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====== z分数以及其应用======

====== z-scores and their applications ======

如果我们以均值为一个参照点，在单个的分布中，我们可以利用离差来衡量每个原始分数的位置，但如果我们想比较两个或者多个分布中的原始分数的相对位置，离差就变得无法发挥作用了。

If we take the mean as a point of reference, in a single distribution we can use the deviation to measure the position of each raw score, but if we want to compare the relative positions of raw scores in two or more distributions, the deviation becomes useless.

在不同的分布间进行比较时，只依靠原始分数的绝对值是远远不够的，所以为了避免错误，我们引入一个方法：计算标准差。

When making comparisons between different distributions, it is not enough to rely only on the absolute values of the raw scores. To avoid errors, we introduce a method: calculating the standard deviation.

===== 一、z分数与原始分数（z-scores and raw scores） =====
z分数由振幅符号和数值两部分组成。符号的正负表示出了z分数所对应的原始分数是比均值大还是均值小。而z分数的数值表示的是原始分数和均值之间相差几个标准差。

The z-score consists of two parts: the amplitude symbol and the numerical value. The positive or negative sign indicates whether the raw score corresponding to the z-score is larger or smaller than the mean. The numerical value of the z-score indicates how many standard deviations between the raw score and the mean.

那么我们很容易可以得到z分数的计算公式：

Then we can easily get the formula for the z-score:

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由于这些数据都是可以进行代数运算的，所以在已知原分布的均值、标准差以及z分数情况下，我们也可以逆推出原始分数。

As these data are algebraically accessible, we can also invert the original score when the mean, standard deviation, and z-score of the original distribution are known.

===== 二、z分数与标准分布（Z-score and standard distribution） =====

z分数的另一个用处是将整个分布标准化。在总体或样本的均值和标准差都已知的情况下、我们变能将分布中的原始分数都转化为z分数。所得到的新分布就被称为z分数分布，也称标准分布，并称此过程为标准化。z分数分布有三个特征：

The another usage of z-score is to standardize the whole distribution. Once the mean and standard deviation of sample or population are known, the original scores could be changed into z-score. The new distribution is called z-score distribution, or standardized distribution. This process is called standardize. Z-score distribution has 3 main features:

（1）z分数分布的形状和未转换前的原始分布的形状完全相同。

(1)The shape of z-score distribution is the same with the original one.

（2）z分数分布的均值为0。

(2)The mean of z-score distribution is 0.

（3）z分数分布的标准差为1。

(3)The standard variation of z-score distribution is 1.

===== 三、z分数的意义（The significance of z-score） =====


z分数不仅可以为我们提供分数在分布中的位置信息，而且可以使整个分布标准化，这样便于在不同的分布之间进行比较。z分数还可以代表概率，我们只要知道的z分数的区间，就可以计算出相应的落在这个区间的概率。其次，z分数还可以代表变量间的关系。不过如果总体为偏态分布，那么z分数只能帮我们比较不同总体内的分数相对均值的距离，而不再能确定分数的位置。

Z-score not only provides the position information of the score, but also can standardize the whole distribution, making it easy to compare with different distributions. Z-score also represents possibility, once the interval of z-score is known, the possibility of scores in this interval could be calculated. Meanwhile, z-score can also represent the relationship between variables. But if the distribution is a skew distribution, z-score could only help to compare the distance to the mean of different scores, but no longer ascertain the position of scores.