The normal distribution is the most important distribution among probability distributions. In the eyes of mathematicians, it is far higher than other distributions.

Other distributions are special. Only the normal distribution is normal and general. From the name, we can also feel its importance.

What’s interesting is that the normal distribution is not only important but also simple. It is like a symmetrical inverted bell curve, high in the middle and declining on both sides, like a bulging hill.

In the normal distribution curve, the abscissa represents the value range of the random variable. The further to the right, the greater the value of the random variable. The ordinate represents the probability. The bottom probability is 0, the higher the probability, the greater the probability. In this way, find a random point on the curve, determine its abscissa and ordinate, and we will know the probability of this value appearing.

Because this curve is symmetrical on the left and right, the highest point in the middle means that the average value has the greatest probability and the most data. The steep decline on both sides means that it is close to the average value and the more data there is. , the further away from the mean, the less data there is.

Of course, we cannot stop at this rough description. To understand the normal distribution, we must understand its three mathematical properties.

1. The mean is the expectation

In other words, the abscissa of the highest point in the middle of the normal distribution not only represents the average of the random variable, but is also equal to its mathematical expectation. This It has been mathematically proven that in probability theory, the mean and expectation of the normal distribution mean the same thing, and are two expressions of the same thing.

As we said before, mathematical expectation represents long-term value, and now the average is mathematical expectation. That is to say, in the normal distribution, the average represents the value of random events.

Why do we use the average score of the college entrance examination to measure the teaching quality of a high school? Why do we have the average rate of return to measure the income of a fund company? The average represents the value of this random event.

Only in normal distribution, the average value has this meaning. If it is not a normal distribution, the average value has no meaning. For example, in earthquakes, no one has heard of average intensity and average loss. Let’s put it this way.

2. There are very few extreme values

Do you remember the graph of the normal distribution? The closer to the mean, the higher the curve and the greater the probability of occurrence; the further away from the mean, the lower the curve and the smaller the probability of occurrence. This shows that most data in a normal distribution are concentrated around the mean value, with very few extreme values.

The phrase "very few extreme values" has two meanings: the probability of occurrence of extreme values ??is very low, and secondly, the impact of extreme values ??on the mean is very small. Therefore, the normal distribution is very stable. As for height, it generally follows a normal distribution, so even if Yao Ming joins, our average height will not change much.

3. Standard deviation determines fatness and thinness

It is also a normal distribution chart. Some curves are shorter and fatter, and some curves are taller and thinner. Why?

< p> Because the standard deviation is different, the standard deviation is the square root of the variance and can also be used to describe the fluctuation of random variables. In the normal distribution, the larger the standard deviation, the more violent the data fluctuations, and the squatter the bell-shaped curve. The smaller the standard deviation, the more concentrated the data, and the taller and thinner the bell-shaped curve.

Why is the normal distribution simple? Because in the normal distribution, the mean value is equal to the expectation, which determines the highest point of the curve, and the variance determines the weight and the curvature of the curve. Two simple data determine the shape of this curve.

Can different normal distribution curves be compared?

Yes,

First, only the mean values ??are different, so we can compare the good and bad.