Course: 5.1 相对比值 | The Way To Learn

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A. 测深值

Suppose you just purchased your first car, and you are wondering what kind of gas mileage it gets. If you collect the data in the table below from your first 4 tanks of gas, how could you calculate your average gas mileage? Is the mean gas mileage different if it is calculated as the average of the mileages from each tank instead of calculated using the total miles and total gallons?
::假设你刚刚购买了第一辆汽车,而你正在想它能得到什么样的气体里程。如果你从头四罐天然气中收集下面表格中的数据,你如何计算平均天然气里程?如果以每辆油罐的平均里程计算,而不是用总里程和总加仑计算,那么平均天然气里程是不一样的吗?

Gallons of Gas	18.7	17.8	16.3	19.1
Miles Driven	363	347	320	402

After the lesson, we’ll return to this question to review the answer.
::在教训之后, 我们将回到这个问题, 来检讨答案。

Arithmetic Means
::亚学手段

Although there are actually three different relatively common types of mean, or average, the arithmetic mean is far and away the most common method of calculating the “middle” value. The other two, the geometric mean and harmonic mean , are the topics of other lessons.
::尽管实际上有三种不同的相对常见的平均值或平均值,但算术平均值是计算“中值”的最常用方法。其余两种,即几何平均值和调和平均值,则是其他经验教训的主题。

The arithmetic mean is a very important calculation to master, because it is the first step in a multitude of more complex calculations. Fortunately, the mean is relatively easy to calculate:
::算术平均值对于主机来说是一个非常重要的计算方法,因为它是众多更复杂的计算方法的第一步。幸运的是,这个平均值比较容易计算:

Calculate the sum of all of the values in your set
::计算您集中所有值的总和

Divide the sum by the number or count of values in the set
::将总和除以集中的值数或计数

The quotient of the sum of the values divided by the number of values is the mean
::以数值数除以的数值总和的商数是平均值。

In statistics , you will work with the arithmetic means of different groupings of units . Most importantly, you will need to know the difference between the population mean and the sample mean . The population mean is the mean of all of the members of an entire population , and the sample mean is the mean only of the members of a sample or subset of a population.
::在统计中,你会使用不同组别单位的算术方法。最重要的是,你需要了解人口平均值和抽样平均值之间的差别。人口平均值是全部人口所有成员的平均值,抽样平均值是抽样或人口子集中仅有的平均值。

Calculating the Arithmetic Mean
::计算自算值

Calculate the arithmetic mean of the set.
::计算集的算术平均值。

\begin{aligned} \begin{array}{ccccccccccccc} 26 & 5 \\ 27 & 5 & 5 \\ 28 & 0 & 0 & 1 & 1 & 1 & 2 & 4 & 5 & 5 & 8 \end{array} \end{aligned}

$\begin{align*}\begin{array}{c|c c c c c c c c c c c c} 26 & 5\\ 27&5&5\\ 28&0&0&1&1&1&2&4&5&5&8\end{array}\end{align*}$

This set is organized in stem and leaf format, and contains 13 values between 265 and 288.
::该组以干叶和叶的形式组织,包含265至288之间的13个值。

To find the mean, first calculate the sum of the values in the set:
::要找到平均值,首先计算集中的数值总和:

\begin{aligned} 265 + 275 + 275 + 280 + 280 + 281 + 281 + 281 + 282 + 284 + 285 + 285 + 288 = 3, 642 \end{aligned}

$\begin{align*}265+275+275+280+280+281+281+281+282+284+285+285+288=3,642\end{align*}$

Since there are 13 values in the set, the mean is:
::由于该集中有13个数值,其平均值是:

\begin{aligned} \frac{3, 642}{13} = 280.154 \end{aligned}

$\begin{align*}\frac{3,642}{13}=280.154\end{align*}$

Rounded to the nearest whole, the arithmetic mean is 280.
::经过四舍五入到最近的整数,算术平均值为280。

Finding the Arithmetic Means of Data Sets
::查找数据集的自测手段

Find the arithmetic means of the sets:
::查找集的算术方式 :

For each set, find the sum of the values in the set, and divide by the number of values:
::对于每一组, 在集中查找数值的总和, 并除以数值数 :

a) 261, 286, 257, 284, 258, 281, 258, 285, 267, 275, 258, 284, 260, 285, 258, 284
:a) 261、286、257、284、258、281、281、258、285、267、275、258、284、260、285、258、284

The sum of the values is 4,341. There are 16 values.
::数值的总和为4,341,有16个数值。

\begin{aligned} \frac{4, 341}{16} = 271.31 \end{aligned}

$\begin{align*}\frac{4,341}{16}=271.31\end{align*}$

b) 43, 44, 45, 45, 46, 46, 47, 47, 47, 48, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 50, 50, 50, 51
:b) 43、44、45、45、45、46、46、47、47、47、47、48、49、49、49、49、49、49、49、49、49、49、49、49、49、49、49、49、50、50、50、50、51

The sum of the values is 1,198. There are 25 values.
::数值的总和是1,198,有25个数值。

\begin{aligned} \frac{1, 198}{25} = 47.92 \end{aligned}

$\begin{align*}\frac{1,198}{25}=47.92\end{align*}$

c) 14, 28, 42, 56, 23, 67, 12, 45, 93, 52
:c) 14、28、42、56、23、67、12、45、93、52

The sum of the values is 432. There are 10 values.
::数值的总和是432,有10个数值。

\begin{aligned} \frac{432}{10} = 43.2 \end{aligned}

$\begin{align*}\frac{432}{10}=43.2\end{align*}$

Finding the Mean Using a Chart
::使用图表查找平均值

Using the data from the chart, find the mean number of prank calls per five-year interval , and the mean number of reported prank calls per year.
::使用图表中的数据,发现每五年间歇的恶作剧电话平均次数,以及每年报告的恶作剧电话平均次数。

First, find the sum of the number of reported calls:
::首先,找到所报通话次数的总和:

\begin{aligned} 188 + 149 + 161 + 172 + 154 + 172 = 996 c a l l s \end{aligned}

$\begin{align*}188+149+161+172+154+172=996 \ calls\end{align*}$
::188+149+161+172+154+172=996电话

The chart indicates that there are six five-year intervals with recorded values. So, dividing the sum by the number of values, we get:
::图表显示,每五年有6个有记录值的5年间隔。因此,将总和除以数值数,我们得到的是:

\begin{aligned} \frac{996}{6} = 166 c a l l s p e r f i v e y e a r i n t e r v a l \end{aligned}

$\begin{align*}\frac{996}{6}=166 \ calls \ per \ five \ year \ interval\end{align*}$
::9966=每五年间间间166次拨打电话

Note that the value “166” is not the actual count of any of the recorded intervals. The arithmetic mean obviously does not need to be a member of the actual set.
::请注意,“166”值不是任何记录间隔的实际数,计算值显然不必是实际数组的一部分。

The chart spans the years 1970 through 1999 (the end of 1999 is the beginning of 2000), so there are a total of 29 years recorded. Dividing the total number of calls by the number of years yields:
::该图表涵盖1970年至1999年(1999年底为2000年初),因此总共记录了29年。

\begin{aligned} \frac{996}{29} \approx 34 (r o u n d e d) c a l l s p e r y e a r \end{aligned}

$\begin{align*}\frac{996}{29} \approx 34 \ (rounded) \ calls \ per \ year\end{align*}$
::每年99629%34(四通电话)

Earlier Problem Revisited
::重审先前的问题

a) If you collect the data in the table below from your first 4 tanks of gas, how could you calculate your average gas mileage?
:a) 如果从头4个油箱收集下表的数据,如何计算平均油气里程?

b) Is the mean gas mileage different if calculated as the average of the mileages from each tank instead of the total miles and total gallons?
:b) 如果计算出每辆坦克的里程平均数,而不是总里程和加仑总量,平均天然气里程是否不同?

Gallons of Gas	18.7	17.8	16.3	19.1
Miles Driven	363	347	320	402

a) Mileage is measured as miles travelled per gallon, so we can simply total the number of miles and divide by total gallons:
:a) 里程以每加仑行驶的里程计量,因此我们可以简单地将里程总数和除以总加仑数:

\begin{aligned} \frac{1, 432 m i l e s}{71.9 g a l l o n s} = 19.92 a v g m i l e s p e r g a l l o n \end{aligned}

$\begin{align*}\frac{1,432 \ miles}{71.9 \ gallons}=19.92 \ avg \ miles \ per \ gallon\end{align*}$
::1,432英里 71.9加仑=19.92 瓦克/加仑

b) If we calculate the mean mileage for each tank:
:b) 如果我们计算每辆坦克的平均里程数:

\begin{aligned} \frac{363}{18.7} = 19.41 \frac{347}{17.8} = 19.49 \frac{320}{16.3} = 19.63 \frac{402}{19.1} = 21.05 \end{aligned}

$\begin{align*}\frac{363}{18.7}=19.41 \quad \frac{347}{17.8}=19.49 \quad \frac{320}{16.3}=19.63 \quad \frac{402}{19.1}=21.05\end{align*}$

and find the mean of the mileages:
::并发现里程的平均值:

\begin{aligned} \frac{19.41 + 19.49 + 19.63 + 21.05}{4} = \frac{79.58}{4} = 19.90 m i l e s p e r g a l l o n \end{aligned}

$\begin{align*}\frac{19.41+19.49+19.63+21.05}{4}=\frac{79.58}{4}=19.90 \ miles \ per \ gallon\end{align*}$
::19.41+19.49+19.63+21.054=79.584=19.90英里每加仑

we see that the overall mileage appears slightly lower this way.
::整体里程似乎稍低一点。

Can you figure out why this might be?
::你能知道为什么吗?

Examples
::实例

Example 1
::例1

Find the mean of the following data:
::查找下列数据的平均值:

237, 258, 232, 232, 241, 238, 233, 245, 242, 243, 237, 232, 241, 242, 233

To find the mean, add all the numbers and divide by the count of numbers you started with.
::要找到平均值, 添加所有的数字, 并除以从您开始的数字的数数。

\begin{aligned} 237 + 258 + 232 + 232 + 241 + 238 + 233 + 245 + 242 + 243 + 237 + 232 + 241 + 242 + 233 = 3586 \\ = \frac{3586}{15} = 239.07 (rounded) \end{aligned}

$\begin{align*}&237+258+232+232+241+238+233+245+242+243+237+232+241+242+233= 3586 \\ &=\frac{3586}{15}=239.07 \ (\text{rounded})\end{align*}$
::237+258+232+232+232+241+238+238+233+245+242+242+244+243+237+232+241+241+2442+242+242+233=3586=358615=239.07(四舍五入)

Example 2
::例2

Find the mean of the following data:
::查找下列数据的平均值:

22, 30, 24, 42, 24, 42, 18, 32, 24, 48, 24, 45, 28, 46, 22, 42

\begin{aligned} 22 + 30 + 24 + 42 + 24 + 42 + 18 + 32 + 24 + 48 + 24 + 45 + 28 + 46 + 22 + 42 = 513 \\ = \frac{513}{16} = 32.06 (rounded) \end{aligned}

$\begin{align*}&22+30+24+42+24+42+18+32+24+48+24+45+28+46+22+42=513 \\ & = \frac{513}{16}=32.06 \ (\text{rounded})\end{align*}$
::22+30+24+42+42+24+42+42+24+42+18+32+24+24+24+24+45+28+46+22+42+42=513=513=51316=32.06(四舍五入)

Example 3
::例3

Find the mean of the following data:
::查找下列数据的平均值:

199, 200, 205, 216, 219, 220, 221, 221, 222, 222, 224, 227, 227, 228, 229

\begin{aligned} 199 + 200 + 205 + 216 + 219 + 220 + 221 + 221 + 222 + 222 + 224 + 227 + 227 + 228 + 229 = 3280 \\ = \frac{3280}{15} = 218.67 \end{aligned}

$\begin{align*}&199+200+205+216+219+220+221+221+222+222+224+227+227+228+229=3280 \\ & =\frac{3280}{15}=218.67\end{align*}$

Review
::回顾

Airon collected a sample of the five cheapest (out of a population of 10) different stores’ prices on the new laptop he wanted. His sample was $285, $290, $300, $305, and $315. The other five stores’ prices were $345, $380, $435, $480, and $500.
::Airon收集了他想要的新笔记本电脑上五个最廉价的商店价格(在10个人口中)的样本。他的样本为285,290,300,305和315美元。其他五家商店的价格为345,380,435,480和500美元。其样本为285,290,300,305和315美元。其他五家商店的价格为345,380,435,400美元和500美元。

1. What was the arithmetic sample mean of laptop prices among the sample Airon collected?
::1. 所采集的样本中笔记本电脑价格的算术抽样平均值是多少?

2. What was the arithmetic mean of laptops that Airon did not add to his sample?
::2. Airon没有在他的样本中添加笔记本电脑的算术含义是什么?

3. What was the arithmetic population mean of laptop prices?
::3. 笔记本电脑价格的算术人口平均值是多少?

Questions 4 – 6 refer to the numbers: 4, 7, 2, 5, 7, 8, 4, 9, 5, 6, 2, and 12
::问题4-6涉及数字:4、7、2、5、7、8、8、4、9、5、6、2和12

4. What is the arithmetic mean of the numbers?
::4. 数字的算术含义是什么?

5. What is the arithmetic mean of the odd numbers?
::5. 奇数的算术含义是什么?

6. What is the arithmetic mean of the even numbers?
::6. 偶数的算术含义是什么?

Chet’s class has 7 boys, ages 15, 16, 14, 15, 17, 14, and 17 years, and 6 girls, ages 15, 16, 14, 14, 15, and 16 years.
::切特班有7名男孩,年龄分别为15岁、16岁、14岁、15岁、17岁、14岁和17岁,6名女孩,年龄分别为15岁、16岁、14岁、14岁、14岁、15岁和16岁。

7. What is the arithmetic population mean of Chet’s class?
::7. Chet班级的算术人数意味着什么?

8. What is the arithmetic mean of girls in Chet’s class?
::8. 切特班女生的算术含义是什么?

9. What is the arithmetic mean of boys in Chet’s class?
::9. 切特班男生的算术含义是什么?

10. What is the arithmetic sample mean of a sample of students consisting of the youngest and oldest boys and youngest and oldest girls (a total of four students)?
::10. 由最年幼和最年长的男孩以及最年幼和最年长的女孩(共有4名学生)组成的抽样学生的算术抽样方法是什么?

Donna works in the customer service department at a large corporation. The employees in her department earn the following incomes: $24,560, $32,540, $29,540, $39,490, $42,100, $27,000, and $35,750.
::Donna在一家大型公司的客户服务部门工作,其雇员的收入如下:24 560美元、32 540美元、29 540美元、39 490美元、42 100美元、27 000美元和35 750美元。

11. What is the mean income of the department?
::11. 该部门的平均收入是多少?

12. What is the mean income above $30,000?
::12. 超过30 000美元的平均收入是多少?

13. What is the mean income below $30,000?
::13. 30 000美元以下的平均收入是多少?

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。

Section outline

General

A. 测深值

Arithmetic Means ::亚学手段

Calculating the Arithmetic Mean ::计算自算值

Finding the Arithmetic Means of Data Sets ::查找数据集的自测手段

Finding the Mean Using a Chart ::使用图表查找平均值

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

Review ::回顾

Review (Answers) ::回顾(答复)

Arithmetic Means
::亚学手段

Calculating the Arithmetic Mean
::计算自算值

Finding the Arithmetic Means of Data Sets
::查找数据集的自测手段

Finding the Mean Using a Chart
::使用图表查找平均值

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

Review
::回顾

Review (Answers)
::回顾(答复)