统计代写|生物统计学作业代写Biostatistics代考| Other Measures of Location

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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|生物统计学作业代写Biostatistics代考|Other Measures of Location

Another useful measure of location is the median. If the observations in the data set are arranged in increasing or decreasing order, the median is the middle observation, which divides the set into equal halves. If the number of observations $n$ is odd, there will be a unique median, the $\frac{1}{2}(n+1)$ th number from either end in the ordered sequence. If $n$ is even, there is strictly no middle observation, but the median is defined by convention as the average of the two middle observations, the $\left(\frac{1}{2} n\right)$ th and $\frac{1}{2}(n+1)$ th from either end. In Section $2.1$ we showed a quicker way to get an approximate value for the median using the cumulative frequency graph (see Figure 2.6).

The two data sets ${8,5,4,12,15,7,28}$ and ${8,5,4,12,15,7,49}$, for example, have different means but the same median, 8 . Therefore, the advantage of the median as a measure of location is that it is less affected by extreme observations. However, the median has some disadvantages in comparison with the mean:

1. It takes no account of the precise magnitude of most of the observations and is therefore less efficient than the mean because it wastes information.
2. If two groups of observations are pooled, the median of the combined group cannot be expressed in terms of the medians of the two component
NUMERICAL METHODS 77
groups. However, the mean can be so expressed. If component groups are of sizes $n_{1}$ and $n_{2}$ and have means $\bar{x}{1}$ and $\bar{x}{2}$ respectively, the mean of the combined group is
$$\bar{x}=\frac{n_{1} \bar{x}{1}+n{2} \bar{x}{2}}{n{1}+n_{2}}$$
3. In large data sets, the median requires more work to calculate than the mean and is not much use in the elaborate statistical techniques (it is still useful as a descriptive measure for skewed distributions).

A third measure of location, the mode, was introduced briefly in Section 2.1.3. It is the value at which the frequency polygon reaches a peak. The mode is not used widely in analytical statistics, other than as a descriptive measure, mainly because of the ambiguity in its definition, as the fluctuations of small frequencies are apt to produce spurious modes. For these reasons, in the remainder of the book we focus on a single measure of location, the mean.

统计代写|生物统计学作业代写Biostatistics代考|Measures of Dispersion

When the mean $\bar{x}$ of a set of measurements has been obtained, it is usually a matter of considerable interest to measure the degree of variation or dispersion around this mean. Are the $x$ ‘s all rather close to $\bar{x}$, or are some of them dispersed widely in each direction? This question is important for purely descriptive reasons, but it is also important because the measurement of dispersion or variation plays a central part in the methods of statistical inference described in subsequent chapters.

An obvious candidate for the measurement of dispersion is the range $R$, defined as the difference between the largest value and the smallest value, which was introduced in Section 2.1.3. However, there are a few difficulties about use of the range. The first is that the value of the range is determined by only two of the original observations. Second, the interpretation of the range depends in a complicated way on the number of observations, which is an undesirable feature.

An alternative approach is to make use of deviations from the mean, $x-\bar{x}$; it is obvious that the greater the variation in the data set, the larger the magnitude of these deviations will tend to be. From these deviations, the variance $s^{2}$ is computed by squaring each deviation, adding them, and dividing their sum by one less than $n$ :
$$s^{2}=\frac{\sum(x-\bar{x})^{2}}{n-1}$$
The use of the divisor $(n-1)$ instead of $n$ is clearly not very important when $n$ is large. It is more important for small values of $n$, and its justification will be explained briefly later in this section. The following should be noted:

1. It would be no use to take the mean of deviations because
$$\sum(x-\bar{x})=0$$
2. Taking the mean of the absolute values, for example
$$\frac{\sum|x-\bar{x}|}{n}$$
is a possibility. However, this measure has the drawback of being difficult to handle mathematically, and we do not consider it further in this book.
The variance $s^{2}$ (s squared) is measured in the square of the units in which the $x^{\prime} s$ are measured. For example, if $x$ is the time in seconds, the variance is measured in seconds squared $\left(\sec ^{2}\right)$. It is convenient, therefore, to have a measure of variation expressed in the same units as the $x$ ‘s, and this can be done easily by taking the square root of the variance. This quantity is the standard deviation, and its formula is
$$s=\sqrt{\frac{\sum(x-\bar{x})^{2}}{n-1}}$$
Consider again the data set
$${8,5,4,12,15,5,7}$$
Calculation of the variance $s^{2}$ and standard deviation $s$ is illustrated in Table $2.9$.

统计代写|生物统计学作业代写Biostatistics代考|Box Plots

The box plot is a graphical representation of a data set that gives a visual impression of location, spread, and the degree and direction of skewness. It also allows for the identification of outliers. Box plots are similar to one-way scatter plots in that they require a single horizontal axis; however, instead of plotting every observation, they display a summary of the data. A box plot consists of the following:

1. A central box extends from the 25 th to the 75 th percentiles. This box is divided into two compartments at the median value of the data set. The relative sizes of the two halves of the box provide an indication of the

Figure 2.12 Crude death rates for the United States, 1988: a combination of one-way scatter and box plots.
distribution symmetry. If they are approximately equal, the data set is roughly symmetric; otherwise, we are able to see the degree and direction of skewness (Figure 2.11).

1. The line segments projecting out from the box extend in both directions to the adjacent values. The adjacent values are the points that are $1.5$ times the length of the box beyond either quartile. All other data points outside this range are represented individually by little circles; these are considered to be outliers or extreme observations that are not typical of the rest of the data.

Of course, it is possible to combine a one-way scatter plot and a box plot so as to convey an even greater amount of information (Figure 2.12). There are other ways of constructing box plots; for example, one may make it vertically or divide it into different levels of outliers.

统计代写|生物统计学作业代写Biostatistics代考|Other Measures of Location

1. 它没有考虑大多数观测值的精确幅度，因此效率低于平均值，因为它浪费了信息。
2. 如果合并两组观察值，则组合组的中位数不能用两个分量
NUMERICAL METHODS 77
组的中位数来表示。但是，平均值可以这样表达。如果组件组的大小n1和n2并且有意味着 $\bar{x} {1}一种nd\bar{x} {2}r和sp和C吨一世在和l是,吨H和米和一种n这F吨H和C这米b一世n和dGr这在p一世sX¯=n1X¯1+n2X¯2n1+n2$
3. 在大型数据集中，中位数需要比平均值更多的工作来计算，并且在复杂的统计技术中用处不大（它仍然可以用作偏态分布的描述性度量）。

统计代写|生物统计学作业代写Biostatistics代考|Measures of Dispersion

s2=∑(X−X¯)2n−1

1. 取偏差的平均值是没有用的，因为
∑(X−X¯)=0
2. 取绝对值的平均值，例如
∑|X−X¯|n
是一种可能。然而，这个度量的缺点是难以用数学方法处理，我们不会在本书中进一步考虑它。
方差s2(s 平方) 以单位的平方来衡量X′s被测量。例如，如果X是以秒为单位的时间，方差以秒的平方为单位测量(秒2). 因此，用与X，这可以通过取方差的平方根来轻松完成。这个量就是标准差，它的公式是
s=∑(X−X¯)2n−1
再次考虑数据集
8,5,4,12,15,5,7
计算方差s2和标准差s如表所示2.9.

统计代写|生物统计学作业代写Biostatistics代考|Box Plots

1. 一个中心框从第 25 个百分位数延伸到第 75 个百分位数。该框在数据集的中值处分为两个隔间。盒子两半的相对大小提供了一个指示

1. 从框伸出的线段在两个方向上延伸到相邻的值。相邻的值是点1.5超出任一四分位数的盒子长度的倍数。此范围之外的所有其他数据点均由小圆圈单独表示；这些被认为是异常值或极端观察值，它们在其余数据中并不典型。

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