# The normal distribution

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