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The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, indicating that data close to the mean occur more frequently than data far from the mean.
The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, indicating that data close to the mean occur more frequently than data far from the mean. When plotted, the normal distribution appears as a "bell curve."
The normal distribution is the most common type of distribution assumed in technical stock market analysis. The standard normal distribution has two parameters: mean and standard deviation. In a normal distribution, the mean (mean), median (midpoint), and mode (most frequent observation) are equal. These values represent the peak point of the distribution, which falls symmetrically around the mean, with its width defined by the standard deviation.
The normal distribution model is the key to the Central Limit Theorem (CLT), which states that means calculated from independent and identically distributed random variables are approximately normally distributed, regardless of the original distribution type of those variables.
The normal distribution is a symmetric distribution. Symmetrical distribution occurs when the dividing line creates two mirror images. Not all symmetric distributions are normal, as some data may take the form of two peaks or a series of hills rather than just the bell-shaped curve that represents a normal distribution.
For all normal distributions, 68.2% of observations will be within plus or minus one standard deviation of the mean; 95.4% will be within plus or minus two standard deviations; and 99.7% will be within plus or minus three standard deviations.
This fact is sometimes called the "rule of thumb" and is a heuristic that describes where most data occurs in a normal distribution. Data that fall outside three standard deviations ("3-sigma") are considered rare events.
Skewness measures how symmetrical a distribution is. The normal distribution is symmetric and has zero skewness. If the distribution skewness of the data set is less than zero, that is, negative skewness (left skewness), the left tail is longer than the right tail; positive skewness (right skewness) means the right tail is longer than the left tail.
Kurtosis measures the thickness of the tails of a distribution. The kurtosis of a normal distribution is 3.0. A distribution with a kurtosis greater than 3.0 exhibits tail data that exceeds the tail of the normal distribution (for example, five or more standard deviations from the mean).
This excess kurtosis is called a leptokurtic tail in statistics, but more colloquially is called a "fat tail." The fat tail phenomenon in financial markets describes what is called tail risk. Distributions with kurtosis less than 3.0 (platykurtic) generally exhibit tails that are less extreme ("skinnier") than the tails of a normal distribution.
The normal distribution follows the following formula. Note that only the values for the mean (μ) and standard deviation (σ) are required.
Among them:
The assumption of normal distribution applies to asset prices and price behavior. Traders might plot price points to fit recent price action to a normal distribution. In this case, the further away the price behavior is from the mean, the greater the likelihood that the asset is overvalued or undervalued. Traders can use standard deviation to suggest potential trades. This type of trade is typically done on very short time frames, as longer time periods make it more difficult to select entry and exit points.
Likewise, many statistical theories attempt to model asset prices and assume that these prices follow a normal distribution. In practice, price distributions tend to have fat tails, so the kurtosis is usually greater than three. The price movements of these assets exceed three standard deviations from the mean much more frequently than would be expected if a normal distribution was assumed. Even if an asset follows a normal distribution over a period of time, its past performance cannot truly reflect the future.
Many natural phenomena appear to be normally distributed. For example, the average height of a person, including men and women, is approximately 175 centimeters (5 feet 9 inches).
As the chart below shows, most people fall within this average. Taller and shorter people appear less frequently in the population. As a rule of thumb, 99.7% of people will fall within plus or minus three standard deviations of the mean, which is between 154 centimeters (5 feet 0 inches) and 196 centimeters (6 feet 5 inches). People above or below this range will be rare (0.15% of the total population each).
The normal distribution describes a plot of data that is symmetric around the mean value, where the width of the curve is defined by the standard deviation. It appears visually as a "bell curve."
The normal distribution is technically called a Gaussian distribution, but it was called "normal" in 19th-century scientific publications and showed that many natural phenomena appear to "normally deviate" from the mean. This concept of "normal variation" was proposed by naturalist Sir Francis Galton in his 1889 book Natural Heredity.
Although the normal distribution is a statistical concept, its application in finance may be limited because financial phenomenasuch as expected stock market returnsdo not exactly conform to a normal distribution. Prices tend to be log-normally distributed, which is right-skewed and have thicker tails. Therefore, when making forecasts, relying too much on the bell curve can lead to unreliable results. Although most analysts are aware of this limitation, overcoming it is relatively difficult because it is often unclear which statistical distribution should be chosen as a proxy.
The normal distribution, also known as the Gaussian distribution, is a probability distribution that appears as a "bell curve" when plotted. The normal distribution describes a plot of data that is symmetric around the mean value, where the width of the curve is defined by the standard deviation. It appears visually as a "bell curve."
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