Example 1
IQ scores: \(\mu = 100, \sigma = 15, x = 115 \Rightarrow z = 1, P(X < 115) = 84.13\%\)
Normal Distribution
Calculate normal distribution probabilities and z-scores
Results
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Normal Distribution
The normal distribution (also called Gaussian distribution) is a continuous probability distribution that is symmetric around its mean. It is one of the most important distributions in statistics.
- Mean (μ): The center of the distribution
- Standard Deviation (σ): Measures the spread of the distribution
- Z-Score: Number of standard deviations a value is from the mean: z = (x - μ) / σ
- 68-95-99.7 Rule: Approximately 68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ
The normal distribution is widely used in natural and social sciences to represent real-valued random variables whose distributions are not known. Many statistical tests assume normality.
\(z = \frac{x - \mu}{\sigma}, f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\)
Worked Examples
Example 2
Heights (cm): \(\mu = 170, \sigma = 10, x = 180 \Rightarrow z = 1, P(X > 180) = 15.87\%\)