Let x be a random variable, and if e {[x-e (x)] 2} exists, then e {[x-e (x)] 2} is called the variance of x, and is recorded as D(X) or DX. That is, d (x) = e {[x-e (x)] 2}, σ (x) = d (x) 0.5 (the same dimension as x) is called standard deviation or mean square deviation.
From the definition of variance, the following commonly used calculation formulas can be obtained:
D(X)=E(X^2)-[E(X)]^2
Several important properties of variance (assuming that every variance has wow.
(1) Let c be a constant, then D(c)=0.
(2) If X is a random variable and C is a constant, then D (CX) = (C 2) D (X).
(3) Let x and y be two independent random variables, then D(X+Y)=D(X)+D(Y).
(4) The necessary and sufficient condition for d (x) = 0 is that x takes the constant value c with the probability of 1, that is, P{X=c}= 1, where e (x) = c.
What is the standard deviation? Formula for calculating standard deviation?
Each number in a set of data is divided by the number of data, and the square of the difference of the average value of this set of data is the variance of this set of data, and the square of the variance is the standard deviation. If the average value of data 1, 2, 3, 4 and 5 is 3, the variance can be calculated as [( 1-3) 2+(2-3)].
What is variance?
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.
In mathematics, e {[x-E(X)] 2} is generally used to measure the deviation degree of random variable X from its mean value E (x), which is called the variance of X.
definition
Let x be a random variable, and if e {[x-e (x)] 2} exists, then e {[x-e (x)] 2} is called the variance of x, and is recorded as D(X) or DX. That is, d (x) = e {[x-e (x)] 2}, σ (x) = d (x) is called standard deviation or mean square deviation for .5 (the same dimension as x).
From the definition of variance, the following commonly used calculation formulas can be obtained:
D(X)=E(X^2)-[E(X)]^2
S 2 = [(x 1-x pull) 2+(x2-x pull) 2+(x3-x pull) 2+…+(xn-x pull) 2]/n
Several important properties of variance (assuming that each variance exists).
(1) Let c be a constant, then D(c)=0.
(2) If X is a random variable and C is a constant, then D (CX) = (C 2) D (X).
(3) Let x and y be two independent random variables, then D(X+Y)=D(X)+D(Y).
(4) The necessary and sufficient condition for d (x) = 0 is that x takes the constant value c with the probability of 1, that is, P{X=c}= 1, where e (x) = c.
What are the concepts of variance and standard deviation?
Standard deviation (standard deviation)
The average value of the distance between each data and the average value (the deviation of the average value), that is, the square root of the sum of the squares of deviation. Expressed by σ. So the standard deviation is also an average.
The standard deviation is the arithmetic square root of variance.
The standard deviation can reflect the degree of dispersion of the data set. If the average value is the same, the standard deviation may be different.
For example, six students in Group A and Group B all took the same Chinese exam. Group A scored 95, 85, 75, 65, 55, 45, and Group B scored 73, 72, 765, 438+0, 69, 68, 67. The average score of the two groups is 70, but the standard deviation of group A is 17.08, and that of group B is 2. 16, which shows that the gap between students in group A is much larger than that in group B. ..
Standard deviation is also called standard deviation, or experimental standard deviation.
This function is described in detail in the STDEVP function in EXCEL, and the word "standard deviation" is used in the Chinese version of EXCEL. However, standard deviation is usually used in Chinese textbooks in China.
The formula is shown in the figure.
postscript
In EXCEL, the STDEVP function is another standard deviation mentioned in the following notes, namely population standard deviation. In some places of traditional Chinese, it may be called "mother's standard deviation"
There are two definitions of arc, which are used in different occasions:
If it is a population, the root sign of the standard deviation formula is divided by n,
If it is a sample, the root sign of the standard deviation formula is divided by (n- 1).
Because we are exposed to a large number of samples, we usually divide by the root sign (n- 1).
What are the concepts of variance and mean square deviation? What's the difference?
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.
In mathematics, e {[x-E(X)] 2} is generally used to measure the deviation degree of random variable X from its mean value E (x), which is called the variance of X. 。
definition
Let x be a random variable, and if e {[x-e (x)] 2} exists, then e {[x-e (x)] 2} is called the variance of x, and is recorded as D(X) or DX. That is, d (x) = e {[x-e (x)] 2.
From the definition of variance, the following commonly used calculation formulas can be obtained:
D(X)=E(X^2)-[E(X)]^2
Several important properties of variance (assuming that each variance exists).
(1) Let c be a constant, then D(c)=0.
(2) If X is a random variable and C is a constant, then D (CX) = (C 2) D (X).
(3) Let x and y be two independent random variables, then D(X+Y)=D(X)+D(Y).
(4) The necessary and sufficient condition for d (x) = 0 is that x takes the constant value c with the probability of 1, that is, P{X=c}= 1, where e (x) = c. 。
What is the standard deviation?
The standard deviation is the arithmetic square root of variance. The standard deviation can reflect the degree of dispersion of the data set. If the average value is the same, the standard deviation may be different. For example, six students in Group A and Group B all took the same Chinese exam. Group A scored 95, 85, 75, 65, 55, 45, and Group B scored 73, 72, 765, 438+0, 69, 68, 67. The average score of the two groups is 70, but the standard deviation of group A is 17.08, and that of group B is 2. 16, which shows that the gap between students in group A is much larger than that in group B. The standard deviation is also called standard deviation, or experimental standard deviation. This function is described in detail in the STDEVP function in EXCEL, and the word "standard deviation" is used in the Chinese version of EXCEL. However, standard deviation is usually used in Chinese textbooks in China. P.S. In EXCEL, the STDEVP function is another standard deviation mentioned in the following notes, namely population standard deviation. In some places of traditional Chinese, it may be called "standard deviation of population", because there are two definitions, which are used in different occasions: in the case of population, the root number of standard deviation formula is divided by n, and in the case of sample, the root number of standard deviation formula is divided by (n- 1). Because we are exposed to a large number of samples, the root sign is generally divided by (n- 1), and the foreign exchange term: standard deviation refers to. Standard deviation is used to evaluate possible price changes or fluctuations. The greater the standard deviation, the wider the price fluctuation range, and the greater the fluctuation of financial instruments such as stocks. Elaboration and application In simple terms, the standard deviation is a measurement concept of the degree to which a group of values deviate from the average. A large standard deviation means that most values differ greatly from their average values; A smaller standard deviation means that these values are closer to the average. For example, the average values of * * {0,5,9, 14} and {5,6,8,9} of two groups of numbers are all 7, but the standard deviation of the second * * * is smaller. Standard deviation can be used to measure uncertainty. For example, in physical science, the standard deviation of measured values * * * represents the accuracy of these measurements when repeated. When determining whether the measured value conforms to the predicted value, the standard deviation of the measured value plays a decisive role: if the measured average value is too far from the predicted value (compared with the standard deviation value at the same time), it is considered that the measured value is contradictory to the predicted value. This is easy to understand, because if the measured value falls outside a certain numerical range, it can be reasonably inferred whether the predicted value is correct or not. Standard deviation is applied to investment and can be used as an index to measure the stability of return. The greater the standard deviation, the higher the risk, because the income is far from the past average. On the contrary, the finer the standard deviation, the more stable the income and the less the risk. In the real world, it is unrealistic to find a total true standard deviation except in some special cases. In most cases, population standard deviation is estimated by randomly sampling a certain number of samples and calculating the standard deviation of the samples.