Course: 10.1 平均值、中值、模式和范围

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平均值、中值、模式和范围
An amusement park is designing a new section for children over 3 years old and under 8 years old. As part of their research, they used a survey of the heights and weights of a thousand children in that age group. Which measure of central tendency should they use to accommodate the greatest number of children on a roller coaster?
::一家游乐公园正在为3岁以上和8岁以下儿童设计一个新区,作为他们研究的一部分,他们利用该年龄组1 000名儿童的身高和体重调查。他们应该用哪种中心趋势衡量在过山车上容纳最多儿童?

In this concept, you will learn to understand Measures of Central Tendency by finding the mean, , mode and range of a data set .
::在这个概念中,你会通过找到一套数据集的平均值、模式和范围,学会了解中央定时的度量。

Statistical Measures
::统计措施统计统计措施

In the real world, there are many situations in which a large group of data is collected. In order to make sense of the data, statistical measures are used. These measures help to generalize a group of data, make inferences about it, and compare it with other groups of data.
::在现实世界中,收集大量数据的情况很多,为了了解数据,采用了统计措施,这些措施有助于将一组数据加以归纳,对其作出推论,并与其他一组数据进行比较。

Statistical measures include mean, median, mode and range . Depending on the situation, certain measures may be more helpful than others in interpreting data. The mean, median, and mode are three common measures of central tendency ; they are three mathematical tools frequently used to analyze data.
::统计措施包括中位数、中位数、模式和范围。根据情况,某些措施在解释数据方面可能比其他措施更有帮助。中位数、中位数和模式是三大常见的中心趋势衡量标准;它们是三个经常用来分析数据的数学工具。

The mean , commonly referred to as the average , is the sum of all the data items divided by the number of data items.
::通常称为平均数的平均值是所有数据项目除以数据项目数的总和。

The median is the middle number in a set of data that is ordered from least to greatest. If there is an even number of data, you take the average of the middle two numbers to find the median.
::中位数是一组数据中的中位数, 从最小到最大排序。如果数据数量偶数, 则用中间两个数字的平均值来找到中位数。

The mode is the number that occurs most often.
::模式是最常见的数字。

Let’s look at an example.
::让我们举个例子。

A manager at a small movie theater was analyzing the number of people who came to the movies during the week. Over nine days, he found the following data: 81, 89, 92, 85, 93, 62, 85, 105, and 90. Find the mean, median, and mode of the data.
::一个小型电影院的经理正在分析一周来看电影的人数。 9天后,他发现了以下数据:81、89、92、85、93、62、85、105和90。找出数据的平均值、中位数和模式。

First, find the mean. Remember that the mean is the same as the average. To find the mean you must add all of the data items and divide by the number of items.
::首先, 找到平均值。记住平均值与平均值相同。要找到平均值, 您必须添加所有的数据项, 并除以项目数量。

$\begin{aligned} \begin{array}{rcl} mean & = & \frac{81 + 89 + 92 + 85 + 93 + 62 + 85 + 105 + 90}{9} \\ = & \frac{782}{9} \\ = & 86.8 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{mean} &=& \frac{81+89+92+85+93+62+85+105+90}{9} \\ &=& \frac{782}{9} \\ &=& 86.8 \end{array}\end{align*}$
::=81+89+92+85+93+62+85+105+909=7829=86.8

Next, find the median. The median is the middle number when the data is ordered from lowest to highest. When you reorder the data from least to greatest you get:
::下一步,请找到中位数。中位数是数据从最低到最高排序时的中位数。当您将数据从最低重新排序到最高排序时,您可以得到 :

$\begin{aligned} \begin{array}{rcl} 62, 81, 85, 85, 89, 90, 92, 93, 105 \\ ↑ \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} & 62, 81, 85, 85, 89, 90, 92, 93,105 \\ & \uparrow \end{array}\end{align*}$

The middle number, 89, is the median.
::中间数字89是中位数。

Next, find the mode. The mode is the number that occurs most often.
::下一步, 找到模式。模式是最常出现的数字。

In this case, 85 occurs two times and all of the other numbers only once.
::在这种情况下,85起是两次,所有其他数字只有一次。

The number 85 is the mode.
::85号是模式

The answer is that the mean is 87 (rounded up since you are finding the mean number of people), the median is 89 and the mode is 85.
::答案是,平均值是87(因为你发现平均人数,所以四舍五入),中位数是89, 模式是85。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the new section of the amusement park.
::早些时候,你得到一个问题关于娱乐公园的新部分。

In order to attract more customers, they should accommodate as many children as possible. For this reason, they should use the range which will include heights of children from the shortest to the tallest.
::为了吸引更多的顾客,他们应当收容尽可能多的儿童,为此,他们应当使用从最短到最高的儿童身高的幅度。

Example 2
::例2

Find the mean, median, mode and range of the following data set.
::查找下列数据集的平均值、中位数、模式和范围。

12, 13, 15, 18, 22, 25, 30, 31, 32, 34, 40

First, find the mean. To find the mean you must add all of the data items and divide by the number of items.
::首先, 找到平均值。要找到平均值, 您必须添加所有的数据项, 并除以项目数量。

$\begin{aligned} \begin{array}{rcl} mean & = & \frac{12 + 13 + 15 + 18 + 22 + 25 + 30 + 31 + 32 + 34 + 40}{11} \\ = & \frac{272}{11} \\ = & 24.73 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{mean} &=& \frac{12+13+15+18+22+25+30+31+32+34+40}{11} \\ &=& \frac{272}{11} \\ &=& 24.73 \end{array}\end{align*}$
::=12+13+15+18+22+25+30+31+32+34+4011=27211=24.73

Next, find the median. The median is the middle number when the data is ordered from lowest to highest. When you reorder the data from least to greatest you get:
::下一步,请找到中位数。中位数是数据从最低到最高排序时的中位数。当您将数据从最低重新排序到最高排序时,您可以得到 :

$\begin{aligned} \begin{array}{rcl} 12, 13, 15, 18, 22, 25, 30, 31, 32, 34, 40 \\ ↑ \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} & 12,13,15,18, 22, 25, 30, 31, 32, 34, 40 \\ & \uparrow \end{array}\end{align*}$

The middle number, 25, is the median.
::中间数字是25,中间数字是中位数。

Next, find the mode. The mode is the number that occurs most often.
::下一步, 找到模式。模式是最常出现的数字。

In this case, there is no number that occurs more often.
::在这种情况下,没有更经常出现的数字。

There is no mode.
::没有模式。

Then, find the range. The range is the largest number minus the smallest number in the data set.
::然后,找到范围。这个范围是数据集中最大数量减去最小数量。

$\begin{aligned} \begin{array}{rcl} range & = & 40 - 12 \\ = & 28 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{range} &=& 40 - 12 \\ &=& 28 \\ \end{array}\end{align*}$
::范围=40-12=28

The answer is that the mean is 24.73, the median is 25, there is no mode, and the range is 28.
::答案是,平均值是24.73,中位数是25,没有模式,范围是28。

Given the following data set, answer the questions found in the examples below.
::鉴于以下数据集,回答以下示例中的问题。

7, 9, 3, 5, 11, 1, 8, 6, 1, 5

Example 3
::例3

Find the mean of the data set.
::查找数据集的平均值。

To find the mean you must add all of the data items and divide by the number of items.
::要找到平均值, 您必须添加所有的数据项, 并除以项目数量。

$\begin{aligned} \begin{array}{rcl} mean & = & \frac{7 + 9 + 3 + 5 + 11 + 1 + 8 + 6 + 1 + 5}{10} \\ = & \frac{56}{10} \\ = & 5.6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{mean} &=& \frac{7+9+3+5+11+1+8+6+1+5}{10} \\ &=& \frac{56}{10} \\ &=& 5.6 \end{array}\end{align*}$
::平均值=7+9+3+5+5+11+1+8+6+1+510=5610=5.6

The answer is 5.6.
::答案是5.6。

Therefore the mean for this data set is 5.6.
::因此,这一数据集的平均值是5.6。

Example 4
::例4

Find the median of the data set.
::查找数据集的中位数。

To find the median you must order the data set from lowest to highest. When you reorder the data from least to greatest you get:
::要找到中位数, 您必须命令数据组从最低到最高。当您将数据从最小重新排序到最大时, 您可以得到 :

$\begin{aligned} \begin{array}{rcl} 1, 1, 3, 5, 5, 6, 7, 8, 9, 11 \\ ↑ \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} &1,1, 3, 5, 5, 6, 7, 8, 9,11 \\ &\uparrow \end{array}\end{align*}$

The middle number is between 5 and 6. Therefore you need to take the average of these two numbers.
::中间数字在5到6之间,所以你需要使用这两个数字的平均数。

$\begin{aligned} \frac{5 + 6}{2} = 5.5 \end{aligned}$
$\begin{align*}\frac{5+6}{2} = 5.5\end{align*}$

The answer is 5.5.
::答案是5.5。

Therefore the median is 5.5.
::因此中位数是5.5。

Example 5
::例5

Find the mode of the data set.
::查找数据集的模式。

To find the mode you need to find the number that occurs most often.
::要找到最常发生的数字, 您需要找到模式。

In this case, both 1 and 5 occur twice.
::在这种情况下,1和5发生两次。

Therefore, the data set is bimodal (two modes) with modes of 1 and 5.
::因此,数据集是双模式(两种模式),模式为1和5。

The answer is 1 and 5.
::答案是1和5

Review
::回顾

Find the mean, median, mode and range. Round all answers to the nearest tenths place. Notice that each problem has four answers.
::查找平均值、中位数、模式和范围。将所有答案回合到最近的十分位。注意每个问题都有四个答案。

13, 18, 24, 21, 16, 24, 14, 17, 24

1. Mean
::1. 平均值

2. Median
::2. 中中

3. Mode
::3. 模式

4. Range
::4. 范围

116, 137, 120, 75, 98, 98, 137, 139, 139

5. Mean
::5. 平均值

6. Median
::6. 中中

7. Mode
::7. 模式

8. Range
::8. 范围

22, 24, 25, 30, 32, 34, 37, 22, 22, 38, 40

9. Mean
::9. 平均值

10. Median
::10. 中

11. Mode
::11. 模式

12. Range
::12. 范围

123, 150, 163, 150, 163, 150, 180, 200, 201

13. Mean
::13. 中

14. Median
::14. 中

15. Mode
::15. 模式

16. Range
::16. 范围

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。

Resources
::资源

Section outline

General

平均值、中值、模式和范围

Statistical Measures ::统计措施统计统计措施

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Given the following data set, answer the questions found in the examples below. ::鉴于以下数据集,回答以下示例中的问题。

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Statistical Measures
::统计措施统计统计措施

Examples
::实例

Example 1
::例1

Example 2
::例2

Given the following data set, answer the questions found in the examples below.
::鉴于以下数据集,回答以下示例中的问题。

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源