# Normal Distribution

# Definition and Properties of Normal Distribution

**Normal distribution**, also known as Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable.- The
**graph**of the normal distribution is bell-shaped and symmetrical, centered around the mean. The spread is determined by the standard deviation. - Normal distribution can be described by two parameters: the
**mean (μ)**and the**standard deviation (σ)**. - The
**total area**under the normal distribution graph is equal to 1, corresponding to the total probability of all possible outcomes.

# Standard Normal Distribution

- The
**standard normal distribution**is a specific type of normal distribution where mean (μ) is 0 and standard deviation (σ) is 1. - Any normal distribution can be converted to a standard normal distribution using the
**standardisation**formula:**Z = (X - μ) / σ**, where Z is the standardised value, X is an observation from the original normal distribution, μ and σ are the mean and standard deviation of the original normal distribution respectively.

# Using the Normal Distribution Tables

- Normal distribution tables provide the probabilities for the standard normal distribution, usually the cumulative probability from the mean to Z (Z-table).
- The
**cumulative probability**from negative infinity to any point Z under the standard normal curve is given by the Z-table. - The area under the curve between two values can be found by subtracting the smaller Z-value from the larger Z-value.

# Applications of Normal Distribution

- Normal distribution can be applied in
**real-world scenarios**such as measuring physical characteristics, examining test scores, or investigating environmental data. - The
**Central Limit Theorem**states that the sum of many independent and identically distributed random variables tends towards a normal distribution, irrespective of the shape of their individual distributions, provided the expected value and variance are defined and finite.

# Examples

- For instance, if you know the average height of a population (μ) and the standard deviation (σ) and want to calculate the probability of randomly selecting someone taller than a certain height (X), the standardisation formula could be used to convert to a Z-score, followed by utilising the Z-table for cumulative probability.
- Similarly, finding a confidence interval for a certain percentage around the mean can be achieved by referring to the Z-table for corresponding Z-scores and then converting these back to values in terms of the original distribution.