Normal Distribution Calculator
Calculate the probability of a value occurring in a normal distribution.
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Table of Contents
Normal Distribution Formula
The normal distribution (also known as the Gaussian distribution) is a continuous probability distribution. The probability density function is:
Where:
- x is the value
- μ is the mean
- σ is the standard deviation
- π is pi (approximately 3.14159)
- e is Euler's number (approximately 2.71828)
How to Calculate Normal Distribution
To calculate the probability in a normal distribution:
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1Identify the value (x), mean (μ), and standard deviation (σ)
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2Calculate the z-score: z = (x - μ) / σ
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3Use the cumulative normal distribution function to find the probability
Understanding Normal Distribution
Key characteristics of normal distribution:
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1Bell Shape:
The distribution is symmetric and bell-shaped, with the highest point at the mean.
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268-95-99.7 Rule:
About 68% of values fall within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
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3Applications:
Used in natural and social sciences to represent real-valued random variables whose distributions are not known.
Practical Examples
Example 1 Test Scores
A class has test scores with mean 75 and standard deviation 10
x = 85, μ = 75, σ = 10
Probability ≈ 0.8413
About 84.13% of students scored 85 or below.
Example 2 Height Distribution
Adult male heights with mean 70 inches and standard deviation 3 inches
x = 73, μ = 70, σ = 3
Probability ≈ 0.8413
About 84.13% of men are 73 inches or shorter.
Example 3 IQ Scores
IQ scores with mean 100 and standard deviation 15
x = 115, μ = 100, σ = 15
Probability ≈ 0.8413
About 84.13% of people have an IQ of 115 or below.