### 统计代写|生物统计代写biostatistics代考|Describing a Population with Parameters

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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|生物统计代写biostatistics代考|Proportions and Percentiles

Populations are often summarized by listing the important percentages or proportions associated with the population. The proportion of units in a population having a particular characteristic is a parameter of the population, and a population proportion will be denoted by $p$. The population proportion having a particular characteristic, say characteristic $A$, is defined to be
$$p=\frac{\text { number of units in population having characteristic A }}{N}$$
Note that the percentage of the population having characteristic A is $p \times 100 \%$. Population proportions and percentages are often associated with the categories of a qualitative variable or with the values in the population falling in a specific range of values. For example, the distribution of a qualitative variable is usually displayed in a bar chart with the height of a bar representing either the proportion or percentage of the population having that particular value.
Example 2.12
The distribution of blood type according to the American Red Cross is given in Table $2.4$ in terms of proportions.

An important proportion in many biomedical studies is the proportion of individuals having a particular disease, which is called the prevalence of the disease. The prevalence of a disease is defined to be
Prevalence $=$ The proportion of individuals in a well-defined population having the disease of interest
For example, according to the Centers for Disease Control and Prevention (CDC) the prevalence of smoking among adults in the United States in January through June 2005 was $20.9 \%$. Proportions also play important roles in the study of survival and cure rates, the occurrence of side effects of new drugs, the absolute and relative risks associated with a disease, and the efficacy of new treatments and drugs.

## 统计代写|生物统计代写biostatistics代考|Parameters Measuring Centrality

The two parameters in the population of values of a quantitative variable that summarize how the variable is distributed are the parameters that measure the typical or central values in the population and the parameters that measure the spread of the values within the population. Parameters describing the central values in a population and the spread of a population are often used for summarizing the distribution of the values in a population; however, it is important to note that most populations cannot be described very well with only the parameters that measure centrality and the spread of the population.

Measures of centrality, location, or the typical value are parameters that lie in the “center” or “middle” region of a distribution. Because the center or middle of a distribution is not easily determined due to the wide range of different shapes that are possible with a distribution, there are several different parameters that can be used to describe the center of a population. The three most commonly used parameters for describing the center of a population are the mean, median, and mode. For a quantitative variable $X$.

• The mean of a population is the average of all of the units in the population, and will be denoted by $\mu$. The mean of a variable $X$ measured on a population consisting of $N$ units is
$$\mu=\frac{\text { sum of the values of } X}{N}=\frac{\sum X}{N}$$
• The median of a population is the 50 th percentile of the population, and will be denoted by $\tilde{\mu}$. The median of a population is found by first listing all of the values of the variable $X$, including repeated $X$ values, in ascending order. When the number of units in the population (i.e., $N$ ) is an odd number, the median is the middle observation in the list of ordered values of $X$; when $N$ is an even number, the median will be the average of the two observations in the middle of the ordered list of $X$ values.
• The mode of a population is the most frequent value in the population, and will be denoted by $M$. In a graph of the probability density function, the mode is the value of $X$ under the peak of the graph, and a population can have more than one mode as shown in Figure 2.8.

The mean, median, and mode are three different parameters that can be used to measure the center of a population or to describe the typical values in a population. These three parameters will have nearly the same value when the distribution is symmetric or mound shaped. For long-tailed distributions, the mean, median, and mode will be different, and the difference in their values will depend on the length of the distribution’s longer tail. Figures $2.12$ and $2.13$ illustrate the relationships between the values of the mean, median, and mode for long-tail right and long-tail left distributions.

## 统计代写|生物统计代写biostatistics代考|Measures of Dispersion

While the mean, median, and mode of a population describe the typical values in the population, these parameters do not describe how the population is spread over its range of values. For example, Figure $2.16$ shows two populations that have the same mean, median, and mode but different spreads.

Even though the mean, median, and mode of these two populations are the same, clearly, population I is much more spread out than population II. The density of population II is greater at the mean, which means that population II is more concentrated at this point than population I.

When describing the typical values in the population, the more variation there is in a population the harder it is to measure the typical value, and just as there are several ways of measuring the center of a population there are also several ways to measure the variation in a population. The three most commonly used parameters for measuring the spread of a population are the variance, standard deviation, and interquartile range. For a quantitative variable $X$

• the variance of a population is defined to be the average of the squared deviations from the mean and will be denoted by $\sigma^{2}$ or $\operatorname{Var}(X)$. The variance of a variable $X$

measured on a population consisting of $N$ units is
$$\sigma^{2}=\frac{\text { sum of all(deviations from } \mu)^{2}}{N}=\frac{\sum(X-\mu)^{2}}{N}$$

• the standard deviation of a population is defined to be the square root of the variance and will be denoted by $\sigma$ or $\operatorname{SD}(X)$.
$$\operatorname{SD}(X)=\sigma=\sqrt{\sigma^{2}}=\sqrt{\operatorname{Var}(X)}$$
• the interquartile range of a population is the distance between the 25 th and 75 th percentiles and will be denoted by IQR.
$$\mathrm{IQR}=75 \text { th percentile }-25 \text { th percentile }=X_{75}-X_{25}$$

## 统计代写|生物统计代写biostatistics代考|Proportions and Percentiles

p= 人口中具有特征 A 的单位数 ñ

## 统计代写|生物统计代写biostatistics代考|Parameters Measuring Centrality

• 总体的平均值是总体中所有单位的平均值，表示为μ. 变量的平均值X在由以下人员组成的总体上测量ñ单位是
μ= 的值的总和 Xñ=∑Xñ
• 人口的中位数是人口的第 50 个百分位，表示为μ~. 通过首先列出变量的所有值来找到总体的中位数X，包括重复X值，按升序排列。当人口中的单位数（即，ñ) 是奇数，中位数是 的有序值列表中的中间观察值X; 什么时候ñ是偶数，中位数将是有序列表中间的两个观察值的平均值X价值观。
• 人口的众数是人口中出现频率最高的值，记为米. 在概率密度函数图中，众数是X如图 2.8 所示，一个总体可以有多个众数。

## 统计代写|生物统计代写biostatistics代考|Measures of Dispersion

• 总体的方差定义为与均值的平方偏差的平均值，并表示为σ2或者曾是⁡(X). 变量的方差X

σ2= 所有的总和（偏离 μ)2ñ=∑(X−μ)2ñ

• 总体的标准差定义为方差的平方根，表示为σ或者标清⁡(X).
标清⁡(X)=σ=σ2=曾是⁡(X)
• 人口的四分位距是第 25 和第 75 个百分位数之间的距离，用 IQR 表示。
我问R=75 百分位数 −25 百分位数 =X75−X25

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