统计学专业英语翻译讲课讲稿
统计学专业英语翻译
汉译英Population 总体，样本总体sample 样本，标本parameter 限制因素median 中位数odd 奇数，单数even 偶数range 极差variance 方差standard deviation 标准差Covariance 协方差empty event 空事件product event 积事件conditional probability 条件概率Random variable 随机变量binominal distribution 二项式分布uniform distribution 均匀分布Poisson distribution 泊松分布residual 残差central limit theorem 中心极限定律英译汉descriptive statistics 描述统计学mathematical statistics 数理统计学inductive statistics 归纳统计学Inferential statistics 推断统计学dimension 维，维数continuous variable 连续变量ordinal variable 有序变量nominal variable 名义变量dichotomous 两分的；二歧的discrete variable 离散变量categorical variable 分类变量location 定位，位置，场所dispersion 分散mean 均值unimodal单峰的multimodal 多峰的chaotic 无秩序的grouped data 分组数据frequency distribution频数分布cumulative frequency 累加频数tallying 计算Uniformly distribution 均匀分布histogram 直方图frequency polygon 频率多边图rectangle 矩形Percentile 百分位数quartile 四分位数interquartile range 四分位数间距simple event 简单事件Compound event 复合事件mutually exclusive 互斥的，互补相交的complementary event 对立事件Independent 独立的joint probability function 联合概率函数jacobian雅克比行列式Law of large numbers大数定律point estimate 点估计estimate 估计值statistic 统计量optimality 最优性Unbiased estimate 无偏估计量efficient estimate 有偏估计量unbiasedness无偏性efficience有效性Consistent estimate 一致估计量asymptotic properties 渐近性质Confidence interval 置信区间interval estimation 区间估计null hypothesis 原假设alternative hypothesis 备择假设significance level 显著性水平power function 幂函数testing procedures 检验方法test statistic 检验统计量rejection region 拒绝区域acceptance region 接受区域critical region 临界区域first-derivatives 一阶导数second-derivatives 二阶导数Likelihood ratio 似然比dependent variable因变量unexplanatory variable未解释变量independent variable自变量Error term 误差项regression coefficients 回归系数Sum of squared residuals 残差平方和Marginal probability function 边际概率函数joint probability density function 联合概率密度函数Marginal probability density function边际概率密度函数stochastically independent 随机独立的Mutually independently distribution 相互独立的分布independently and identically distribution 独立同分布的likelihood function 似然函数maximum likelihood estimator 最大似然估计量maximum likelihood estimate 最大似然估计值log-likelihood function 对数似然函数ordinary least squares estimation/estimate/estimator 普通最小二乘估计/估计值/估计量linear unbiased estimator 线性无偏估计第三章、概念与符号[An index]把指数定义成是对一组相关变量之中变化进行测算的一个实数。
关于统计的英语演讲
关于统计的英语演讲English:Statistics play a crucial role in understanding, interpreting, and making decisions based on data in various fields. Whether in economics, sociology, medicine, or any other discipline, statistics provide a framework for analyzing information and drawing meaningful conclusions. Statistical methods allow us to summarize and describe complex datasets, identify patterns and trends, and evaluate the likelihood of outcomes. From basic descriptive statistics like mean, median, and mode to advanced inferential techniques such as regression analysis and hypothesis testing, statisticians employ a range of tools to extract insights from data. Moreover, statistics help in decision-making by quantifying uncertainty and risk, guiding policy formulation, and assessing the effectiveness of interventions. In today's data-driven world, where information overload is common, statistical literacy is increasingly important for individuals to navigate through the plethora of data and discern credible insights from noise. Therefore, incorporating statistical education into various curricula and promoting data literacy initiatives are essential steps towards fostering a society equipped toharness the power of statistics for informed decision-making and societal advancement.中文翻译：统计在理解、解释和基于各个领域的数据做出决策中发挥着至关重要的作用。
基础统计学培训讲座(英文版)(ppt 31页)
1-18
Types of Variables
A continuous variable can assume any value within a specified range.
Examples are: The pressure in a tire, the weight of a pork chop, or the height of students in a class. Typically, continuous variables are the result of measuring something.
1-11
Types of Statistics
Inferential Statistics: The methods used to determine something about a population, based on a sample.
EXAMPLE 3: Wine tasters sip a few drops of wine to make a decision with respect to all the wine waiting to be released for sale.
统计学专业英语翻译
汉译英Population 总体，样本总体 sample 样本，标本 parameter 限制因素median 中位数 odd 奇数，单数 even 偶数range 极差 variance 方差 standard deviation 标准差Covariance 协方差 empty event 空事件 product event 积事件conditional probability 条件概率 Random variable 随机变量 binominal distribution 二项式分布uniform distribution 均匀分布 Poisson distribution 泊松分布 residual 残差central limit theorem 中心极限定律英译汉descriptive statistics 描述统计学 mathematical statistics 数理统计学 inductive statistics 归纳统计学Inferential statistics 推断统计学 dimension 维，维数 continuous variable 连续变量ordinal variable 有序变量 nominal variable 名义变量 dichotomous 两分的；二歧的discrete variable 离散变量 categorical variable 分类变量 location 定位，位置，场所dispersion 分散 mean 均值 unimodal 单峰的multimodal 多峰的 chaotic 无秩序的 grouped data 分组数据frequency distribution频数分布 cumulative frequency 累加频数 tallying 计算Uniformly distribution 均匀分布 histogram 直方图 frequency polygon 频率多边图rectangle 矩形 Percentile 百分位数 quartile 四分位数interquartile range 四分位数间距 simple event 简单事件Compound event 复合事件 mutually exclusive 互斥的，互补相交的 complementary event 对立事件Independent 独立的 joint probability function 联合概率函数 jacobian 雅克比行列式Law of large numbers大数定律 point estimate 点估计 estimate 估计值statistic 统计量 optimality 最优性 Unbiased estimate 无偏估计量 efficient estimate 有偏估计量unbiasedness 无偏性 efficience 有效性 Consistent estimate 一致估计量asymptotic properties 渐近性质 Confidence interval 置信区间 interval estimation 区间估计null hypothesis 原假设 alternative hypothesis 备择假设 significance level 显著性水平power function 幂函数 testing procedures 检验方法 test statistic 检验统计量rejection region 拒绝区域 acceptance region 接受区域 critical region 临界区域first-derivatives 一阶导数 second-derivatives 二阶导数 Likelihood ratio 似然比dependent variable因变量 unexplanatory variable未解释变量 independent variable自变量Error term 误差项 regression coefficients 回归系数 Sum of squared residuals 残差平方和Marginal probability function 边际概率函数 joint probability density function 联合概率密度函数Marginal probability density function边际概率密度函数 stochastically independent 随机独立的Mutually independently distribution 相互独立的分布 independently and identically distribution 独立同分布的likelihood function 似然函数 maximum likelihood estimator 最大似然估计量maximum likelihood estimate 最大似然估计值 log-likelihood function 对数似然函数ordinary least squares estimation/estimate/estimator 普通最小二乘估计/估计值/估计量linear unbiased estimator 线性无偏估计第三章、概念与符号[An index]把指数定义成是对一组相关变量之中变化进行测算的一个实数。
统计学(中英文)_ch01
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 1-12
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Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 1-8
Inferential Statistics 推断统计
Estimation 估计 e.g., Estimate the population mean weight using the sample mean weight 例如：利用采样的平均重量估计人口的平均体 重 Hypothesis testing 假设检验 e.g., Test the claim that the population mean weight is 120 pounds 例如：根据测试的要求，人口平均体重是120 磅
英文翻译乃自己所做， 英文翻译乃自己所做，有错误 之处请自行查证。 之处请自行查证。
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 1-1
Business Statistics, A First Course
Defined descriptive vs. inferential statistics 描述性统计和推理统计 Reviewed data types 回顾数据类型
♦ ♦ ♦ ♦
Categorical vs. Numerical data 绝对的和数值的数据 Discrete vs. Continuous data 离散的和连续的数据
统计学英文版教材课件
Combining Events
There are some important ways in which events can be combined that we will encounter repeatedly throughout this course. Suppose we have two events, A and B .
For example, A ∪ B = {1, 3, 4, 5}.
S A 1 5 2
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Introduction
Intersection, Union and Complement
Complement
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Introduction
Definitions
Probabilities of Outcomes
The probability of an outcome occurring on a single trial is written as P (Oi ). Probabilities associated with the outcomes in a sample space must satisfy two important requirements:
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Introduction
Events
Events
A simple event is an individual outcome from the sample space. An event is a collection of one or more simple events (or outcomes).
关于统计学的英文介绍
关于统计学的英文介绍【中英文版】Introduction to StatisticsStatistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It plays a crucial role in various fields, including economics, biology, psychology, and many more. By utilizing statistical methods, we can draw meaningful conclusions and make informed decisions based on the information extracted from the data.统计学是一门研究数据的收集、分析、解释、呈现和组织方法的数学分支。
它在经济学、生物学、心理学等多个领域发挥着至关重要的作用。
通过运用统计方法，我们可以从数据中提取有意义的信息，并据此做出明智的决策。
The beauty of statistics lies in its ability to simplify complex phenomena into quantifiable measures, enabling us to understand patterns, trends, and relationships within the data. Fundamental concepts such as mean, median, and mode help us summarize and describe data, while techniques like hypothesis testing and regression analysis allow us to make predictions and draw inferences.统计学的魅力在于它能将复杂的现象简化为可量化的指标，使我们能够理解数据中的模式、趋势和关系。
基础统计学(英文版)(ppt 31页)
1-3
Why study statistics?
• Numerical info is everywhere
– But how do we know if conclusions reported are accurate?
• Statistical techniques are used to make decisions that affect our lives
1-18
Types of Variables
A continuous variable can assume any value within a specified range.
Examples are: The pressure in a tire, the weight of a pork chop, or the height of students in a class. Typically, continuous variables are the result of measuring something.
1-15
Types of Variables
For a Qualitative or Attribute variable the characteristic being studied is nonnumeric.
统计学课件英语
1
Financial Accounting and Accounting Standards
LEARNING OBJECTIVES
After studying this chapter, you should be able to: 1. Identify the major financial statements and other means of financial reporting. 2. Explain how accounting assists in the efficient use of scarce resources. 3. Identify the objective of financial reporting. 4. Explain the need for accounting standards. 5. Identify the major policy-setting bodies and their role in the standard-setting process.
in making decisions in their capacity as capital providers.
Illustration 1-1 Capital Allocation Process
It promotes productivity, encourages innovation, and provides an efficient and liquid market for buying and selling securities and obtaining and granting credit.
1-6
LO 2 Explain how accounting assists in the efficient use of scare resources.
统计学英文教学课件Chapter 1(商科)What is Statistics
Statistics Terminology
Population The collection of all possible individuals, objects, or measurements of interest
Sample A portion, or part, of the population of interest
• How to do it:
– Use random number table or random number generator, such as Excel • Assign numbers to population • Select n random numbers • Sample population elements that correspond to the random numbers
Methods of organizing, summarizing, and presenting data in an informative way.
Graphical & Tabular [Ch. 2] Numerical [Ch. 3]
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Descriptive Statistics – Graphical
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Levels of Measurement
Ordinal Data are rank-ordered, order is meaningful, differences between rankings not meaningful
Examples: Sports rankings, Earthquake magnitude [Richter scale]
SAMPLE
统计学外文翻译学习资料
外文翻译原文名称：Fundamentals_of_StatisticsMeasures of Central Tendency and Location: mean, median, mode, percentiles, quartiles and deciles.x sorted x53 5355 5370 5358 5564 5757 5753 5869 6457 6868 6953 70The Measures of Central Tendency are Mean, Median and ModeMean →x-bar or x→ for a given variable, it is the sum of the values divided by the number of values (∑x i/n). In this case, we have n = 11. So we need to add all of the values together and divide by 11. ∑ = 657, x= 59.73Median→ the number in a distribution of a variable’s response where one half of the values are above and one half of the values are below. To find the median, we first need to put our data in ascending order (smallest to largest). Then we can determine the median…if the valu e of n is odd, it issimply the middle observation, but if the value of n is even, it is the average of the two middle observations.In this case, n is odd, so the median will be the middle observation of our sorted values (the 6th value) (57)Mode→the value that occurs most frequently. If there are two different values most frequently occurring, the data are said to be bi-modal. If there are more than two modes, and the distribution is said to be multi-modal. In this case, the value that occurs most often is 53. So, the mode is 53.The measures of location are Percentile, Quartile and DecilePercentile → the p th percentile is a value such that at least p percent of the observations are less than or equal to this value and at least (100 – p) percent of the observations are greater than orequal to this value. To calculate percentiles, we use indices (i).i = (p/100) n for p1, p2, p3,…p99If the answer is a whole number (an integer), then i is the average of (P/100)n and 1+ (P/100)n.If the index number is not a whole number, we ALWAYS round up. The positionof the index is the next whole number (integer) greater than the computed index.For example:i(p50)= (50/100)11 = 5.5...this rounds up to 6So, we would count from the lowest value of the sorted data to the index number (6).Since the calculated i was not a whole number we had to round up to find the valuewhere at least 50% of the values are equal to or lower than this value and at least 50%are equal to or higher than this value. In this case, the value of the 50th percentileis the 6th value...57 … Does this look familiar? → The 50th percentile is the samething as the median.What does it tell us? In this distribution, AT LEAST 50% of the observations areLESS THAN OR EQUAL TO 57 AND AT LEAST 50% of the observations areGREATER THAN OR EQUAL TO 57.i(p80)= (80/100)11 = 8.8...this round up to 9. The 9th value is 68.Again, since the index number is not a whole number, we round up. So, we would count from the lowest value of the sorted data to the index number (9). In this case, the value of the80th percentile is 68.Since this dataset has 11 observations, we won’t have any instances where our calculated index number is a whole number. However, if we just remove our value of 70 and create anew distribution, we will be able to see an example...53 53 53 55 57 57 58 64 68 69i(p30)= (30/100)10 = 3...this is a whole number, so we must take the 3rd and 4th values and average them to find the 30th percentile. (53 + 55)/2 = 54So, the value of the 30th percentile is 54.Return to our original data distribution ...Quartiles –are special cases of percentiles…Q1 = P25, Q2 = P50, Q3 = P75,These three values divide the distribution into 4 equal quartersi(Q1)= (25/100)11 = 2.75...this rounds to 3, so Q1 is the 3rd value (53)i(Q2)= (50/100)11 = 5.5...this round to 6, so Q2 is the 6th value (57)i (Q3) = (75/100)11 = 8.25...this rounds to 9, so Q3 is the 9th value (64)Measures of Dispersion or V ariability : Range, interquartile range (IQR), variance, standard deviation and coefficient of variation.Range = This tells us how wide the span is from the maximum value to the minimum value. (Max –Min) = Range. In this instance, the range is 69 - 53 = 16.Interquartile Range (IQR) = This tells us how wide the span is in the middle 50% of the data. (Q3 – Q1) = IQR. In this case ... 64 – 53 = 11We use the formula:1)(2--∑n x x = s 2The variance for these data is 454.18. For our purposes here, the computation of variance is just a step towards the computation of the standard deviation.Sample standard deviation (s ) is the positive square root of the variance.74.642.45=+= sSo the formula for sample standard deviation is…1)(2--∑n x xPopulation Variance (σ2)→uses the same formula in the numerator, but N instead of n-1 in thedenominator. Since we rarely have information about the entire population, we almostalways use the formula for sample variance, s 2.Population Standard Deviation: σ = 2σ…since we rarely have information from the entire population, we use the formula for sample standard deviation, s .Coefficient of Variation: 100⎪⎭⎫⎝⎛x s tells us what percent the sample standard deviation is of the sample meanThis number is “relative” and is only of use in comparing the distribution of two or more variables.Suppose I have two samples, and I want to know which sample has more variability…If both samples have the same mean, the one with the higher standard deviation will have the greater variability. However, if they have different means, I need to calculate the coefficient of variation to determine which one has the most variability. xbar = 458, s = 112 versus xbar = 687, s = 192Standardized Data and Detecting OutliersZ -score: z =sxx -The z-score tells us how many standard deviations a value is from the mean. We can look at a picture of what a z-score tells us. In the Normal Curve…the mean is at the highest point and the curve tails off symmetrically in both directions.The sign of the z-score tells us which direction the value is from the mean on the Normal Curve. Negative values will be to the left, and positive values will be to the right.Standardizing Scores:Standard Normal Curve …the mean is zero, and the standard deviation is 1. The distribution is bell-shaped and symmetrical. The area under the curve is 1, and the tails of the curve extend out infinitely. They never actually touch the horizontal axis. The highest point on the curve is at the meanReturn to our data …let’s calculate the z -scores for each of the values…Empirical Rule →used when the distribution is assumed to known to be approximately normal. → Approximately 68% of the values will fall within 1 sd of the mean → Approximately 95% of the values will fall within 2 sd of the mean→ Approximately 99.9% of the values will fall within 3 sd of the meanChebyshev’s Theorem → doesn’t require that the data have a normal distributionSays that at least (1 – 1/z2) values will fall within z standard deviations of the mean.1-1/12 = 0, 1-1/22 = .75, 1-1/32 = .88889, 1-1/42 = .9375, 1-1/52 = .96 →We can’t make any assumptions about the percent of values that are within 1 sd of the meanBut…→ At least 75% of the values will fall within 2 sd of the mean→ At least 88.9% of the values will fall within 3 sd of the meanWe use Chebyshev’s Theorem to estimate the variation in a distribution when→n < 30, or→ the shape of the distribution is unknown, or→the distribution is assumed to be non-normal.Outliers:suspect or extreme values of data that must be identified and scrutinized. If they are instances of incorrectly entered data, they should be corrected. If the value wasentered correctly and it is a valid number, it should remain in the dataset as part of the initial analysis.When we use the z-score method for identifying outliers, we assume that any value that has a z-score with an absolute value greater than 3.0 (that is less than -3.0 or greater than +3.0) is an outlier. Before we proceed with data analysis, we need to examine all outliers for accuracy. If we determine that the value is valid, we often run two sets of analysis. One with the outlier, and one without.Another way to identify outliers…Related to IQR is the Five number summary…minimum, Q1, Q2, Q3, & maximum. These values feed into upper and lower limits, and we graph them in a box plot.Use the box plot… The advantage of the boxplot is that it is not influenced by outliers orextreme values as are Z-scores.Box Plots –Whiskers show the range of data within the inner fences3(IQR) 133(IQR) below Q1below Q1(IQR) above Q3above Q3 (Lower Outer & Inner Fences) (Upper Inner & OuterFences)Any values between the inner and outer fences are “unusual,” and any values out beyond the outer fences are “outliers.”Advantage of using the box plot method as well as the z-score method...the box plot method is not influenced by extreme values in the same way that the mean and the standard deviation are....it is saidto be a more conservative method of evaluating outliers.外文翻译原文课题名称：统计基础趋势和位置的划分： 意思是说，中位数，众数，百分位数，四分位数和十分位数。
应用统计学 第一课 英文
2
Keep in Touch
If you have a question, you are STRONGLY encouraged to ASK IT IN CLASS. You are probably not the only one that needs the question answered, and other students may benefit from your questions, too. Check our course website regularly. Announcements, assignments, and solutions etc will be posted online regularly. Drop by during office hours or book another time to see me in my office.
x P(x)
0.00 0.05 0.12 0.30 0.52
E ( X ) xi P( xi )
i 1
8
= 4.71 rooms
5
6 7 Total
0.20
0.15 0.26 1.00
1.00
0.90 1.82 = 4.71
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Example: Bed and Breakfast
Business Data Analysis
73-102 Lecture 01
1
Agenda
Go through Course Outline (on CLEW) Brief Introduction to Statistics Review of Random Variable Review of Normal Distribution
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汉译英Population 总体，样本总体sample 样本，标本parameter 限制因素median 中位数odd 奇数，单数even 偶数range 极差variance 方差standard deviation 标准差Covariance 协方差empty event 空事件product event 积事件conditional probability 条件概率Random variable 随机变量binominal distribution 二项式分布uniform distribution 均匀分布Poisson distribution 泊松分布residual 残差central limit theorem 中心极限定律英译汉descriptive statistics 描述统计学mathematical statistics 数理统计学inductive statistics 归纳统计学Inferential statistics 推断统计学dimension 维，维数continuous variable 连续变量ordinal variable 有序变量nominal variable 名义变量dichotomous 两分的；二歧的discrete variable 离散变量categorical variable 分类变量location 定位，位置，场所dispersion 分散mean 均值unimodal 单峰的multimodal 多峰的chaotic 无秩序的grouped data 分组数据frequency distribution频数分布cumulative frequency 累加频数tallying 计算Uniformly distribution 均匀分布histogram 直方图frequency polygon 频率多边图rectangle 矩形Percentile 百分位数quartile 四分位数interquartile range 四分位数间距simple event 简单事件Compound event 复合事件mutually exclusive 互斥的，互补相交的complementary event 对立事件Independent 独立的joint probability function 联合概率函数jacobian 雅克比行列式Law of large numbers大数定律point estimate 点估计estimate 估计值statistic 统计量optimality 最优性Unbiased estimate 无偏估计量efficient estimate 有偏估计量unbiasedness 无偏性efficience 有效性Consistent estimate 一致估计量asymptotic properties 渐近性质Confidence interval 置信区间interval estimation 区间估计null hypothesis 原假设alternative hypothesis 备择假设significance level 显著性水平power function 幂函数testing procedures 检验方法test statistic 检验统计量rejection region 拒绝区域acceptance region 接受区域critical region 临界区域first-derivatives 一阶导数second-derivatives 二阶导数Likelihood ratio 似然比dependent variable因变量unexplanatory variable未解释变量independent variable自变量Error term 误差项regression coefficients 回归系数Sum of squared residuals 残差平方和Marginal probability function 边际概率函数joint probability density function 联合概率密度函数Marginal probability density function边际概率密度函数stochastically independent 随机独立的Mutually independently distribution 相互独立的分布independently and identically distribution 独立同分布的likelihood function 似然函数maximum likelihood estimator 最大似然估计量maximum likelihood estimate 最大似然估计值log-likelihood function 对数似然函数ordinary least squares estimation/estimate/estimator 普通最小二乘估计/估计值/估计量linear unbiased estimator 线性无偏估计第三章、概念与符号[An index]把指数定义成是对一组相关变量之中变化进行测算的一个实数。
从概念上讲，指数可以用来比较随时间或者空间或随两者同时变化的量。
指数用来测算随时间变化的价格与数量，也可以用来衡量不同厂商、行业、地区或国家的水平差异。
价格指数可以指消费者物价投入与产出、价格进出口价格等等，而数量指数可以测算一个厂商或行业随时间变化或者不同厂商在产出商品以及所用投入上的数量变化。
[Index number have]指数在经济学上有着悠久的且与众不同的历史，一些最重要的贡献归功于早在十九世纪晚期的Laspeyres 和Paasche 的研究。
Laspeyres 和Paasche 公式仍旧被全世界一些国家的统计局所广泛应用。
但是，正是Irving Fisher 的工作以及他的著作——在1992年出版的《编制指数》——认识到使用许多统计公式生成适当的指数的可能性。
rnquist o T 指数（1936）在生产率测量中起到重要作用。
Diewert 和Nakamura （1993）书中的第二章提供了极好的阐明指数构建的历史背景。
符号[We use the following]在一这章中我们自始至终地使用下述符号。
设mj p 和mj q 分别表示第m 种（m=1,2,...,M ）商品在第j （j=s,t ）个时期中的价格与数量。
为了不失一般性，s 与t 除了可以表示时期之外，还可以指两家厂商，而数量可以是投入量，也可以是产出量。
[Conceptually,all]从概念上讲，所有指数测算了来自于一个参考时期的一组变量水平的变化。
参考时期由“基期”表示，用于计算指数的时期称为“现期”。
设st I 表示以s 为基期以t 的现期t的综合指数。
类似地，设st V ，st P 和st Q 分别表示价值指数、价格指数以及数量指数。
综合指数问题[The value change from]从时期s 到时期t 的价值变化是在时期s 与t 的商品价值之比值，价值有各自的价格衡量。
因而 N i is is N i itit st q p q p V 11指数st V 测算了从时期s 到t 的M 种商品集合数量价值的变化。
显然，st V 是两种成分即价格变化与数量变化的结果。
尽管st V 容易测算时，但是要剔除价格变化与数量变化的影响就非常难。
我们想要踢除这种影响，因此，比如说，使数量成分可以用于测算数量的变化。
[if we are operating]如果我们在单一商品的世界里处理问题，那么这种分解就很容易做到我们有s t s t s s t t st q q p p q p q p V 其中比值s t p p 与s t q q 测算了相关的价格变化与数量变化，从而不存在指数问题。
[In gennerally]通常当商品数M ≥2时，我们就有了综合问题。
相对价格pmt/pms 测算了第m 种商品价格水平的变化。
相对数量qmt/qms 测算了第m 种商品数量水平的变化。
[Now the]现在的问题是，如何将这m 种不同价格（数量）变化的测算合成一个简单实数，称之为价格（数量）指数。
这个问题有点类似于选择一种合适的中心趋势来测量，在下面两节中我们简要地阐明测算价格指数与数量指数变化的一些最常用的公式。
第四章、非模型处理缺失数据[if no models]倘若没有模型可以利用，则人们直接分析可用数据，或者分析经过基于非模型估算之后的数据。
只利用可用数据[Listwise deletion]成列删除或完整个案分析意指，删除数据中有缺失值的一个或多个变量的那种观测值。
在MCAR 假设下，经过成列删除之后，所保留的样本仍是源自最初总体的一个随机样本；因此，基于该样本的估计是一致的。
不过，其标准误差将会扩大，因为所用信息甚少，若回归元个数很多，则成列删除的总效果导致总观测值会剧烈减少。
这激发人们脱离那种对拥有高比例缺失观测值的变量进行分析，可是，由该种方法所产生的结果却潜在的使人误导。
[if MCAR]如果MCAR 得不到满足且缺失数据仅仅是MAR ，那么估计将是有偏的。
因而，成列删除对违背MCAR 而言是不稳健的。
不过，成列删除对回归分析中各个自变量违背MAR 而言是稳健的。
也就是，当任何回归元出现缺失数据的概率并不依赖于因变量之值。
简约地讲，成列删除是可接受的，如果归因于缺失数据的不完全情况构成了各种各样种情况的比例很小，比如说5%或更少。
更重要的是，成列删除之后的样本是说研究总体的代表。
[Pairwise deletion]成对删除或可用案例分析，时常被认为是比成列删除更好的一种方法。
其思想是估计（x1.x2）的联合样本矩时，运用观测值（x1i,x2i ）的全部可能对，并且估计边缘矩时运用个体变量的全部观测值。
因而，在线性回归中，在成对删除下我们运用回归元的所可能对估计（X'X ）与（X'y ），而在成列删除下，要在删除任何拥有缺失观测值的全部情况后才能估计 （X'X ）与（X'y ）。
很明显，在成对删除下，我们损失较少信息。
这里建议要运用最大信息量去估计个体概括统计量，诸如均值与协方差，然后使用这些概括统计量去计算回归估计。
[There are two]成对删除有两个重要局限性：（1）一般的讲，估计标准误差与检验统计量都是有偏的；2所得到的回归元协方差矩阵（X'X ）可能不是正定的。
不用模型的估算[There are a number of]统计软件经常执行一系列专门或勉强证明合理的方法。