
PDF：f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
CDF：F(x) = ∫₋∞ˣ f(t)dt (Approximated by Error Function)
Z-Score：Z = (x - μ) / σ
Standard normal distribution: μ=0, σ=1. Normal distribution is the most important and widely used probability distribution in statistics.
⚠Standard deviation must be greater than 0. CDF values use numerical approximation (error function), accurate to 6 decimal places.
What is Normal Distribution?
Normal distribution (Gaussian distribution) is the most common continuous probability distribution in nature and statistics. It features a symmetric bell-shaped curve centered at the mean, with standard deviation determining width. The sum of many independent random variables approximates a normal distribution (Central Limit Theorem).
Bell Curve
Symmetric bell-shaped curve with maximum density at the mean. 68% of data within μ±σ, 95% within μ±2σ.
Standardization
Convert any normal distribution to N(0,1) using Z=(x-μ)/σ for easy lookup and probability calculation.
Central Limit Theorem
For large sample sizes, the sampling distribution of the sample mean approximates a normal distribution, forming the basis of statistical inference.
6σ Management
Six Sigma quality management uses normal distribution, requiring process deviation ≤1.5σ and defect rate ≤3.4ppm.
💡 Example: Standard normal distribution N(0,1), calculate probability at x=1. Z=(1-0)/1=1. PDF f(1)=1/√(2π)×e^(-0.5)≈0.2420. CDF F(1)=Φ(1)≈0.8413 (84.13% of data ≤1). 68-95-99.7 Rule: P(|Z|<1)≈68%, P(|Z|<2)≈95%, P(|Z|<3)≈99.7%.
What is the 68-95-99.7 rule for normal distribution?▼
In a normal distribution: approximately 68% of data falls within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations. This is the foundation of statistical quality control (6σ management).
What is a Z-score (standard score)?▼
Z = (x-μ)/σ, representing how many standard deviations a raw data point x is from the mean μ. The Z-score converts any normal distribution to the standard normal distribution N(0,1) for easy probability calculations and data comparison.
What is the difference between PDF and CDF?▼
PDF (Probability Density Function) shows the density at a specific x-value; probability equals the area under the curve. CDF (Cumulative Distribution Function) shows the probability that a variable is ≤ x, ranging from 0 to 1. CDF equals 0.5 at x = μ.
What are the practical applications of normal distribution?▼
Normal distribution is the most common natural distribution, widely used in: biological data (height/weight), measurement error analysis, test score standardization, quality control, financial risk calculation, machine learning (Gaussian Naive Bayes), and statistical inference.
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