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.

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.

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.

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.