Standard Normal Distribution Notes

Standardizing Normal Distributions

• There are infinite normal distributions, but standardizing them allows us to treat them as one.
• Standardization is crucial for confidence intervals and hypothesis tests because the required values are derived from the standardized normal distribution.

The Standard Normal Distribution

• The standard normal distribution is a specific type of normal distribution obtained through transformation.
• Normal distributions can vary with different means ($\mu$) and standard deviations ($\sigma$).
• Standardization transforms these different normal distributions into a single, standard form.

Transformation

• Starting with a normal distribution x, which has mean $(\mu)$
• Transformation process:
  • Subtract the mean ($\mu$) from each value of x.
  • Divide the result by the standard deviation ($\sigma$).
• Equation: $z = \frac{x - \mu}{\sigma}$

Properties of the Standard Normal Distribution

• The resulting distribution, z, is the standard normal distribution.
• Mean: 0
• Variance: 1
• Standard Deviation: 1

Usefulness

• The standard normal distribution is widely applicable in various statistical analyses.
• Key applications include:
  • Constructing confidence intervals.
  • Performing hypothesis testing.