算术平均值的捷径

用外行的话来说，统计是收集、分类、检查、解释并最终以可理解的方式呈现信息的过程，以便人们可以形成意见并在必要时采取行动。例子：

• 老师收集学生的成绩，按升序或降序排列，计算平均班级成绩，或确定有多少学生不及格并告诉他们，以便他们可以开始更加勤奋地学习。
• 政府官员正在收集人口普查数据并将其与以前的记录进行比较，以查看人口增长是否得到控制。
• 分析一个国家信奉特定宗教的人数。
• 分析一个国家信奉特定宗教的人数。

算术平均值

算术平均值，通常称为平均值，是通过将数据中的数字相加并将总和除以观测值的数量来确定给定数据集合的。它是最广泛使用的集中趋势方法。计算算术平均值的直接方法就是所谓的。

什么是统计学中的捷径？

解决方案：

Short-Cut Method

The short-cut approach is used whenever the data values are huge and the calculation is time-consuming. When using the short- cut method to get the arithmetic mean, the stages are as follows:

• Select one observation from the data set and use it as the series’ assumed mean. Because it is impossible to choose one observation from the class intervals while working with grouped data, one must first compute the class marks of the intervals’ mid-points and designate one as the presumed mean
• Next, determine deviations from the expected mean (A) by subtracting the assumed mean from all other data. d = X − A.
• Take the total of the numbers produced by multiplying the deviations obtained with the frequencies.
• Apply the formula $x̄=a+\frac{Σdf}{Σf}$, where Σdf is the sum of all the deviations multiplied by respective frequencies.
• The arithmetic mean of the given data set is the number produced in this way.

Thus the formula for the calculation of arithmetic mean by short- cut method is:

$x̄=a+\frac{Σdf}{Σf}$

示例问题

问题1：使用捷径法计算以下数据集的算术平均值：

Marks	Number of students
0 – 10	5
10 – 20	12
20 – 30	14
30 – 40	10
40 – 50	5

解决方案：

$x̄=a+\frac{Σdf}{Σf}$

= 25 + 40/49

= 25 + 0.81

x̄ = 25.81

问题2：使用捷径法计算以下数据集的算术平均值：

Marks	f	m	d = m – a	fd
0 – 10	5	5	5 – 25 = −20	−100
10 – 20	12	15	15 – 25 = −10	−120
20 – 30	14	A = 25	25 – 25 = 0	0
30 – 40	10	35	35 – 25 = 10	100
40 – 50	8	45	45 – 25 = 20	160
	Σf = 49			Σdf = 40

解决方案：

Mean = X̄ = $a+\frac{Σdf}{Σf}$

= 45 + 280/40

= 45 + 7

x̄ = 52

问题3：使用捷径法计算以下数据集的算术平均值：

Marks	Number of Students
10 – 20	5
20 – 30	3
30 – 40	4
40 – 50	7
50 – 60	2
60 – 70	6
70 – 80	13

解决方案：

Mean = X̄ = $a+\frac{Σdf}{Σf}$

= 25 + (-90)/160

x̄ = 24.44

问题4：使用捷径法计算以下数据集的算术平均值：

Marks	f	m	d = m – a	fd
10 – 20	5	15	−30	−150
20 – 30	3	25	−20	−60
30 – 40	4	35	−10	−40
40 – 50	7	A = 45	0	0
50 – 60	2	55	10	20
60 – 70	6	65	20	120
70 – 80	13	75	30	390
	Σf = 40			Σdf = 280

解决方案：

Wages	f	m	d = m – A	fd
3-6	10	4.5	-6	-60
6-9	20	7.5	-3	-60
9-12	30	A =10.5	0	0
12-15	40	13.5	3	120
15-18	50	16.5	6	300
	Σf = 150			Σdf = 300

Mean = X̄ = $a+\frac{Σdf}{Σf}$

= 10.5 + (3000)/150

x̄ = 12.5

问题 5：使用捷径法计算以下数据集的算术平均值：75、68、80、56、92。

解决方案：

x	d = x – A
75	7
A = 68	0
80	12
56	-12
92	24
	Σd = 31

Since the given series is individual and not discrete, the formula for mean using short- cut method would be as follows: 

Mean = X̄ = $a+\frac{Σd}{n}$, where n is the number of observations.

= 68 + 31/5

x̄ = 74.2

问题 6：使用捷径法计算以下数据集的算术平均值。假设 a = 8。

Wages	Number of Workers
0 – 10	22
10 -20	38
20 – 30	46
30 – 40	35
40 – 50	19

解决方案：

d	f	fd
-2	4	-8
-1	8	-8
0	13	0
1	20	20
2	11	24
	Σf = 56	Σdf = 28

Mean = X̄ = $a+\frac{Σdf}{Σf}$

= 8 + (28)/56

x̄ = 8.5

问题 7：用捷径法计算下列数据的算术平均值：

Wages	f	m	d = m – a	fd
0 – 10	22	5	-20	−440
10 -20	38	15	-10	−380
20 – 30	46	a = 25	0	0
30 – 40	35	35	10	350
40 – 50	19	45	20	380
	Σf = 160			Σdf = -90

解决方案：

x	f	m	d = m – A	fd
40-45	6	42.5	-10	-60
45-50	18	47.5	-5	-90
50-55	12	A = 52.5	0	0
55-60	3	57.5	5	15
60-65	1	62.5	10	10
		Σf = 40		Σfd = -125

Mean = X̄ = $a+\frac{Σdf}{Σf}$

= 52.5 + (-125)/40

x̄ = 49.37