Variance and standard deviation

Variance and standard deviation

The sum of squares of the differences between each data and the average sample is called the sample variance. The arithmetic square root of the sample variance is called the sample standard deviation. Both the sample variance and the sample standard deviation measure the fluctuation size of a sample. The larger the sample variance or sample standard deviation, the larger the fluctuation of the sample data.

In mathematics, E {[X-E (X)] ^ 2} is used to measure the deviation between random variable X and mean E (X), which is called the variance of X.

Definition
If E {[X-E (X)] ^ 2} exists, E {[X-E (X)] ^ 2} is called the variance of X, as D (X) or DX. That is, D (X) = E {[X-E (X)] ^ 2}, while σ (X) = D (X) ^ 0.5 (having the same dimension as X) the standard deviation or mean variance.

The following commonly used calculation formula can be obtained based on the variance definition:
D (X) = E (X ^ 2)-[E (X)] ^ 2

Several important properties of variance (each of the following variance exists ).
(1) If c is a constant, D (c) = 0.
(2) If X is a random variable and c is a constant, D (cX) = c ^ 2D (X) exists ).
(3) If X and Y are two independent random variables, D (X + Y) = D (X) + D (Y ).
(4) a sufficient and necessary condition for D (X) = 0 is that X takes the constant value c as 1, that is, P {X = c} = 1, where E (X) = c.

Standard Deviation)

The mean of the distance (mean deviation) from the mean of each data, which is the squared deviation and the percentile after the mean. Expressed in σ. Therefore, the standard deviation is also an average.

The standard deviation reflects the degree of discretization of a dataset. If the mean is the same, the standard deviation may not be the same.

For example, there are 6 students in group A and group B in the same language test. The scores in group A are 95, 85, 75, 65, 55, and 45, the scores of group B are 73, 72, 71, 69, 68, and 67. The mean of both groups is 70, but the standard deviation of group A is 17.08, and the standard deviation of group B is 2.16, it indicates that the gap between the students in group A is much larger than that between the students in group B.

In simple terms:
Variance is the mean of the square of the difference between the actual value and the expected value, while the standard deviation is the square root of the variance.
Variance: is the mean of the square of the difference between each data and the mean, that is, s ^ 2 = 1/n [(x1-x _) ^ 2 + (x2-x _) ^ 2 +... + (xn-x _) ^ 2].
In layman's terms, it is the degree of deviation from the center! Used to measure the fluctuation of a batch of data (that is, the size of the deviation from the average of this batch of data ).
When the sample size is the same, the larger the variance, the larger the data fluctuation, the more unstable.

Formula for Calculating standard deviation: