The Gaussian or normal probability distribution role is very important in statistics. It was investigated by researchers/ persons interested in gambling or in the distribution of errors made by people observing astronomical events. The normal probability distribution is important in other fields such as social sciences, behavioural statistics, business and management sciences, and engineering and technologies.
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Importance of Normal Distribution
Some of the important reasons for the normal probability distribution are:
- Many variables such as (weight, height, marks, measurement errors, IQ, etc.) are distributed as the symmetrical bell-shaped normal curve approximately.
- Many inferential procedures (parametric tests: confidence intervals, hypothesis testing, regression analysis, etc.) assume that the variables follow the normal distribution.
- All probability distributions approach a normal distribution under some conditions.
- Even if a variable is not normally distributed, a distribution of sample sums or averages on that variable will be approximately normally distributed if the sample size is large enough.
- The mathematics of a normal curve is well-known and relatively simple. One can find the probability that a score randomly sampled from a normal distribution falls within the interval $a$ and $b$ by integrating the normal probability density function (PDF) from $a$ to $b$. This is equivalent to finding the area under the curve between $a$ and $b$ assuming a total area of one.
- Due to the Central Limit Theorem, the average of many independent random variables tends to follow a normal probability distribution, regardless of the original distribution of the variables.
Probability Density Functions of Normal Distribution
The probability density function known as the normal curve. $F(X)$ is the probability density, aka the height of the curve at value $X$.
$$F(X) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(X-\mu)^2}{2\sigma^2} }$$
There are two parameters in the PDF of normal distribution, (i) the mean and (ii) the standard deviation. Everything else in the PDF of normal distribution on the right-hand side is a constant. There is a family of normal probability distribution with respect to their means and their standard deviations.
Standard Normal Probability Distribution
One can work with normal curve, even if one don’t know about integral calculus. One can use computer to compute the area under the normal curve or make use of the normal curve table. The normal curve table (standard normal table) is based on the standard normal curve ($Z$), which has a mean of 0 and a variance of 1. To use a standard normal curve table, one need to convert the raw score to $Z$-scores. A $Z$-score is the number of standard deviations ($\sigma$ or $s$) a score is above or below the mean of a reference distribution.
$$Z_X = \frac{X-\mu}{\sigma}$$
For example, suppose one wish to know the percentile rank of a score of 90 on an IQ test with $\mu = 100$ and $\sigma=10$. The $Z$-score will be
$$Z=\frac{X-\mu}{\sigma} = \frac{90-100}{10} = -1$$
One can either integrate the normal cure from $-\infty$ to $-1$ or use the standard normal table. The probability or area under the curve on the left of $-1$ is 0.1587 or 15.87%.
Key Characteristics of Normal Probability Distribution
- Symmetry: In normal probability distribution, the mean, median, and mode are all equal and located at the center of the curve.
- Spread: In normal distribution, the spread of the data is determined by the standard deviation. A larger standard deviation means that the curve is wider, and a smaller standard deviation means a narrower curve.
- Area under the Normal Curve: The total area under the normal curve is always equal to 1 or 100%.
Real-Life Applications of Normal Distribution
The following are some real-life applications of normal probability distribution.
- Natural Phenomena:
- Biological Traits: Many biological traits, such as weight, height, and IQ scores, tend to follow a normal distribution. This helps us to understand the typical range of values for different biological traits and identify outliers.
- Physical Measurements: Errors in measurements often follow a normal distribution. This knowledge is crucial in fields like engineering and physics for quality control and precision.
- Statistical Inference:
- Hypothesis Testing: The normal distribution is used extensively in hypothesis testing to determine the statistical significance of the results. By understanding the distribution of sample means, one can make inferences about population parameters.
- Confidence Intervals: Normal distribution helps calculate confidence intervals, which provide a range of values within which a population parameter is likely to fall with a certain level of confidence.
- Machine Learning and Artificial Intelligence:
- Feature Distribution: Many machine learning (ML) algorithms assume that features in data follow a normal distribution. The normality assumption about machine learning algorithms can influence the choice of algorithms and the effectiveness of models.
- Error Analysis: The normal distribution is used to analyze the distribution of errors in machine learning models, helping to identify potential biases and improve accuracy.
- Finance and Economics:
- Asset Returns: While not perfectly normal, many financial assets, such as stock prices, follow an approximately normal distribution over short time periods. The assumption of normality is used in various financial models and risk assessments.
- Economic Indicators: Economic indicators such as GDP growth rates and inflation rates often exhibit a normal distribution, allowing economists to analyze trends and make predictions.
- Quality Control:
- Process Control Charts: In manufacturing and other industries, normal distribution is used to create control charts that monitor the quality of products or processes. By tracking the distribution of measurements, one can identify when a process is going out of control.
- Product Quality: Manufacturers use statistical quality control methods based on normal distribution to ensure that products meet quality standards.
- Everyday Life:
- Standardized Tests: The standardized Test scores, such as SAT and GRE, are often normalized to a standard normal distribution, allowing for comparisons between different test-takers.
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