The “bell” curve

## Normal distribution

As the number of observations of numerical data increase, they tend to form a bell-shaped curve called the **normal distribution** or Gaussian distribution.

**Properties of the normal distribution**

- Symmetrical around the mean
- Continuous, so the probability is measured as the area under the curve
- Single mode
- Mean is the highest point

A **standard normal **curve has a mean of zero and a standard deviation of one

**Z distribution and Z transformation**

The **Z distribution** can be used to calculate probabilities using the standard normal distribution. It describes the probability that a random draw from the standard normal distribution is greater than a given value

A non-standard normal can be transformed to a standard normal using the **Z transformation**.

**Law of large numbers**

**Law of large numbers**:** **As you increase the sample size, mean and standard deviation remains the same, but standard errors become smaller.

**Central limit theorem: **the mean of a large random sample from any population is approximately normally distributed