标准方差的计算公式

标准方差的计算公式

标准方差的计算公式：每一个数与这个数列的平均值的差的平方和，除以这个数列的项数，再开根号。
下面做一下解释：
1、数据分布离平均值越近，标准方差越小；数据分布离平均值越远，标准方差越大。
2、标准方差为0，意味着数列中每一个数都相等。
3、序列中每一个数都加上一个常数，标准方差会保持不变。
4、序列中每一个数都乘以不为零的数n，标准方差扩大n倍。

先求出总的平均数！例：共给出a.b.c.d.e五位数。它们的平均数求得为x,该组数的方差S=1/5[(a-x)"+(b-x)"+(c-x)"+(d-x)"+(e-x)"]{其中 " 为平方}
请采纳，谢谢

1。求每一个数与这个样本数列的数学平均值之间的差，称均差；
2。计算每一个差的平方，称方差；
3。求它们的总和，再除以这个样本数列的项数得到均方差；
4。再开根号得到标准方差！
标准方差主要和分母（项数）、分子（无极性偏差）有直接关系！
这里的偏差为每一个数与平均值的差异，平方运算后以去除正负极性。
为保持单位一致，再开方运算。
几个适用的理解：1.数据整体分布离平均值越近，标准方差就越小；
数据整体分布离平均值越远，标准方差越大。
（标准方差和差异的正相关）
2.特例，标准方差为0，意味着数列中每一个数都相等。
（一组平方数总和为零时，每一个平方数都必须为零）
3.序列中每一个数都加上一个常数，标准方差保持不变！
（方差本身是数值和平均值之间作比较，常数已被相互抵消。）
Standard deviation of a probability distribution or random variable
The standard deviation of a (univariate) probability distribution is the same as that of a random variable having that distribution.
The standard deviation σ of a real-valued random variable X is defined as:
egin{array}{lcl} sigma & = &sqrt{operatorname{E}((X - operatorname{E}(X))^2)} = sqrt{operatorname{E}(X^2) - (operatorname{E}(X))^2},, end{array}
where E(X) is the expected value of X (another word for the mean), often indicated with the Greek letter μ.
Not all random variables have a standard deviation, since these expected values need not exist. For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined because its E(X) is undefined.
[edit] Standard deviation of a continuous random variable
Continuous distributions usually give a formula for calculating the standard deviation as a function of the parameters of the distribution. In general, the standard deviation of a continuous real-valued random variable X with probability density function p(x) is
sigma = sqrt{int (x-mu)^2 , p(x) , dx},,
where
mu = int x , p(x) , dx,,
and where the integrals are definite integrals taken for x ranging over the range of X.
[edit] Standard deviation of a discrete random variable or data set
The standard deviation of a discrete random variable is the root-mean-square (RMS) deviation of its values from the mean.
If the random variable X takes on N values extstyle x_1,dots,x_N (which are real numbers) with equal probability, then its standard deviation σ can be calculated as follows:
1. Find the mean, scriptstyleoverline{x}, of the values.
2. For each value xi calculate its deviation (scriptstyle x_i - overline{x}) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance σ2.
5. Take the square root of the variance.
This calculation is described by the following formula:
sigma = sqrt{frac{1}{N} sum_{i=1}^N (x_i - overline{x})^2},,
where scriptstyle overline{x} is the arithmetic mean of the values xi, defined as:
overline{x} = frac{x_1+x_2+cdots+x_N}{N} = frac{1}{N}sum_{i=1}^N x_i,.
If not all values have equal probability, but the probability of value xi equals pi, the standard deviation can be computed by:
sigma = sqrt{frac{sum_{i=1}^N p_i(x_i - overline{x})^2}{sum_{i=1}^N p_i}},,and
s = sqrt{frac{N" sum_{i=1}^N p_i(x_i - overline{x})^2}{(N"-1)sum_{i=1}^N p_i}},,
where
overline{x} =frac{ sum_{i=1}^N p_i x_i}{sum_{i=1}^N p_i},,
and N" is the number of non-zero weight elements.
The standard deviation of a data set is the same as that of a discrete random variable that can assume precisely the values from the data set, where the point mass for each value is proportional to its multiplicity in the data set.