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Normal distribution and standard deviation

The normal distribution, often referred to as the "bell-shaped curve", is the most well-known and widely used of all probability distributions. Because the normal distribution is very close to many natural phenomena, it has developed into a reference standard for many probability problems. Several sets of data follow a normal distribution: for example, the height of adults; The weight of the baby Classroom examination results The returns of the stock market and the beads in the Galton board.

As proved by the Galton board, the random path of 3,000 beads is approximately a bell-shaped curve each time. Standard deviation (a) measures the degree to which all data points converge around the average value. The shape of the normal distribution is determined by the mean and the standard deviation. The narrower the bell-shaped curve is, the smaller the standard deviation is. When the bell-shaped curve is wide, the standard deviation is large.

As shown in the figure, approximately two-thirds of the data in the bell-shaped curve fall within one standard deviation of the average value. Approximately 95% of the data are within two standard deviations, and approximately 99.7% are within three standard deviations.

In the Galton board experiment, each time the ball drops a layer, it may fall to the left or right. These factors determine the final landing point of the ball. Similarly, many random phenomena in nature have various interfering factors that affect the possible distribution of an event. However, under the combined influence of these factors, as long as the sample size is large enough, these random phenomena often follow a normal distribution. This is also a kind of mathematical beauty, isn't it? This is also an important application of the central limit theorem in life.

Quality control: Simulate the cumulative effect of production errors and explain the normal distribution phenomenon of product parameters.

Financial model: By analogy with the random walk of asset prices, it provides physical intuition for the Black-Shulls option pricing model.

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