课程： 5.3 调和平均 | The Way To Learn

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调温中

Can you define the difference between the arithmetic, geometric, and harmonic means? Can you think of a situation when each might be appropriately used?
::您能定义算术、几何和口音之间的差别吗 ? 您能想到一种情况下, 每种方法都可以被适当使用吗 ?

By the end of this lesson you will!
::到这一课结束时,你会!

The Harmonic Mean
::调和均势

The harmonic mean is a ‘middle number’ you can use when you are worried that the arithmetic mean of your set would be skewed by a few very large values (as in figure 1), and/or when particularly small values are disproportionately important. The harmonic mean is also useful for calculations of average rates, or finding other sorts of weighted averages .
::调和值是一个“ 中间数字 ” 。当你担心你的一组的算术值会被几个非常大的数值(如图1所示)和/或当特别小的数值特别重要时,你就可以使用调和值。调和值对于计算平均汇率或找到其他加权平均值也有用。

The harmonic mean can be a much better average number than the arithmetic mean.
::口音平均值可能比算术平均值要好得多。

Of the three mean value calculations we discuss in this chapter, the harmonic mean is always the lowest value if calculated using the same data .
::在本章讨论的三种平均值计算中,如果使用相同数据计算,和谐值总是最低的。

Calculating the harmonic mean is a bit more complex than the geometric mean , but it quickly gets easier with practice:
::与几何平均值相比, 计算口音的平均值要复杂一些,

Count the number of values in your data set . This number becomes your numerator.
::计算数据组中的值数。这个数字成为你的分子。

Calculate the sum of the inverses of the data values, this sum becomes your denominator.
::计算数据值的逆数之和,此数值成为你的分母。

Divide the numerator by the denominator, the resulting quotient is the harmonic mean of the values.
::将分子除以分母,由此得出的商数是数值的调和平均值。

\begin{aligned} \frac{n u m b e r o f v a l u e s}{(\frac{1}{v a l u e 1}) + (\frac{1}{v a l u e 2}) + (\frac{1}{v a l u e 3}) \dots (\frac{1}{v a l u e n})} \end{aligned}

$\begin{align*}\frac{number \ of \ values}{\left(\frac{1}{value \ 1}\right) + \left(\frac{1}{value \ 2}\right) + \left(\frac{1}{value \ 3}\right) \cdots \left(\frac{1}{value \ n}\right)}\end{align*}$
::数值数( 1值 1) +(1值 2)+(1值 3)\\\\( 1值 n)

For example, to find the harmonic mean of the set {1, 2, 3}:
::例如,要找到设置 {1, 2, 3} 的调和平均值 :

First count the values, there are 3. The number 3 becomes the numerator of your final calculation.
::首先计算值,有3,3,3,3 成为你最后计算结果的分子。

Sum the inverses of the values: $\begin{align*}\frac{1}{1} + \frac{1}{2} + \frac{1}{3} = \frac{6}{6} + \frac{3}{6} + \frac{2}{6} = \frac{11}{6}\end{align*}$ . $\begin{align*}\frac{11}{6}\end{align*}$ becomes the denominator of your final calculation.
::和数值的反数: 11+12+13=66+36+26=116。 116 成为最后计算时的分母。

Final calculation:
- Divide the numerator (3) by the denominator $\begin{align*}\left(\frac{11}{6}\right)\end{align*}$ :
  ::将分子(3)除以分母(116):
::最后计算:将分子(3)除以分母(116):

\begin{aligned} \frac{3}{\frac{11}{6}} & = \frac{18}{11} = 1.636 \\ 1.636 is the & harmonic mean of {1, 2, 3} \end{aligned}

$\begin{align*}\frac{3}{\frac{11}{6}} &= \frac{18}{11} = 1.636 \\ 1.636 \ \text{is the } & \text{harmonic mean of} \ \{1, 2, 3\}\end{align*}$
::3116=1811=1.6361.636是 {1,2,3} 的口音平均值

This type of harmonic mean assumes un-weighted values, such as the same distance travelled at different rates. To calculate the harmonic mean of differently weighted values ( the weighted harmonic mean ), such as when calculating the average speed of a trip made of segments with different distances and different rates, (see Example C), the process is nearly the same:
::这种类型的调和平均值假设了非加权值,例如以不同速度行驶的距离。要计算不同加权值的调和平均值(加权调和平均值),例如计算由距离和速度不同的段段进行的旅行的平均速度时(见例C),这一过程几乎相同:

First calculate the sum of the weights of your values (as opposed to just counting the values, as with the basic harmonic mean), this number becomes your numerator.
::首先计算数值的重量总和(而不是仅仅计算数值,就像基本口音平均值一样),这个数字变成你的分子。

Second, find the sum of the values (weights divided by amounts – NOT the inverses); this number becomes your denominator.
::第二,找到数值的总和(重量除以数量,而不是反数);这个数字成为你的分母。

Divide the numerator by the denominator, the resulting quotient is the weighted harmonic mean of the values.
::将分子除以分母,由此得出的商数是数值的加权调和平均值。

As a formula, this looks like: $\begin{align*}WHM =\frac{(\sum w)}{\left(\sum \frac{w}{a}\right)} \end{align*}$ Where $\begin{align*}w\end{align*}$ is the weight (the numerator if your values are fractional) and $\begin{align*}a\end{align*}$ is the amount (the denominator if your values are fractional).
::作为公式, 这看起来像 : WHM= (ww)( wa) w 是重量( 如果数值是分数, 则是分子) , a 是数量( 如果数值是分数, 则是分数 ) 。

Calculating the Harmonic Mean
::计算调和平均值

Calculate the harmonic mean of $\begin{align*}x\end{align*}$ .
::计算 x 的口音平均值。

\begin{aligned} x = {4, 15, 17, 5, 22} \end{aligned}

$\begin{align*}x=\{4, 15, 17, 5, 22\}\end{align*}$ Follow the steps above for the un-weighted harmonic mean:
::X4、15、17、5、22}按照以上步骤,未加权口音中值:

The number of values is 6, this is the numerator
::值数为 6 。这是分子

The sum of the inverses is $\begin{align*}\frac{1}{4} + \frac{1}{15} + \frac{1}{17} + \frac{1}{5} + \frac{1}{22} =.621\end{align*}$ , this is the denominator
::逆数总和是 14+115+117+15+1222=621,这是分母

The harmonic mean is $\begin{align*}\frac{6}{.621} = 9.66 \end{align*}$
::口音平均值为6.621=9.66。

Average Speed
::平均平均速度

Kiera decided to take a quick tour around town on her bike, if the information below describes the speeds she travelled over each equal-length segment, what was her average speed for the trip?
::Kiera决定骑自行车环游城镇, 如果以下信息描述她每个等长段的行驶速度, 她的平均行驶速度是多少?

Segment #1	8 mph	3 mi
Segment #2	12 mph	3 mi
Segment #3	14 mph	3 mi
Segment #4	7 mph	3 mi

There are two ways to solve this:
::有两种方法可以解决这个问题:

1. We could find the average speed for the whole trip by dividing the entire distance by the entire time:
::1. 通过将整个距离除以整个时间,我们可以看到整个行程的平均速度:

Sum the Distances: $\begin{align*}3 \ mi + 3 \ mi + 3 \ mi + 3 \ mi = 12 \ mi\mathbf{ \ total \ distance}\end{align*}$
::距离总和: 3 MI+3 MI+3 MI+3 MI+3 MI= 12 MI 总距离

Sum the times:
1. $\begin{align*}\text{Segment} \ 1 \ \frac{3 \ mi}{8 \ mph}=.375 \ hr\end{align*}$
  ::片段 1 3 mi8 mph= 375 小时
2. $\begin{align*}\text{Segment} \ 2 \ \frac{3 \ mi}{12 \ mph}=.25 \ hr\end{align*}$
  ::第2部分 3 mi12 mph= 25 小时
3. $\begin{align*}\text{Segment} \ 3 \ \frac{3 \ mi}{14 \ mph}=.214 \ hr\end{align*}$
  ::第3段 3 mi14 mph= 214 小时
4. $\begin{align*}\text{Segment} \ 4 \ \frac{3 \ mi}{7 \ mph}=.429 \ hr \end{align*}$
  ::第4部分 3 mi7 mph= 429 小时
5. $\begin{align*}TOTAL \ TIME: .375 + .25 + .214 + .429 = 1.268 \ hrs\mathbf{ \ total \ time}\end{align*}$
  ::总时间:375+25+25+214+429=1.268
::总和时间: 第1部分 3 mi8 mph= 375 hr section 2 3 mi12 mph= 25 hr section 3 3 mi14 mph= 214 hr section 4 3 mi7 mph= 429 小时总时间: 375+ 25+ 214+ 429=1.268

$\begin{align*}Average \ rate: \frac{total \ distance}{total \ time} = \frac{12 \ mi}{1.268 \ hrs} = 9.46 \ mph\end{align*}$
::平均时速:总距离总时间=12 MI1.268小时=9.46英里

2. The other method is to find the harmonic mean of the four rates:
::2. 另一种方法是找到四个汇率的调和平均值:

$\begin{align*}HM = \frac{4}{\frac{1}{8} + \frac{1}{12} + \frac{1}{14} + \frac{1}{7}} = \frac{4}{.423} = 9.46 \ mph\end{align*}$
::HM=418+112+114+17=4.423=9.46毫秒

Obviously, the second method is more efficient!
::显然,第二种方法更有效率!

Using the Weighted Harmonic Mean
::使用加权调力均势

Use the weighted harmonic mean to find the average rate of a traveler who records the following itinerary:
::使用加权口音平均值来找到记录下列行程的旅行人员的平均速度:

1 ^st Segment	230 km	66 kph
2 ^nd Segment	310 km	83 kph
3 ^rd Segment	199 km	72 kph
4 ^th Segment	210 km	77 kph
5 ^th Segment	240 km	91 kph

Because this problem includes segments with varying speeds and varying distances, we will need to use a weighted harmonic mean to calculate the average rate.
::由于这一问题包括速度和距离不同的部分,我们需要使用加权调和平均值来计算平均速率。

Use the formula: $\begin{align*}WHM= \frac{(\sum w)}{\left(\sum \frac{w}{a}\right)}\end{align*}$ Where $\begin{align*}w\end{align*}$ (the weight) is the distance travelled at each rate, and $\begin{align*}a\end{align*}$ (the amount) is the rate for each segment.
::使用公式: WHM=(ww)(wa) w(重量)是按每种速率行走的距离,而a(数量)是每个段段的速率。

First, calculate the sum of the weights (distances):
::首先,计算加权(距离)的总和:

\begin{aligned} 230 + 310 + 199 + 210 + 240 = 1189 \end{aligned}

$\begin{align*}230 + 310 + 199 + 210 + 240 = 1189\end{align*}$

Next, calculate the sum of the distances over the rates (remember that $\begin{align*} time = \frac{distance}{rate}\end{align*}$ ):
::接下来,计算费率的距离总和(记住时间=距离):

\begin{aligned} \frac{230 k m}{66 k p h} + \frac{310 k m}{83 k p h} + \frac{199 k m}{72 k p h} + \frac{210 k m}{77 k p h} + \frac{240 k m}{91 k p h} = 3.48 + 3.735 + 2.764 + 2.727 + 2.637 = 15.343 \end{aligned}

$\begin{align*}\frac{230 \ km}{66 \ kph} + \frac{310 \ km}{83 \ kph} + \frac{199 \ km}{72 \ kph} + \frac{210 \ km}{77 \ kph} + \frac{240 \ km}{91 \ kph} = 3.48 + 3.735 + 2.764 + 2.727 + 2.637 = 15.343\end{align*}$
::23066公里/公里+310公里/公里/公里+199公里/公里+199公里/公里+210公里/公里+210公里/公里/公里+240公里/公里=3.48+3.735+3.735+2.764+2.764+2.727+2.637=15.343

Finally, divide the sum of the weights by the sum of the times.
::最后,将权重之和除以时间之和。

The weighted harmonic mean is $\begin{align*}\frac{1189}{15.343} = 77.5 \ kph \end{align*}$
::加权口音平均值为118915.343=77.5千赫

Earlier Problem Revisited
::重审先前的问题

Can you define the difference between the arithmetic, geometric, and harmonic means? Can you think of a situation when each might be appropriately used?
::您能定义算术、几何和口音之间的差别吗 ? 您能想到一种情况下, 每种方法都可以被适当使用吗 ?

The arithmetic mean is the simple average of a set of values, the sum of the values divided by the number of values. It is appropriate for situations such as the average rate for a journey composed of segments of equal time and distance.
::算术平均值是一组数值的简单平均数,数值除以数值数。它适用于诸如由时间和距离等分组成的行程平均速度等情况。

The geometric rate is the $\begin{align*}n^{th}\end{align*}$ root of the product of the values, and is appropriate for situations such as deriving a single value to represent scores from multiple scales. The single value could then be used to compare an overall ranking of scores.
::几何率是数值产物的 nth 根,适用于产生一个单一值以代表多个尺度的得分等等情形。单一值可用于比较总得分排名。

The harmonic mean is the reciprocal of the arithmetic means of the reciprocals of a set of values. It is useful for calculations such as the average rate for a journey composed of segments of differing times or distances. A weighted harmonic mean can be used to calculate the average rate of a journey composed of differing times and distances.
::调和平均值是一组数值对等的算术手段的对等值。它可用于计算由不同时间或距离段组成的行程的平均速度。可以使用加权调和平均值计算由不同时间和距离组成的行程的平均速度。

Examples
::实例

Example 1
::例1

Find the harmonic mean of the set: $\begin{align*}\{13, 17, 22, 29, 39, 45, 50\}\end{align*}$
::查找集的口音平均值 : {13, 17, 22, 29, 39, 45, 50}

First, count the values: 7
::首先,计算值: 7

Second, sum the inverses of the values: $\begin{align*}\frac{1}{13} + \frac{1}{17} + \frac{1}{22} + \frac{1}{29} + \frac{1}{39} + \frac{1}{45} + \frac{1}{50} = \frac{4497703}{15862275} = .283\end{align*}$
::其次,加之数值的反数:113+117+122+129+139+145+150=4497703158862275=283

Finally, divide the count by the sum of the inverses: $\begin{align*}\frac{7}{.283} = 24. 74\end{align*}$
::最后,将倒数除以倒数之和:7.283=24.74。

Example 2
::例2

Find the weighted harmonic mean of the set: $\begin{align*}\left \{ \frac{1}{3}, \frac{2}{5}, \frac{2}{7}, \frac{1}{2}, \frac{3}{11}\right \}\end{align*}$
::查找集的加权口音平均值 : {13, 25, 27, 12, 311}

First, sum the numerators: $\begin{align*}1 + 2 + 2 + 1 + 3 = 9\end{align*}$
::首先,总和计数器:1+2+2+2+1+3=9

Second, sum the values: $\begin{align*}\frac{1}{3} + \frac{2}{5} + \frac{2}{7} + \frac{1}{2} + \frac{3}{11} = \frac{4, 139}{2,310} = 1.792\end{align*}$
::第二,总和数值:13+25+27+27+12+12+311=4,1392,310=1.792

Finally, divide the sum of the numerators by the sum of the values: $\begin{align*}\frac{9}{1.792}=5.022\end{align*}$
::最后,将数值之和除以数值之和: 91.792=5.022

Example 3
::例3

Use the weighted harmonic mean to find the average rate of a traveler who records the following itinerary:
::使用加权口音平均值来找到记录下列行程的旅行人员的平均速度:

1 ^st Segment	110 km	23 kph
2 ^nd Segment	230 km	56 kph
3 ^rd Segment	259 km	42 kph
4 ^th Segment	300 km	102 kph
5 ^th Segment	330 km	71 kph

First, find the sum of the distances (the weights of the values):
::首先,找到距离的总和(值的权重) :

$\begin{align*}110 \ km + 230 \ km + 259 \ km + 300 \ km + 330 \ km = 1,229\end{align*}$
::110公里+230公里+259公里+300公里+330公里=1 229公里

Next, find the sum of the times $\begin{align*}\left(\frac{\text{distance}}{\text{rate}} = \text{time}\right)\end{align*}$ :
::下一步, 找到时间的总和( 远程=时间) :

$\begin{align*}\frac{110 \ km}{23 \ kph} + \frac{230 \ km}{56 \ kph} + \frac{259 \ km}{42 \ kph} + \frac{300 \ km}{102 \ kph} + \frac{330 \ km}{71 \ kph} = 4.78 + 4.10 + 6.17 + 2.94 + 4.65 =22.64\end{align*}$
::110公里23公里+230公里56公里+259公里42公里+300公里102公里+330公里71公里=4.78+4.10+6.17+2.94+4.65=22.64

Finally, divide the distance by the time: $\begin{align*}\frac{1,229}{22.65} = 54.26 \ kph\end{align*}$
::最后,将距离除以时间:1,22922.65=54.26千兆赫。

Example 4
::例4

Comparing the basic and weighted harmonic means of this set: $\begin{align*}\left \{ \frac{1}{3}, 5, 3, \frac{3}{5}, 6, \frac{5}{11} \right \}\end{align*}$
::比较这一组基本和加权调音手段:{13,5,3,35,35,6,511}

a) Basic harmonic mean:
:a) 基本口音中值:

First, count the values: there are 6
::首先,计算值:有6个

Second, sum the inverses of the values: $\begin{align*}\frac{3}{1} + \frac{1}{5} + \frac{1}{3} + \frac{5}{3} + \frac{1}{6} + \frac{11}{5} = \frac{227}{30} \ or \ 7.57 \end{align*}$
::其次,加之数值的反数:31+15+13+53+16+115=22730或7.57。

Finally, divide the count by the sum of the inverses: $\begin{align*}\frac{6}{7.57} = 0.792 \end{align*}$
::最后,将倒数除以倒数之和:67.57=0.792

b) Weighted harmonic mean:
:b) 加权口音中值:

First, find the sum of the numerators: $\begin{align*}1 + 5 + 3 + 3 + 6 + 5 = 23 \end{align*}$
::首先,找到乘数的总和:1+5+3+3+3+6+5=23

Second, find the sum of the values: $\begin{align*}\frac{1}{3} + \frac{5}{1} + \frac{3}{1} + \frac{3}{5} + \frac{6}{1} + \frac{5}{11} = 15.39 \end{align*}$
::第二,找到数值的总和:13+51+31+35+61+511=15.39。

Finally, divide the sum of the numerators by the sum of the values: $\begin{align*}\frac{23}{15.39} = 1.494 \end{align*}$
::最后,将数值之和除以数值之和:2315.39=1.494。

Review
::回顾

For questions 1 - 10, calculate the harmonic mean.
::对于问题1-10, 计算出口音的中值。

1. $\begin{align*}\{5, 8, 9, 7, 6, 8, 3, 4\} \end{align*}$

2. $\begin{align*}\{12, 16, 18, 13, 14, 16\} \end{align*}$

3. $\begin{align*}\{23, 24, 26, 28\} \end{align*}$

4. $\begin{align*}\{33, 37, 38, 36, 35, 35, 36, 34\} \end{align*}$

5. $\begin{align*}\left \{ \frac{5}{8}, \frac{13}{21}, \frac{7}{9}, \frac{3}{5}, \frac{11}{23} \right \}\end{align*}$

6. $\begin{align*}\left \{ \frac{6}{7}, \frac{17}{23}, \frac{17}{19}, \frac{6}{11}, \frac{5}{13} \right \}\end{align*}$

7. $\begin{align*}\left \{\frac{2}{5}, \frac{5}{7} \ 2 \frac{3}{4}, \frac{15}{3}, \frac{9}{11}, \frac{14}{17} \right \}\end{align*}$

8. $\begin{align*}\left \{ 3 \frac{3}{4}, 2 \frac{2}{3}, 5 \frac{7}{8}, 3.25, 1 \frac{6}{5} \right \}\end{align*}$

9. $\begin{align*}\{5, 8, 9, 7, 6, 8, 3, 4\} \end{align*}$

10. $\begin{align*}\{12, 16, 18, 13, 14, 16\} \end{align*}$

11. Brian records the following trip data, use it and the basic harmonic mean to find his average rate for the complete trip:
::11. Brian记录下列旅行数据,使用这些数据和基本口音,以寻找他整个旅行的平均费率:

Segment #1	18 mph	4.2 mi
Segment #2	21 mph	4.2 mi
Segment #3	24 mph	4.2 mi
Segment #4	17 mph	4.2 mi

12. Use the weighted harmonic mean to find the average rate of a traveler who records the following itinerary:
::12. 使用加权口音平均数来找到记录下列行程的旅行人员的平均速度:

1 ^st Segment	21.1 km	23 kph
2 ^nd Segment	32.0 km	15.6 kph
3 ^rd Segment	29.5 km	14.2 kph
4 ^th Segment	30.5 km	10.2 kph

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。

章节大纲

常规

调温中

The Harmonic Mean ::调和均势

Calculating the Harmonic Mean ::计算调和平均值

Average Speed ::平均平均速度

Using the Weighted Harmonic Mean ::使用加权调力均势

Earlier Problem Revisited ::重审先前的问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)