节: 零分数 | 9.3 Z-核心 | The Way To Learn

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Using the can give you a good idea of the probability of occurrence of a value that happens to be exactly one, two or three to either side of the mean, but how do you compare the probabilities of values that are in between standard deviations?
::使用该数值可以使您很好地了解一个值的概率, 恰好是平均值的一、二或三, 但您如何比较标准偏差之间的数值概率 ?

Z-Scores
::零分数

Z-scores are related to the Empirical Rule from the standpoint of being a method of evaluating how extreme a particular value is in a given set. You can think of a z-score as the number of standard deviations there are between a given value and the mean of the set. While the Empirical Rule allows you to associate the first three standard deviations with the percentage of data that each SD includes, the z-score allows you to state (as accurately as you like), just how many SD’s a given value is above or below the mean.
::Z 函数从一种方法的角度来评估某一组中某一特定值的极端程度。您可以将z- score 视为某一值和该组值平均值之间的标准偏差数量。虽然经验规则允许您将前三个标准偏差与每个SD包含的数据的百分比挂钩, z- score 允许您( 如你所想的准确性) 说明 SD 的指定值是多少高于或低于平均值。

Conceptually, the z-score calculation is just what you might expect, given that you are calculating the number of SD’s between a value and the mean. You calculate the z-score by first calculating the difference between your value and the mean, and then dividing that amount by the standard deviation of the set. The formula looks like this:
::从概念上讲,z-score的计算正是你所期望的,因为你正在计算一个数值和平均值之间的SD数字。你首先计算数值和平均值之间的差数,然后将数值除以一组标准差。公式是这样的:

$\begin{aligned} z - s c o r e = \frac{(v a l u e - m e a n)}{s t a n d a r d d e v i a t i o n} = \frac{(x - μ)}{σ} \end{aligned}$

::- zscore=(价值-平均值)标准偏差=(x)-

In this lesson, we will practice calculating the z-score for various values. In the next lesson, we will learn how to associate the z-score of a value with the probability that the value will occur.
::在这一教训中,我们将实践计算各种数值的z-score。在下一个教训中,我们将学习如何将数值的z-score与数值发生的可能性联系起来。

Finding the Z-Score
::寻找 Z - S - S - S - S

1. What is the z -score of a value of 27, given a set mean of 24, and a standard deviation of 2?
::1. 数值为27的z分数(按24分制,标准偏差为2分)是多少?

To find the z-score we need to divide the difference between the value, 27, and the mean, 24, by the standard deviation of the set, 2.
::为了找到z-score,我们需要将数值(27)和平均值(24)之间的差额按照一组标准偏差(2)除以。

$\begin{aligned} z s c o r e & = \frac{27 - μ}{σ} \\ \frac{27 - 24}{2} \\ \frac{3}{2} \\ z s c o r e o f 27 & = + 1.5 \end{aligned}$

::27 - 242 32 32 32 z z z z z z z z= 27 27 - 242 32 32 z z z z z z z z z 27 z z = 15

This indicates that 27 is 1.5 standard deviations above the mean.
::这表明27is1.5标准差高于平均值。

2. What is the z -score of a value of 104.5, in a set with $\begin{aligned} μ = 125 \end{aligned}$ and $\begin{aligned} σ = 6.2 \end{aligned}$ ?
::2. 价值为104.5的Z-分数值是多少?

Find the difference between the given value and the mean, then divide it by the standard deviation.
::查找给定值和平均值之间的差,然后用标准偏差将其除以。

$\begin{aligned} z s c o r e & = \frac{104.5 - μ}{σ} \\ \frac{104.5 - 125}{6.2} \\ \frac{- 20.5}{6.2} \\ z s c o r e o f 104.5 & = - 3.306 \end{aligned}$

::104.5-1256.2 至20.56.2z 得分为 104.53.306

Note that the z -score is negative, since the measured value, 104.5, is less than (below) the mean, 125.
::请注意,z-分数为负值,因为测量值为104.5,低于(以下)平均值125。

Finding the Value Represented by a Z-Score
::查找 Z- SCR 代表的值

Find the value represented by a z -score of 2.403, given $\begin{aligned} μ = 63 \end{aligned}$ and $\begin{aligned} σ = 4.25 \end{aligned}$ .
::查找以 z-score 表示的2.403 值,以 63 和4.25 表示。

This one requires that we solve for a missing value rather than for a missing z -score, so we just need to fill in our formula with what we know and solve for the missing value:
::这就要求我们解决一个缺失的价值, 而不是一个缺失的 z -score, 所以我们只需要填充我们的公式以我们所知道的和解决的缺失的价值:

$\begin{aligned} z s c o r e & = \frac{x - μ}{σ} \\ 2.403 & = \frac{x - 63}{4.25} \\ 10.213 & = x - 63 \\ 73.213 & = x \end{aligned}$

::213=x-6373.213=x

73.213 has a z -score of 2.403
::73 213的Z分数为2.403

Earlier Problem Revisited
::重审先前的问题

Using the Empirical Rule can give you a good idea of the probability of occurrence of a value that happens to be right on one of the first three standard deviations to either side of the mean, but how do you compare the probabilities of values that are in between standard deviations?
::使用“经验规则”可以使您很好地了解一个值的发生概率,该值恰好位于平均值的两侧的前三个标准偏差之一,但您如何比较标准偏差之间的数值概率?

The z -score of a value is the count of the number of standard deviations between the value and the mean of the set. You can find it by subtracting the value from the mean, and dividing the result by the standard deviation.
::数值的 z 分数是集值和平均值之间标准差数的计数。您可以通过从平均值中减去值和按标准差除以结果来找到该值。

Examples
::实例

Example 1
::例1

What is the z -score of the price of a pair of skis that cost $247, if the mean ski price is $279, with a standard deviation of $16?
::如果平均滑雪价格为279美元,标准偏差为16美元,那么一双耗资247美元的滑雪车的价格的Z值是多少?

Find first the difference between the measured value and the mean, then divide that difference by the standard deviation:
$\begin{aligned} \frac{247 - 279}{16} \\ \frac{- 32}{16} \\ z -score = - 2 \end{aligned}$

::首先找出测量值与平均值之间的差异,然后用标准差除以这一差异:247-27916-3216z-score%2

Example 2
::例2

What is the z -score of a 5-scoop ice cream cone if the mean number of scoops is 3, with a standard deviation of 1 scoop?
::如果平均勺子数是3, 标准偏差是1勺, 那么5勺冰淇淋锥的z分点是什么?

This one is easy: The difference between 5 scoops and 3 scoops is +2, and we divide that by the standard deviation of 1, so the z-score is +2
::这很简单:5勺和3勺之间的差差是+2,我们用标准差1来划分,所以z分是+2

Example 3
::例3

What is the z -score of the weight of a cow that tips the scales at 825 lbs, if the mean weight for cows of her type is 1150 lbs, with a standard deviation of 77 lbs?
::如果母牛的平均重量为1 150磅,标准偏差为77磅,那么,在825磅的量级上把牛的重量缩压到825磅的牛的重量的Z值是多少?

First find the difference between the measured value and the mean, then divide that difference by the standard deviation:
::首先发现测量值和平均值之间的差,然后用标准偏差将差除以:

$\begin{aligned} \frac{825 l b s - 1150 l b s}{771 l b s} \\ \frac{- 325}{77} \\ z -score = - 4.2207 \end{aligned}$

::825 lbs-1150 lbs771 lbs-32577z-score=4.2207

Example 4
::例4

What is the z -score of a measured value of 0.0034, given $\begin{aligned} μ = 0.0041 \end{aligned}$ and $\begin{aligned} σ = 0.0008 \end{aligned}$ ?
::按 0.0041 和 0.0008 计算, 测量值为 0.0034 的 z 分数是什么 ?

First find the difference between the measured value and the mean, then divide that difference by the standard deviation:
::首先发现测量值和平均值之间的差,然后用标准偏差将差除以:

$\begin{aligned} \frac{0.0034 - 0.0041}{0.0008} \\ \frac{- 0.0007}{0.0008} \\ z -score = - 0.875 \end{aligned}$

::0.0034-0.004100008-0.007000008z-score=0.875

Review
::回顾

Given a distribution with a mean of 70 and standard deviation of 62, find a value with a z-score of -1.82.
::如果分布平均值为70,标准偏差为62,则发现一个值为z-1.22。

What does a z-score of 3.4 mean?
::3.4的Z分数是什么意思?

Given a distribution with a mean of 60 and standard deviation of 98, find the z-score of 120.76.
::分布平均值为60,标准偏差为98,兹分数为120.76。

Given a distribution with a mean of 60 and standard deviation of 21, find a value with a z-score of 2.19.
::如果分布平均值为60,标准差为21,则发现值为Z-分数2.19。

Find the z-score of 187.37, given a distribution with a mean of 185 and standard deviation of 1.
::查找 z -score 共 187.37 , 以 185 平均值和 1 标准差分配。

What does a z-score of -3.8 mean?
::-3.8的Z分数是什么意思?

Find the z-score of 125.18, given a distribution with a mean of 101 and standard deviation of 62.
::查找 z - score 为125.18, 给分布以 101 平均值, 标准偏差为 62 。

Given a distribution with a mean of 117 and standard deviation of 42, find a value with a z-score of -0.94.
::如果分布的平均值为117,标准偏差为42,则找到z-score值为-0.94。

Given a distribution with a mean of 126 and standard deviation of 100, find a value with a z-score of -0.75.
::如果分布的平均值为126,标准偏差为100,则发现值为Z-0.75。

Find the z-score of 264.16, given $\begin{aligned} μ = 188 \end{aligned}$ and $\begin{aligned} σ = 64 \end{aligned}$ .
::查找 z -score 共 264. 16, 给出 188 和 64 。

Find a value with a z-score of -0.2, given $\begin{aligned} μ = 145 \end{aligned}$ and $\begin{aligned} σ = 56 \end{aligned}$ .
::根据 145 和 56, 找到一个 z -score 值为 -0. 2 的值。

Find the z-score of 89.79 given $\begin{aligned} μ = 10 \end{aligned}$ and $\begin{aligned} σ = 79 \end{aligned}$ .
::找到89.79的 z-score 给 10 和 79 。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

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