The normal distribution, also called a Gaussian distribution, has long been widely used in fields as diverse as statistics and finance. It is a probability distribution that describes how the values of an event can tend to cluster around an average or mean value. A normal distribution consists of a bell-shaped curve with the mean (μ) represented by the peak or the highest value of the curve. The standard deviation (σ) of the normal distribution is the measure of the spread of the values around the mean.

Represented by a formula, the normal distribution looks like this:

f(x) = (1/√2πσ)e ^ –(x-μ)² / 2σ² In a normal distribution, 68% of the area under the curve falls within one standard deviation of the mean, 95% of the area under the curve falls within two standard deviations of the mean, and 99.7% of the area under the curve falls within three standard deviations of the mean.

The normal distribution captures how values tend to be distributed in the natural, real-world. For example, IQ test scores, height, blood pressure are all some parameters that are known to be normally distributed. It is also used to assess the probability of rare events and finds its utilization in areas such as risk analysis and portfolio optimization.

Often, the normal distribution is compared to the binomial or Poisson distributions. The main differences among them lie in the shape of the distribution, the number of parameters, and the rules governing how they are used. Binomial distributions are characterized by two number parameters instead of one, while Poisson distributions take more of a “clumping” approach that don’t necessarily adhere to a mean. On the other hand, a normal distribution peaks in the middle, and generally follows a bell-shaped curve.

However, a perfectly normal distribution rarely occurs in practice. Often times events observed in the real world do not follow the probability bell curve associated with normal distributions. Even when natural phenomena and events follow a general trend or pattern that fits into a normal distribution, some features of the data point to the presence of skewness and kurtosis. Skewness measures the degree of asymmetry, while kurtosis measures the peakedness of the distribution. In order for the normal distribution to be valid, the skewness should be zero and the kurtosis should have a value of 3.

In conclusion, the normal distribution is a probability distribution that can be used to describe many naturally-occurring phenomena. It is characterized by a bell-shaped curve and its mean and standard deviation are used to measure the spread of the values around the mean. Although a perfect normal distribution is a rarity, the normal distribution captures how values tend to be distributed in the natural, real-world and is an incredibly useful tool for assessing risk and probability in fields such as finance and statistics.