D(X) refers to variance, and E(X) refers to expectation. E(X) says that the simple point is the average value, and the specific method is to sum and then divide by the quantity. d(x)=e[x-e(x)]^2=e{x^2-2xe(x)+[e(x)]^2}=e(x^2)-2[e(x)]^2+[e(x)]^2。

1, let c be a constant, then D(C)=0 (constant has no fluctuation);

2.D(cx)=C2D(x) (constant square extraction);

Certificate:

D(-X)=D(X), D(-2X)=4D(X) (variance is not negative).

When x and y are independent of each other, the third term is zero.

statistical significance

When the data distribution is scattered (that is, the data fluctuates greatly around the average value), the sum of squares of differences between each data and the average value is large, and the variance is large; When the data distribution is concentrated, the sum of squares of the differences between each data and the average value is very small. Therefore, the greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation.

The average value of the sum of squares of the difference between the data in the sample and the average value of the sample is called sample variance; The arithmetic square root of sample variance is called sample standard deviation. Sample variance and sample standard deviation are both measures of sample fluctuation. The greater the sample variance or standard deviation, the greater the fluctuation of sample data.

Variance and standard deviation are the most important and commonly used indicators to measure discrete trends. Variance is the square of variance of each variable value and the average of its mean, which is the most important method to measure the dispersion degree of numerical data. The standard deviation is the arithmetic square root of variance, expressed by S.