The normal distribution

Basic Understanding

Normal Distribution: A type of continuous probability distribution for a real-valued random variable.
Often referred to as the bell curve due to its bell shape.
Most commonly characterised by its mean (‘μ’) and standard deviation (‘σ’).
Under a normal distribution, data is symmetrically distributed with no skew.
It is derived from the Central Limit Theorem, which states that the sum of a large number of independent and identically distributed variable has an approximate normal distribution.

Features of Normal Distribution

Mean, Mode and Median: In a normal distribution, the mean, mode and median are all equal.
Symmetry: It is symmetric around its mean. The left and right halves of the distribution are mirror images.
Asymptotic: The curve is asymptotic to the x-axis, which means it approaches but never quite touches the x-axis.
Inflection Points: There are inflection points at μ - σ and μ + σ, where the curve changes concavity.

Standard Normal Distribution

It is a special case of a normal distribution where the mean is 0 and the standard deviation is 1.
The random variable ‘Z’ represents the number of standard deviations from the mean, named the ‘Z-score’.
Any given normal distribution can be converted to a standard normal distribution using the formula: Z = (X - μ) / σ.

Applications of Normal Distribution

Used widely in natural and social sciences to represent real-valued random variables with uncertainty.
Forms the basis for the assumption of normality in many statistical tests, including t-tests.
In financial markets, normal distribution is used to model price changes in stocks and other securities.

Probabilities in a Normal Distribution

Approximately 68% of the data lies within one standard deviation from the mean (μ - σ to μ + σ).
Around 95% lies within two standard deviations.
Nearly all (99.7%) data lies within three standard deviations.

Remember, developing a comprehensive understanding of the normal distribution and its properties is crucial for your progression in advanced mathematics. Good luck with your learning!