Normal Distribution probability

• What is the normal distribution?


  A **bell-shaped** probability distribution that is **symmetrical** about the mean, where most data points cluster around the mean.
• What are the key properties of a normal distribution?

  • **Symmetrical** about the mean.
  • Mean, median, and mode are **equal**.
  • **68%** of data lies within $\pm1\sigma$ of the mean.
  • **95%** of data lies within $\pm2\sigma$ of the mean.
  • **99.7%** of data lies within $\pm3\sigma$ of the mean.
• What is the standard normal distribution?


  A normal distribution with a mean of $0$ and a standard deviation of $1$, denoted as $Z \sim N(0, 1)$.
• How do you standardize a normal distribution?


  Use the formula: $Z =$$\frac{X - \mu}{\sigma}$, where:

  • $X$ = data point,
  • $\mu$ = mean,
  • $\sigma$ = standard deviation.
• What is the z-score?


  A measure of how many **standard deviations** a data point is from the mean. Calculated as: $Z =$$\frac{X - \mu}{\sigma}$.
• What is the notation for a normal distribution?

  It is denoted as $N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance.
• What percentage of data lies within 1 standard deviation ($\sigma$) of the mean in a normal distribution?


  Approximately **68%** of the data lies within $\mu \pm \sigma$.
• What percentage of data lies within 2 standard deviations ($2\sigma$) of the mean in a normal distribution?


  Approximately **95%** of the data lies within $\mu \pm 2\sigma$.
• What percentage of data lies within 3 standard deviations ($3\sigma$) of the mean in a normal distribution?


  Approximately **99.7%** of the data lies within $\mu \pm 3\sigma$.
• What is the inverse normal function used for?

  It finds the value of $x$ given a cumulative probability $p$ in a normal distribution, denoted as $X =$$\text{invNorm}(p, \mu, \sigma)$.