What is analysis of variance?

analysis of variance: a method of decomposing the variance of a variable into different parts according to different needs, comparing the sizes between them, and testing the significance with F-test. Also known as "analysis of variance" or "F test", it is used to test the significance of the difference between two or more samples.

the value of f is the ratio of two mean squares [effect term/error term], so it is impossible to have a negative value. The larger the F value [compared with the standard F value of a given significant level], the more obvious the effect [difference] between treatments, and the smaller the error term, the higher the test accuracy.

Extended data:

Analysis of variance, also known as "analysis of variance", was invented by R.A.Fisher to test the significance of the difference between two or more samples. Due to the influence of various factors, the data obtained from the study are fluctuating. The reasons for the fluctuation can be divided into two categories, one is uncontrollable random factors, and the other is controllable factors that influence the results.

The basic principle of variance analysis is that there are two basic sources of the differences between the means of different treatment groups:

(1) The experimental conditions, that is, the differences caused by different treatments, are called the differences between groups. It is expressed by the sum of the squares of the deviations between the mean value of variables in each group and the total mean value, which is denoted as SSb and dfb.

(2)? Random errors, such as differences caused by measurement errors or differences between individuals, are called intra-group differences, which are expressed by the sum of the mean values of variables in each group and the sum of the squares of deviations of variables in the group, and are recorded as SSw, and the degree of freedom in the group dfw.

sum of squares of total deviation SSt = SSb+SSw.

divide the intra-group SSw and inter-group SSb by their degrees of freedom (intra-group dfw =n-m, inter-group dfb=m-1, where n is the total number of samples and m is the number of groups), and get their mean square MSw and MSb. In one case, the processing has no effect, that is, all the samples in each group come from the same population, and MSb/MSw≈1. In another case, the treatment does work, and the mean square between groups is the result of the same error as different treatments, that is, the samples come from different populations. So, msb >; > MSw (much greater than).

the ratio of p>MSb/MSw constitutes an f distribution. Compare the F value with its critical value to infer whether each sample comes from the same population.

variance is a measure of the degree of dispersion when probability theory and statistical variance measure a random variable or a set of data. Variance in probability theory is used to measure the deviation between random variables and their mathematical expectations (that is, the mean). The variance (sample variance) in statistics is the average of the square value of the difference between each sample value and the average of all sample values. In many practical problems, it is of great significance to study variance, that is, deviation degree.

variance is a measure of the difference between the source data and the expected value.

1. let c be a constant, then D(C)=

2. let x be a random variable, and c be a constant, then?

3. Let x and y be two random variables, then

where is the covariance? Especially, when x and y are two unrelated random variables,

this property can be extended to the sum of a finite number of unrelated random variables.

4. The necessary and sufficient condition for D(X)= is that x takes the constant E(X) with a probability of 1, that is?

(D(X)= if and only if the probability of x taking a constant value E(X) is 1. )

Note: It cannot be concluded that X is constant. When X is continuous, X can take a value different from the constant c at any finite point.

Reference: Baidu Encyclopedia-Analysis of Variance