Normal Distribution

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What is normal distribution?

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetrical with respect to the average, showing that data close to the average is more frequent than data far from the average. As a graph, the normal distribution will appear as a bell curve.


Normal distribution

Understanding normal distribution

Normal distribution is the most common type of distribution assumed in technical analysis of stock markets and other types of statistical analysis. The standard normal distribution has two parameters: the mean and the standard deviation. For a normal distribution, 68% of the observations are within +/- one standard deviation from the mean, 95% are within +/- two standard deviations and 99.7% are within + – three standard deviations.

The normal distribution model is motivated by the central limit theorem. This theory states that means computed from independent random variables and identically distributed have approximately normal distributions, regardless of the type of distribution from which the variables are sampled (provided it has a finite variance). The normal distribution is sometimes confused with the symmetrical distribution. Symmetric distribution is where a dividing line produces two mirror images, but the actual data may be two bumps or a series of hills in addition to the bell curve which indicates normal distribution.

Key points to remember

  • Normal distribution is the appropriate term for a probability bell curve.
  • The normal distribution is a symmetric distribution, but not all symmetric distributions are normal.
  • In reality, most price distributions are not perfectly normal.

Asymmetry and kurtosis

Actual data rarely, if ever, follows a perfect normal distribution. The asymmetry and kurtosis coefficients measure the difference between a given distribution and a normal distribution. Asymmetry measures the symmetry of a distribution. The normal distribution is symmetrical and has an asymmetry of zero. If the distribution of a data set has an asymmetry less than zero or a negative asymmetry, the left tail of the distribution is longer than the right tail; a positive asymmetry implies that the right tail of the distribution is longer than the left.

The kurtosis statistic measures the thickness of the tail ends of a distribution relative to the tails of the normal distribution. Distributions with significant kurtosis present tail data exceeding the tails of the normal distribution (for example, five or more standard deviations from the average). Low kurtosis distributions show tail data that is generally less extreme than the tails of the normal distribution. The normal distribution has a kurtosis of three, which indicates that the distribution has neither a fat nor fine tail. Therefore, if an observed distribution has kurtosis greater than three, the distribution would have heavy tails compared to the normal distribution. If the distribution has a kurtosis of less than three, it is said to have fine tails compared to the normal distribution.

How normal distribution is used in finance

The normal distribution assumption is applied to asset prices as well as to price action. Traders can plot prices over time to adapt recent price action to normal distribution. The more the price action deviates from the average, in this case, the more likely that an asset is overvalued or undervalued. Traders can use standard deviations to suggest potential trades. This type of negotiation is generally done on very short deadlines, because longer deadlines make the choice of entry and exit points much more difficult.

Likewise, many statistical theories attempt to model asset prices by assuming that they follow a normal distribution. In reality, price distributions tend to have a fat tail and, therefore, a kurtosis greater than three. These assets experienced price movements greater than three standard deviations above the average more often than expected under the assumption of a normal distribution. Even if an asset has gone through a long period of time when it corresponds to a normal distribution, there is no guarantee that past performance will really inform future prospects.

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