Statistics 简明教程

Statistics - Normal Distribution

正态分布是在其中大多数值聚集在范围中部，其余值向两端对称递减的数据集的排列方式。身高就是一个简单例子，它遵循正态分布模式：大多数人的身高都是平均身高，比平均身高高的人和矮的人数相当，极少数（且大致相等）的人要么非常高，要么非常矮。以下是一个正态分布曲线的示例：

A normal distribution is an arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme. Height is one simple example of something that follows a normal distribution pattern: Most people are of average height the numbers of people that are taller and shorter than average are fairly equal and a very small (and still roughly equivalent) number of people are either extremely tall or extremely short.Here’s an example of a normal distribution curve:

normal distribution1

正态分布的图形表示有时称为钟形曲线，因为它呈喇叭状。根据总体分布的不同，其精确形状可能有所不同，但峰值始终位于中间，曲线始终是对称的。在正态分布中，均值、众数和中位数都相同。

A graphical representation of a normal distribution is sometimes called a bell curve because of its flared shape. The precise shape can vary according to the distribution of the population but the peak is always in the middle and the curve is always symmetrical. In a normal distribution the mean mode and median are all the same.

Formula

其中——

Where −

  1. ${\mu}$ = Mean

  2. ${\sigma}$ = Standard Deviation

  3. ${\pi \approx 3.14159}$

  4. ${e \approx 2.71828}$

Example

Problem Statement:

Problem Statement:

对每日出行时间进行的调查得出了以下结果（单位：分钟）：

A survey of daily travel time had these results (in minutes):

26

33

65

28

34

55

25

44

50

36

26

37

43

62

35

38

45

32

28

34

平均值为 38.8 分钟，标准差为 11.4 分钟。将值转换为 z 分数，并准备正态分布图。

The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes. Convert the values to z - scores and prepare the Normal Distribution Graph.

Solution:

Solution:

我们一直在使用的 z 分数公式：

The formula for z-score that we have been using:

其中——

Where −

  1. ${z}$ = the "z-score" (Standard Score)

  2. ${x}$ = the value to be standardized

  3. ${\mu}$ = mean

  4. ${\sigma}$ = the standard deviation

要转换 26：

To convert 26:

首先减去平均值：26-38.8 = -12.8，

First subtract the mean: 26-38.8 = -12.8,

然后除以标准差：-12.8/11.4 = -1.12

Then divide by the Standard Deviation: -12.8/11.4 = -1.12

所以 26 是 -1.12 以标准差计量的均值偏差

So 26 is -1.12 Standard Deviation from the Mean

以下是前三个转换方式。

Here are the first three conversions.

Original Value

Calculation

Standard Score (z-score)

26

(26-38.8) / 11.4 =

-1.12

33

(33-38.8) / 11.4 =

-0.51

65

(65-38.8) / 11.4 =

-2.30

…​

…​

…​

并且以下为其图表表示：

And here they graphically represent:

normal distribution2