A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric around its mean, meaning that the data is equally likely to fall on either side of the mean. It is a fundamental concept in statistics and probability theory, playing a crucial role in various scientific and mathematical fields. Here’s a comprehensive overview of the normal distribution:

# 1. Characteristics of the Normal Distribution:

## a. Symmetry:

- The normal distribution is symmetric, meaning that if you were to draw a vertical line at the mean, the two halves of the distribution would mirror each other.

## b. Bell-Shaped Curve:

- The graph of a normal distribution forms a bell-shaped curve, with the highest point at the mean.

## c. Measures of Central Tendency:

- The mean, median, and mode are all equal in a perfectly normal distribution.

## d. Standard Deviation:

- The spread or dispersion of the distribution is determined by the standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

## e. Empirical Rule:

- The empirical rule, also known as the 68–95–99.7 rule, states that a high percentage of observations fall within certain standard deviation ranges.

## f. Z-Score:

- The Z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution.

# 2. Probability Density Function (PDF):

- The probability density function for the normal distribution is given by the formula:
*f*(*x*∣*μ*,*σ*) = 1/*σ*2*π exp(-(x-μ)*² / 2*σ*²) *μ*is the mean,*σ*is the standard deviation, and*x*is a particular value.

# 3. Standard Normal Distribution:

- A standard normal distribution is a normal distribution with a mean (

) of 0 and…*μ*