z-score

A z-score (aka, a standard score) indicates how many standard deviations an element is from the mean. A z-score can be calculated from the following formula.

z = (X - μ) / σ

where z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation.

Here is how to interpret z-scores.

A z-score less than 0 represents an element less than the mean.
A z-score greater than 0 represents an element greater than the mean.
A z-score equal to 0 represents an element equal to the mean.
A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc.
A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc.
If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3.

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z分数（z-score），也叫标准分数（standard score），标准化变量，是一个数与平均数的差再除以标准差的过程。在统计学中，标准分数是一个观测或数据点的值高于被观测值或测量值的平均值的标准偏差的符号数。

z分数可以回答这样一个问题："一个给定分数距离平均数多少个标准差?"在平均数之上的分数会得到一个正的标准分数，在平均数之下的分数会得到一个负的标准分数。 z分数是一种可以看出某分数在分布中相对位置的方法。

z分数能够真实的反应一个分数距离平均数的相对标准距离。如果我们把每一个分数都转换成z分数，那么每一个z分数会以标准差为单位表示一个具体分数到平均数的距离或离差。将成正态分布的数据中的原始分数转换为z分数，我们就可以通过查阅z分数在正态曲线下面积的表格来得知平均数与z分数之间的面积，进而得知原始分数在数据集合中的百分等级。一个数列的各z分数的平方和等于该数列数据的个数，并且z分数的标准差和方差都为1.平均数为0.

z分数常用于标准化考试的z - test——模拟学生的t检验，而不是估计其参数。由于了解整个总体的情况很复杂，所以t检验被广泛应用。

此外，标准分数可用于计算预测区间。一个预测区间[L,U]，由一个较低的端点指定的L和一个上端点指定的U组成，这是一个区间，未来的观察值X将在高概率γ伽玛的区间内，即