Mean and Variance of a Normal Distribution
Mean and Variance of a Normal Distribution
Definition of Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean.

The formula for a normal distribution is given by the function: f(x) = (1/σ√2π) e^−((x−μ)^2/2σ^2)

In the formula, ‘x’ is the variable, ‘μ’ is the mean, ‘σ’ is the standard deviation, ‘e’ is the base of natural logarithms and ‘π’ is a constant approximately equal to 3.14159.
Mean and Variance of Normal Distribution

The mean (μ) of a normal distribution is the central value about which data points are distributed. It is the peak of the distribution’s curve.

The variance (σ^2) is a measure of the dispersion of the distribution, indicating how much values deviate from the mean.

This specific normal distribution, with mean 0 and variance 1, is referred to as the standard normal distribution.

All normal distributions can be transformed into the standard normal distribution by calculating the Zscore for each value. The Zscore is calculated using the formula: Z = (X  μ) / σ.
Key Features of Normal Distribution
 The normal distribution follows the empirical rule (also known as the 689599.7 rule), which states that within a normal distribution:
 Approximately 68% of the data falls within one standard deviation (σ) of the mean (μ).
 Around 95% falls within two standard deviations.
 Approximately 99.7% falls within three standard deviations.
 The normal distribution is important in statistics due to the central limit theorem, which states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, no matter the shape of the population distribution.
Examples

If you have a data set that consists of the heights of a large number of adults, the data would be normally distributed with a mean at the average adult height and a standard deviation related to the average difference in height among adults.

The grading of standardized exam scores often assumes a normal distribution to determine passing and failing cutoffs. For instance, a Zscore of 1.0 represents a value that’s one standard deviation from the mean.