### 统计代写|Generalized linear model代考广义线性模型代写|Models of Central Tendency and Variability

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|Generalized linear model代考广义线性模型代写|Models of Central Tendency

Models of central tendency are statistical models that are used to describe where the middle of the histogram lies. For many distributions, the model of central tendency is at or near the tallest bar in the histogram. (We will discuss exceptions to this general rule in the next chapter.) These statistical models are valuable in helping people understand their data because they show where most scores tend to cluster together. Models of central tendency are also important because they can help us understand the characteristics of the “typical” sample member.

Mode. The most basic (and easiest to calculate) statistical model of central tendency is the mode. The mode of a variable is the most frequently occurring score in the data. For example, in the Waite et al. (2015) study that was used as an example in Chapter 3, there were four males (labeled as Group 1) and nine females (labeled as Group 2). Therefore, the mode for this variable is 2, or you could alternatively say that the mode sex of the sample is female. Modes are especially easy to find if the data are already in a frequency table because the mode will be the score that has the highest frequency value.

Calculating the mode requires at least nominal-level data. Remember from Chapter 2 that nominal data can be counted and classified (see Table 2.1). Because finding the mode merely requires counting the number of sample members who belong to each category, it makes sense that a mode can be calculated with nominal data. Additionally, any mathematical function that can be

performed with nominal data can be performed with all other levels of data, so a mode can also be calculated for ordinal, interval, and ratio data. This makes the mode the only model of central tendency that can be calculated for all levels of data, making the mode a useful and versatile statistical model.

One advantage of the mode is that it is not influenced by outliers. An outlier (also called an extreme value) is a member of a dataset that is unusual because its score is much higher or much lower than the other scores in the dataset (Cortina, 2002). Outliers can distort some statistical models, but not the mode because the mode is the most frequent score in the dataset. For an outlier to change the mode, there would have to be so many people with the same extreme score that it becomes the most common score in the dataset. If that were to happen, then these “outliers” would not be unusual.

The major disadvantage of the mode is that it cannot be used in later, more complex calculations in statistics. In other words, the mode is useful in its own right, but it cannot be used to create other statistical models that can provide additional information about the data.

## 统计代写|Generalized linear model代考广义线性模型代写|Models of Variability

Statistical models of central tendency are useful, but they only communicate information about one characteristic of a variable: the location of the histogram’s center. Models of central tendency say nothing about another important characteristic of distributions – their variability. In statistics variability refers to the degree to which scores in a dataset vary from one another. Distributions with high variability tend to be spread out, while distributions with low variability tend to be compact. In this chapter we will discuss four models of variability: the range, the interquartile range, the standard deviation, and the variance.

Range. The range for a variable is the difference between the highest and the lowest scores in the dataset. Mathematically, this is:
$$\text { Range }=\text { Highest Score – Lowest Score } \quad \text { (Formula 4.6) }$$
The advantage of the range is that it is simple to calculate. But the disadvantages should also be apparent in the formula. First, only two scores (the highest score and the lowest score) determine the range. Although this does provide insight into the span of scores within a dataset, it does not say much (if anything) about the variability seen among the typical scores in the dataset. This is especially true if either the highest score or the lowest score is an outlier (or if both are outliers).
The second disadvantage of the range as a statistical model of variability is that outliers have more influence on the range than on any other statistical model discussed in this chapter. If either the highest score or the lowest score (or both) is an outlier, then the range can be greatly inflated and the data may appear to be more variable than they really are. Nevertheless, the range is still a useful statistical model of variability, especially in studies of growth or change, where it may show effectively whether sample members become more alike or grow in their differences over time.
A statistical assumption of the range is that the data are interval-or ratio-level data. This is because the formula for the range requires subtracting one score from another – a mathematical operation that requires interval data at a minimum.

Interquartile Range. Because the mean is extremely susceptible to the influence of outliers, a similar model of variability was developed in an attempt to overcome these shortfalls: the interquartile range, which is the range of the middle $50 \%$ of scores. There are three quartiles in any dataset, and these three scores divide the dataset into four equal-sized groups. These quartiles are (as numbered from the lowest score to the highest score) the first quartile, second quartile, and third quartile. ${ }^{1}$ There are four steps to finding the interquartile range:

1. Calculate the median for a dataset. This is the second quartile.
2. Find the score that is halfway between the median and the lowest score. This is the first quartile.
3. Find the score that is halfway between the median and the highest score. This is the third quartile.
4. Subtract the score in the first quartile from the third quartile to produce the interquartile range.

## 统计代写|Generalized linear model代考广义线性模型代写|Using Models of Central Tendency and Variance Together

Models of central tendency and models of variance provide different, but complementary information. Neither type of model can tell you everything you need to know about a distribution. However, when combined, these models can provide researchers with a thorough understanding of their data. For example, Figure $4.2$ shows two samples with the same mean, but one has a standard deviation that is twice as large as the other. Using just the mean to understand the two distributions would mask the important differences between the histograms. Indeed, when reporting a model of central tendency, it is recommended that you report an accompanying model of variability (Warne et al., 2012; Zientek,Capraro, \& Capraro, 2008). It is impossible to fully understand a distribution of scores without a knowledge of the variable’s central tendency and variability, and even slight differences in these values across distributions can be important (Voracek, Mohr, \& Hagmann, 2013).

This chapter discussed two types of descriptive statistics: models of central tendency and models of variability. Models of central tendency describe the location of the middle of the distribution, and models of variability describe the degree that scores are spread out from one another.

There were four models of central tendency in this chapter. Listed in ascending order of the complexity of their calculations, these are the mode, median, mean, and trimmed mean. The mode is calculated by finding the most frequent score in a dataset. The median is the center score when the data are arranged from the smallest score to the highest score (or vice versa). The mean is calculated by adding all the scores for a particular variable together and dividing by the number of scores. To find the trimmed mean, you should eliminate the same number of scores from the top and bottom of the distribution (usually $1 \%, 5 \%$, or $10 \%$ ) and then calculate the mean of the remaining data.

There were also four principal models of variability discussed in this chapter. The first was the range, which is found by subtracting the lowest score for a variable from the highest score for that same variable. The interquartile range requires finding the median and then finding the scores that are halfway between (a) the median and the lowest score in the dataset, and (b) the median and the highest score in the dataset. Score (a) is then subtracted from score (b). The standard deviation is defined as the square root of the average of the squared deviation scores of each individual in the dataset. Finally, the variance is defined as the square of the standard deviation (as a result, the variance is the mean of the squared deviation scores). There are three formulas for the standard deviation and three formulas for the variance in this chapter. Selecting the appropriate formula depends on (1) whether you have sample or population data, and (2) whether you wish to estimate the population standard deviation or variance.

No statistical model of central tendency or variability tells you everything you may need to know about your data. Only by using multiple models in conjunction with each other can you have a thorough understanding of your data.

## 统计代写|Generalized linear model代考广义线性模型代写|Models of Variability

范围 = 最高分 – 最低分  （公式 4.6）

1. 计算数据集的中位数。这是第二个四分位数。
2. 找到中位数和最低分数之间的分数。这是第一个四分位数。
3. 找到中位数和最高分之间的分数。这是第三个四分位数。
4. 从第三个四分位数中减去第一个四分位数的分数以产生四分位数范围。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。