## Understanding Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

## Characteristics of Normal Distribution

A normal distribution is determined by two parameters: the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow.

All normal distributions are symmetric and have bell-shaped density curves with a single peak. A fundamental property of the normal distribution is that the area under the curve for the entire set of possible outcomes (i.e., all possible values the random variable can take) is equal to 1, ensuring that it represents a true probability distribution.

## The Empirical Rule

The empirical rule, or the 68-95-99.7 rule, is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively.

According to the empirical rule: - Approximately 68% of the data falls within one standard deviation of the mean. - Approximately 95% of the data falls within two standard deviations of the mean. - Approximately 99.7% of the data falls within three standard deviations of the mean.

## Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It occurs when a normal random variable has a mean of zero and a standard deviation of one. It is used to calculate z-scores, which measure how many standard deviations above or below the mean a particular data point is.

## Applications of Normal Distribution

Normal distribution is widely used in understanding distributions of factors in the population. It applies to a variety of scientific disciplines. In psychology, for example, it can be used to measure the intelligence quotient (IQ) where the mean score is 100. In finance, it can be used to model asset prices, portfolio returns, and to conduct hypothesis testing.

## Central Limit Theorem

The central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normally distributed, regardless of the underlying distribution. This is a key concept in probability and statistics and underscores the importance of the normal distribution in these fields.

## Limitations of Normal Distribution

While normal distribution is a powerful tool for statistical analysis, it has its limitations. Real-world data often has skewness or kurtosis that deviates from that of a normal distribution. This means that the normal distribution may not accurately describe the behavior of real-world data, especially if it is skewed or has extreme values (outliers).

## Conclusion

Normal distribution is a fundamental concept in statistics that enables us to perform various types of statistical analyses and hypothesis testing. It is a good approximation for the distribution of many random variables encountered in practice and forms the basis for the parametric inferential statistical methods. However, it is important to assess the fit of the normal distribution to your data before applying any statistical methods that assume normality.