5.9 变化的系数

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  变化的系数

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  Suppose you were given three different sets of data , one with a variance of 3.2 and mean of 9.2, another with a variance of 16 and mean of 45, and the third with a variance of 155 and mean of 2100. If you were asked which set was the least centrally clustered, how could you find out?
  ::假设给了你三套不同的数据,一套是3.2和9.2的差数,另一套是16和45的差数,第三套是155和2100的差数。如果你被问及哪一套是最小集中集中的,你怎么知道呢?

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  Coefficient of Variation 
  ::变化的系数

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  In a prior lesson, we touched on the idea that variance is calculated as a single value, but that the level of clustering that it represents depends on the mean of the data. One measure that accounts for the differences between means when comparing variance is called the coefficient of variation , which is defined as:
  ::在以前的一个教训中,我们触及到一个概念,即差异是作为一个单一值计算的,但其所代表的分组水平取决于数据的平均值。 在比较差异时计算手段之间的差异的一个尺度是变量系数,该系数的定义是:

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  $$\begin{array}{r}\frac{\sigma }{\mu }\times 100=CV\mathrm{\%}\end{array}$$

  
  ::=100=CV%

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  Where  $\begin{array}{r}\sigma =\text{standard deviation}\end{array}$ , $\begin{array}{r}\mu =\text{arithmetic mean}\end{array}$ , and $\begin{array}{r}CV\mathrm{\%}=\text{coefficient of variation}\end{array}$
  ::标准偏差、 振量平均值 和 CV 差异系数

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  Recall that $\begin{array}{r}\sigma \end{array}$ , the standard deviation , is simply the square root of $\begin{array}{r}{\sigma }^2\end{array}$ , the variance.
  ::回顾标准偏差 {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

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  There are many ways to compare the relative spread of different data sets, and we will review some of them in more detail in later lessons, particularly in the chapter on ANOVA .
  ::有许多方法可以比较不同数据集的相对分布,我们将在以后的教训中,特别是在关于ANOVA的一章中,更详细地审查其中一些数据集。

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  Finding the Coefficient of Variance Percentage 
  ::找出差异系数

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  1. What is the  $\begin{array}{r}CV\mathrm{\%}\end{array}$ of a data set with a variance of 23.91 and mean of 283?
  ::1. 以23.91和283为平均值的23.91和283为差异的一组数据集的CV%是多少?

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  Recall that  $\begin{array}{r}CV\mathrm{\%}\end{array}$ (coefficient of variance percentage) is equal to 100 times the ratio of the standard deviation to the mean. This means that we should start by finding the standard deviation.
  ::回顾CV%(差异百分比的系数)等于标准偏差与平均值之比的100倍。 这意味着我们应该首先找到标准偏差。

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  • $\begin{array}{r}\sigma =\sqrt{{\sigma }^2}\end{array}$ So the standard deviation would be $\begin{array}{r}\sqrt{23.91}\end{array}$ , or $\begin{array}{r}4.89\end{array}$
    ::=2 标准差为23.91或4.89
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  • $\begin{array}{r}CV\mathrm{\%}=\frac{4.89}{283}\times 100=1.728\mathrm{\%}\end{array}$
    ::CV4.89283×100=1.728%

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  2. What is the  $\begin{array}{r}CV\mathrm{\%}\end{array}$ of the data in the table below?
  ::2. 下表中CV占数据的百分比是多少?

  Spinner	Frequency
  1	4
  2	9
  3	5
  4	8
  5	9
  6	10
  7	7

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  First find the population .
  ::首先找到人口。

  • $\begin{array}{r}\mu =\frac{4+9+5+8+9+10+7}{52}=7.43\end{array}$
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  • $\begin{array}{r}\text{Sum of squared deviances}=32.615\end{array}$
    ::平方偏离=32.615
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  • $\begin{array}{r}\text{Variance}=\frac{32.615}{7}=4.659\end{array}$
    ::差异=32.6157=4.659
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  • $\begin{array}{r}\text{Standard deviation}=\sqrt{4.659}=2.16\end{array}$
    ::标准差=4.659=2.16

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  $$\begin{array}{r}CV\mathrm{\%}=\frac{2.16}{7.43}\times 100=29.07\mathrm{\%}\end{array}$$

  
  ::CV=2.167.43×100=29.07%

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  Comparing Coefficients of Variation 
  ::比较变化系数

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  Which population data set has the highest and which the lowest coefficient of variation?
  ::哪些人口数据集是最高的,哪些是最低的变异系数?

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  $\begin{array}{r}x=\{14,16,17,19,16,19\}y=\{22,24,27,24,29,35,31\}z=\{41,44,47,44,40,49,52\}\end{array}$
  ::14,16,17,19,16,19}22,24,27,24,29,35,31}41,44,47,44,40,49,52}

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  First find the mean and standard deviation of each set:
  ::首先找到每组的平均值和标准偏差 :

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  • Set $\begin{array}{r}x\end{array}$ :
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    • $\begin{array}{r}\text{Mean}:\frac{101}{6}=16.83\end{array}$
      ::平均值:1016=16.83
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    • $\begin{array}{r}\text{Variance}:3.139\end{array}$
      ::差异:3.139
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    • $\begin{array}{r}\text{Standard Deviation}:\sqrt{3.139}=1.772\end{array}$
      ::标准偏离:3.139=1.772
    
    ::标准偏离:3.139=1.772
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  • Set $\begin{array}{r}y\end{array}$ :
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    • $\begin{array}{r}\text{Mean}:\frac{192}{7}=27.43\end{array}$
      ::平均值:1927=27.43
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    • $\begin{array}{r}\text{Variance}:17.96\end{array}$
      ::差额:17.96
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    • $\begin{array}{r}\text{Standard Deviation}:\sqrt{17.96}=4.24\end{array}$
      ::标准偏离:17.96=4.24
    
    ::标准偏差:17.96=4.24
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  • Set $\begin{array}{r}z\end{array}$ :
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    • $\begin{array}{r}\text{Mean}:\frac{317}{7}=45.29\end{array}$
      ::平均值:31777=45.29
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    • $\begin{array}{r}\text{Variance}:15.92\end{array}$
      ::差额:15.92
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    • $\begin{array}{r}\text{Standard Deviation}:\sqrt{15.92}=3.99\end{array}$
      ::标准偏离:15.92=3.99
    
    ::兹:平均值:3177=45.29 差异:15.92 标准偏离:15.92=3.99

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  Divide the standard deviation of each set by its mean, multiply by 100, and compare the percent coefficients of variation:
  ::将每一标准差除以平均值,乘以100,并比较变动系数百分率:

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  • Coefficient of variation set $\begin{array}{r}x:\frac{1.772}{16.83}=10.53\mathrm{\%}\end{array}$
    ::变异集集的系数x:1.77216.83=10.53%
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  • Coefficient of variation set $\begin{array}{r}y:\frac{4.24}{27.43}=15.46\mathrm{\%}\end{array}$
    ::变异系数y:4.2427.43=15.46%
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  • Coefficient of variation set $\begin{array}{r}z:\frac{3.99}{45.29}=8.8\mathrm{\%}\end{array}$
    ::变异设定系数z:3.9945.29=8.8%

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  Set  $\begin{array}{r}z\end{array}$ has the lowest coefficient of variation and set  $\begin{array}{r}y\end{array}$ has the highest.
  ::Set z 的变差系数最低, y 的变差系数最高。

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  Earlier  Problem Revisited
  ::重审先前的问题

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  Suppose you were given three different sets of data, one with a variance of 3.2 and mean of 9.2, another with a variance of 16 and mean of 45, and the third with a variance of 155 and mean of 2100. If you were asked which set was the least centrally clustered, how could you find out?
  ::假设给了你三套不同的数据,一套是3.2和9.2的差数,另一套是16和45的差数,第三套是155和2100的差数。如果你被问及哪一套是最小集中集中的,你怎么知道呢?

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  By finding the square root of the variance (the standard deviation), and dividing the standard deviation by the mean, you can find the coefficient of variation. Comparing the coefficients of variation allows you to directly compare the data clustering of each set, since a higher  $\begin{array}{r}CV\mathrm{\%}\end{array}$ means the data is more spread out.
  ::通过找到差异的平方根(标准偏差),并将标准偏差除以平均值,您可以找到变差系数。比较变差系数可以直接比较每组数据集的数据组合,因为更高的CV%意味着数据更加分散。

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  Examples
  ::实例

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  Example 1
  ::例1

  3240, 3260, 3250, 3280, 3280, 3300, 3310, 3270

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  Start by finding the mean and the standard deviation:
  ::首先找到平均值和标准偏差:

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  • Arithmetic mean : $\begin{array}{r}\frac{26,190}{8}=3273.75\end{array}$
    ::相对值: 26,1908=3273.75
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  • Find the variance (here I am using “mean of squares minus the square of mean”) :
    • $\begin{array}{r}\frac{{3240}^2+{3260}^2+{3250}^2+{3280}^2+{3280}^2+{3300}^2+{3310}^2+{3270}^2}{8}=10,717,937.5\end{array}$
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    • Subtract the squared mean $\begin{array}{r}({3273.75}^2=10,715,802.25)\end{array}$ to get the variance: $\begin{array}{r}10,717,937.5-10,715,802.25=2135.25\end{array}$
      ::减去平方平均值(3273.752=10,715,802.25)以得出差额:10,717,937.5-10,715,802.25=2135.25。
    
    ::找出差异(此处我使用的是“平方平均值减去平方平均值”): 32402+32602+32502+32502+32802+32802+33002+33002+33102+327028=10,717,937.8=10,717,937.5减去平方平均值(3273.752=10,715,802.25),得出差异:10,717,937-5-10,715-10,715,802.25=2135.25。
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  • The square root of the variance is the standard deviation : $\begin{array}{r}\sqrt{2135.25}=46.209\end{array}$
    ::差异的平方根是标准差:2135.25=46.209

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  Divide the standard deviation by the mean, and multiply by 100 to get $\begin{array}{r}CV\mathrm{\%}\end{array}$
  ::将标准差除以平均值,乘以100以获得 CV%

  • $\begin{array}{r}\frac{46.209}{3273.75}\times 100=1.4115\mathrm{\%}\end{array}$

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  Example 2
  ::例2

  34.4. 34.7, 34.7, 34.6, 34, 34.1, 31, 31.3 

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  Find the mean and standard deviation:
  ::找出平均值和标准偏差:

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  • Arithmetic mean : $\begin{array}{r}\frac{34.4+34+34.7+34.6+34+34.1+31+31.3}{8}=33.5125\end{array}$
    ::亚性值: 34.4+34+34+34.7+34.6+34+34.1+31+31.38=33.5125
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  • Standard deviation (square root of the “mean of squares minus square of mean”):
    ::标准偏差(“正方减去正方平方”的平方根) :

  $\begin{array}{r}\sqrt{\frac{({34.4}^2+{34}^2+{34.7}^2+{34.6}^2+{34}^2+{34.1}^2+{31}^2+{31.3}^2)}{8}-{33.5125}^2}=1.388\end{array}$

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  Divide the standard deviation by the mean, and multiply by 100 to get $\begin{array}{r}CV\mathrm{\%}\end{array}$
  
  ::将标准差除以平均值,乘以100以获得 CV%

  • $\begin{array}{r}\frac{1.388}{33.5125}\times 100=4.0414\mathrm{\%}\end{array}$

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  Example 3
  ::例3

  898.22, 990.6, 992, 996.9, 981.1, 986, 975

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  Find the mean and standard deviation:
  ::找出平均值和标准偏差:

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  • Arithmetic mean :   $\begin{array}{r}\frac{989.22+990.6+992+996.9+981.1+986+975}{7}=987.26\end{array}$
    ::亚性值: 989.22+990.6+992+996.9+981.1+986+9757=987.26
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  • Standard deviation : 
    ::标准偏差 :

  $\begin{array}{r}\sqrt{\frac{{989.22}^2+{990.6}^2+{992}^2+{996.9}^2+{981.1}^2+{986}^2+{975}^2}{7}-{987.26}^2}=6.764\end{array}$

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  Divide the standard deviation by the mean and multiply by 100 to get $\begin{array}{r}CV\mathrm{\%}\end{array}$ :
  ::将标准差除以平均值,乘以100以获得 CV% :

  • $\begin{array}{r}\frac{6.764}{987.26}\times 100=.685\mathrm{\%}\end{array}$

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  Review 
  ::回顾

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  Find the coefficient of variation %:
  ::找出变差系数%:

  1. 10, 11.1, 10.33, 10.63, 11, 11.2, 11.36, 10.46
  2. 275, 280.7, 283, 279, 284.2, 280, 282
  3. 7100.5, 7080, 7065.9, 7100, 7096, 7112, 7116.1
  4. 37, 35.3, 32.7, 34, 36, 36.2, 33.3, 33.8
  5. 3607, 3600, 3604, 3631, 3606
  6. 702, 704, 712, 716, 721, 716, 722
  7. 3370, 3300.5, 3366, 3306.6, 3310, 3336, 3301.3
  8. 34.4, 34, 34.7, 34.6, 34, 34.1, 31, 31.3
  9. 989.22, 990.6, 992, 996.9, 981.1, 986, 985
  10. 10.2, 16.34, 10.33, 10.63, 10.2, 10.44, 16.36, 10.46
  11. 3240, 3260, 3250, 3280, 3280, 3300, 3310, 3270

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  Review (Answers)
  ::回顾(答复)

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  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 。