مجاني عبر الإنترنتالآلات الحاسبة
Normal Distribution Calculator
Calculate probabilities under the normal distribution curve. Find P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b) for any mean and standard deviation.
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Query Type
Bell Curve (μ=0, σ=1)
-4μ−2σμ=0μ+2σ4
Empirical Rule (68-95-99.7)
Within 1σ (μ±σ)
68.27%
-1 to 1
Within 2σ (μ±2σ)
95.45%
-2 to 2
Within 3σ (μ±3σ)
99.73%
-3 to 3
الميزات الرئيسية — Normal Distribution Calculator
P(X ≤ x), P(X ≥ x), and P(a ≤ X ≤ b)
Custom mean (μ) and standard deviation (σ)
Interactive bell curve visualization
Empirical rule (68-95-99.7)
Standard normal (μ=0, σ=1) supported
لماذا تستخدم هذه الأداة — Normal Distribution Calculator
مجاني عبر الإنترنت Normal distribution calculatorBell curve calculator — للجميعسريع وسهل Gaussian distribution calculatorProbability normal distribution — مجاني 100%مجاني عبر الإنترنت Standard normal distribution calculator
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الأسئلة الشائعة — Normal Distribution Calculator
What is the normal distribution?
The normal distribution (bell curve, Gaussian distribution) is a symmetric, continuous probability distribution defined by its mean (μ) and standard deviation (σ). Many natural phenomena follow this distribution: heights, test scores, measurement errors. It is fundamental to statistics and the central limit theorem.
What is the empirical rule (68-95-99.7 rule)?
For a normal distribution: approximately 68% of data falls within 1 standard deviation of the mean (μ±σ), 95% within 2 standard deviations (μ±2σ), and 99.7% within 3 standard deviations (μ±3σ). This is also called the three-sigma rule and is used to identify outliers.
What is the standard normal distribution?
The standard normal distribution has μ=0 and σ=1. Any normal distribution can be standardized by converting values to z-scores: Z = (X−μ)/σ. This allows use of standard z-tables. This calculator works for any μ and σ — it automatically handles the conversion.