Normal Distribution Calculator

Normal Distribution (Bell Curve):

The normal distribution is the most important probability distribution in statistics. It describes data that clusters around a central value with a symmetric bell shape. This calculator finds the probability of values falling below, above, or between any two points.

Probability Density Function:

f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²))

What each variable means:

• Mean (μ) — the center of the distribution, where the peak of the bell curve sits.
• Standard Deviation (σ) — measures the spread. A larger σ means a wider, flatter curve.
• Value (x) — the point at which you want to calculate the probability.
• Z-score — how many standard deviations a value is from the mean: z = (x - μ) / σ.

Key properties (the 68-95-99.7 rule):

• 68.27% of data falls within ±1σ of the mean
• 95.45% of data falls within ±2σ of the mean
• 99.73% of data falls within ±3σ of the mean

When to use this calculator: Use it for quality control, test score analysis, scientific measurements, or any situation where data follows a bell-shaped pattern. For example, human heights, exam scores, and manufacturing tolerances often follow normal distributions.

Practical example: Exam scores have a mean of 75 and standard deviation of 10. What percentage of students scored below 85? Z = (85 - 75) / 10 = 1.0. P(X < 85) = 84.13%, meaning about 84% of students scored below 85.

Standard Normal Distribution: When μ = 0 and σ = 1, it is called the standard normal distribution. Any normal distribution can be converted to standard normal using z-scores, which is how statistical tables and this calculator work internally.

Tips: If your data is skewed or has heavy tails, the normal distribution may not be appropriate. Always visualize your data first. The calculator uses a numerical approximation of the cumulative distribution function, which is accurate to several decimal places.