标准方差的计算公式
标准差的计算公式：
标准差，中文环境中又常称均方差，但不同于均方误差（mean squared error，均方误差是各数据偏离真实值的距离平方的平均数，也即误差平方和的平均数，计算公式形式上接近方差，它的开方叫均方根误差，均方根误差才和标准差形式上接近）。
标准差是离均差平方和平均后的方根，用σ表示。假设有一组数值X1,X2,X3,......XN（皆为实数），其平均值（算术平均值）为μ，公式如图：
扩展资料：
标准误表示的是抽样的误差。因为从一个总体中可以抽取出无数多种样本，每一个样本的数据都是对总体的数据的估计。标准误代表的就是当前的样本对总体数据的估计，标准误代表的就是样本均数与总体均数的相对误差。
标准误是由样本的标准差除以样本容量的开平方来计算的。从这里可以看到，标准误更大的是受到样本容量的影响。样本容量越大，标准误越小，那么抽样误差就越小，就表明所抽取的样本能够较好地代表总体。
参考资料来源：百度百科-标准差
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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。
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:
\begin{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 \textstyle 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, \scriptstyle\overline{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.
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