Okay, here is the translation following your instructions: Experimental Design and Statistics

Author: Fang Ping et al.
Publisher:
Publish Date: 2005-01-01
Features: Chapter Statistics Basics
Experimental design and statistics apply the principles of mathematical statistics and error theory to analyze and interpret quantitative relationships in environmental and resource sciences, helping researchers design experiments correctly and scientifically analyze experimental results, thereby revealing the truth of environmental and resource issues. It falls within the scope of statistics. This chapter focuses on introducing the basic concepts of population, sample, and error, the statistical characteristics of population and sample, and the significance and properties of probability distribution and sampling distribution, laying a foundation for further learning experimental design and various statistical analysis methods.
I. Population and Sample
1. Common Statistical Terms
Population refers to the entire set of specific objects under study in an experimental research. It is a collection of many objectively existing units with certain common properties. Each unit constituting the population is called an individual. The value obtained by examining a certain trait of an individual (e.g., weighing, measuring, counting, or analyzing) is called an observation value. The number of individuals in the population (N) is called the population size. The population has the following three characteristics:
(1) Homogeneity. Individuals in the same population must have the same property in one aspect to be grouped together as a population with the same property. Homogeneity is not absolute but relative and varies with the research purpose. For example, the central sugar content of watermelon is an important quality indicator of watermelon. In the study of screening high-quality watermelon varieties, the central sugar content of each watermelon variety is an individual of the population. When studying the effect of nitrogen application on the quality of the watermelon variety Zhenmi No., the central sugar content of Zhenmi No. watermelon at each nitrogen level is a population unit.
(2) Variability. Under the premise of homogeneity, different units within the population generally exhibit differences, known as variability. This homogeneity and variability are determined by the objectivity of things. In other words, all objective things are a unity of opposites between homogeneity and variability. A statistical population would not exist without homogeneity, and it would be unnecessary without variability. The statistical study of a population is essentially the study of the variability among individuals that make up these populations. For example, to study the pollution status of cadmium in soil in a certain region, it is necessary to study the variation in soil cadmium content across different plots in the region. On the basis of homogeneity, studying the degree of variation, central tendency, and patterns of the population is one of the important tasks of statistical analysis.
(3) Large Quantity. The purpose of statistical research is to reveal the objective laws of things, and such laws can only be reflected in the general connections among a large number of things. Therefore, a population is composed of a large number of individuals. If the population size is infinite, it is called an infinite population. For example, continuous bodies such as the atmosphere, water bodies, and soil bodies are infinitely divisible, so the number of population units is infinite. If the population size is finite, it is called a finite population. For example, the national population at a certain moment is a very large finite population. The study of a population can be conducted through a comprehensive census, such as a national population census or a soil census. However, a comprehensive census is often costly and difficult to implement, especially for infinite or large finite populations. Additionally, some observation methods are destructive, so even if the population size is not very large, it is not feasible to examine all individuals. Therefore, in most cases, sampling surveys are used.
2. Sample
A set of individuals drawn from the population is called a sample. Inferring the properties of the population from sample characteristics is the basic method of statistical analysis. For this purpose, the sample must be representative of the population, which requires sampling to meet the requirements of random sampling:
(1) Equal probability, meaning each individual has an equal chance of being selected in each sampling.
(2) Independence, meaning each sampling does not affect the chance of selection for subsequent samplings.
The number of individuals in a sample is called the sample size, usually denoted as \( n \). A sample with \( n \geq 30 \) is called a large sample; a sample with \( n < 30 \) is called a small sample. Sometimes, large and small samples have different statistical distribution characteristics.
II. Variables and Data
Due to the influence of many random factors, there is generally variability among individuals within the population, and the observed values will also exhibit fluctuations. For example, if a field is divided into equal-area plots and the same variety of rice is planted with the same field management, the yield of each plot will vary due to the unevenness of soil fertility and other random factors at harvest. Similarly, when repeating 10 measurements of cadmium content in a soil sample, the results will exhibit certain fluctuations due to the influence of factors such as testing instruments, testing conditions, and the operator's skills. Such quantities that are influenced by many random factors and exhibit fluctuations are called random variables or random numbers, often abbreviated as variables, and are commonly represented by uppercase letters such as \( x \), \( y \), \( z \), etc. For example, crop yield, biological oxygen demand in water bodies, and mercury content in soil all belong to random variables. A set of specific observed values of a random variable is called data. For example, measuring the total nitrogen content of 10 soil samples will obtain the data on the total nitrogen content of these 10 soil samples. Due to differences in testing or survey methods, tools, techniques, and subjects, the nature of the obtained data varies greatly. Some are quantitative data, while others are descriptive data. Therefore, these data can be divided into two major categories based on their nature or characteristics.
1. Quantitative Attributes
Quantitative attributes refer to the properties of the objects of testing or survey that can be measured or counted. For example, the area of environmental pollution, the concentration of pollutants, the concentration of nitrate in groundwater, and the number of bacteria in water bodies. Based on the nature of the differences between data of individuals, quantitative attribute data can be further divided into two types: continuous variation and discontinuous variation.
Continuous variation data refers to data where the differences between individuals are very small. When the population is large enough, with the improvement of measurement accuracy, these differences can reach any arbitrary precision that is humanly measurable. Generally, this type of data is obtained through weighing, measuring, measuring, or analyzing. The precision of the values depends on the precision of the measuring tools. Discontinuous variation data refers to data obtained through counting methods, where the values are limited to non-negative integers. For example, the number of living organisms in the environment is always counted in natural numbers. Although the population is infinite, the difference between two groups of organisms will not be less than 1.
2. Qualitative Attributes
In environmental and resource research, some attributes of observed or surveyed objects cannot be measured but can be observed, such as the different colors and odors of pollutants. These are qualitative attributes. Qualitative attributes can be quantified using the assignment method, where different numerical values are assigned to different categories of a property. For example, different colors of polluted water bodies can be assigned different numerical values, such as red as -1, yellow as 0, and green as 1, and so on. After assigning numerical values to qualitative attributes, statistical methods can be further used to analyze the studied environmental or resource issues.
III. Parameters and Statistics
Under the premise of homogeneity, a population has the characteristics of variability and large quantity. Therefore, in statistics, some parameters are used to reflect the characteristics of the variability or central tendency of individuals within the population, such as the population mean, population variance, and population standard deviation, which are collectively referred to as population parameters, or simply parameters. Correspondingly, some indicators calculated from sample data, which are used to describe the variability or central tendency of individuals within the sample, such as the sample mean and sample standard deviation, are called sample statistics, or simply statistics.