Unveiling the Normal Distribution: Understanding Its Crucial Role in Statistics
Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, indicating that data near the mean are more likely to occur than data far from the mean. This distribution is graphically represented by a ‘bell curve.’
Key Takeaways
- The normal distribution, commonly referred to as the bell curve, is a cornerstone of statistical analysis.
- In a normal distribution, the mean is zero and the standard deviation is one. It has zero skew and a kurtosis of three.
- Symmetry characterizes normal distributions, but not all symmetrical distributions are normal.
Properties of an Exquisite Normal Distribution
The normal distribution is the most frequently assumed type of distribution in technical stock market analysis. Both its symmetry and its distinct characteristics make it the foundation of many statistical models. Key parameters include the mean (average), median (midpoint), and mode (most frequent observation), all of which are equal.
Moreover, the normal distribution plays a crucial role in the Central Limit Theorem (CLT), which states that means of independent, identically distributed variables tend to follow a normal distribution, regardless of the original distribution.
The Empirical Rule
For all normal distributions:
- 68.2% of observations fall within plus or minus one standard deviation of the mean.
- 95.4% fall within plus or minus two standard deviations.
- 99.7% fall within plus or minus three standard deviations.
This is often referred to as the ’empirical rule,’ a heuristic that guides the understanding of data dispersion in normal distributions.
Unpacking Skewness
Skewness gauges the symmetry of data distribution. A perfectly normal distribution is symmetric and has zero skewness. Negative skewness implies a longer left tail, while positive skewness signifies a longer right tail.
Understanding Kurtosis
Kurtosis indicates the ’tailedness’ of a distribution. The normal distribution has a kurtosis of three. Distributions with a higher kurtosis (>3) show ‘heavy tails,’ meaning more extreme data points than a normal distribution. Lower kurtosis (<3) indicates ’light tails.’
The Mathematical Essence
The normal distribution is mathematically described by:
where:
- x = variable or data point
- μ = mean
- σ = standard deviation
Applying Normal Distribution in Finance
In finance, the normal distribution model helps in asset pricing and determining deviations. Prices deviating significantly from the mean suggest over or undervaluation. However, most financial data exhibit ‘fat tails,’ representing extreme events more frequently than a normal distribution would predict.
Real-World Example: Human Heights
Height among the human population is normally distributed, with an average height of 175 cm. Most people fall within three standard deviations of this mean (154 cm to 196 cm), while extremely tall or short individuals are rare.
Conclusion: Embracing Normal Distribution
Normal distribution is emblematic in science and finance, visualized through the bell curve. While it provides robust analytical insights, its limitations, especially in financial applications, require cautious application. In a world where data may not always conform to norms, understanding normal distribution and its quirks empowers more informed decisions.
Related Terms: Probability Distribution, Central Limit Theorem, Skewness, Kurtosis, Standard Deviation, Symmetrical Distribution.
References
- Boston University. “The Central Limit Theorem”.
- DePaul University. “NORMAL Distribution: Origin of the name”.
