Calculate Z-Score & Probability with Step-by-Step Derivation
Mean (μ)
Std Dev (σ)
Value (x)
Results
Z-Score
P(X ≤ x)
P(X > x)
P(|X-μ| > |x-μ|)
Step-by-Step Derivation
Normal Distribution Formula
PDF: f(x) = (1/√(2πσ²)) × e^(-(x-μ)²/(2σ²))
Z-score: z = (x - μ) / σ
P(X ≤ x) = Φ(z) = cumulative distribution function
Normal distribution is symmetric, bell-shaped, and defined by mean μ and standard deviation σ.
⚠σ must be positive. Results use standard normal CDF approximation.
What is Normal Distribution?
The normal (or Gaussian) distribution is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Characteristics
Bell-shaped, symmetric, mean=median=mode
Parameters
μ (mean), σ (standard deviation)
68-95-99.7 Rule
68% within μ±σ, 95% within μ±2σ, 99.7% within μ±3σ
Normal (Gaussian) distribution is a continuous probability distribution symmetric around the mean. Many natural phenomena follow this bell-shaped curve.
What is z-score?▼
Z-score measures how many standard deviations a value is from the mean. z = (x - μ) / σ. Used to standardize normal distributions.
What is standard normal distribution?▼
Standard normal has μ=0 and σ=1. Any normal distribution can be standardized by subtracting mean and dividing by standard deviation.
How to calculate probability from z-score?▼
Use the cumulative distribution function Φ(z). P(X ≤ x) = Φ(z) where z = (x-μ)/σ. Tables or calculators provide Φ(z) values.
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