### 统计代写|生物统计学作业代写Biostatistics代考|DESCRIPTIVE METHODS FOR CONTINUOUS DATA

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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|Frequency Distribution

There is no difficulty if the data set is small, for we can arrange those few numbers and write them, say, in increasing order; the result would be sufficiently clear; Figure $2.1$ is an example. For fairly large data sets, a useful device for summarization is the formation of a frequency table or frequency distribution. This is a table showing the number of observations, called frequency, within certain ranges of values of the variable under investigation. For example, taking the variable to be the age at death, we have the following example; the second column of the table provides the frequencies.

Example 2.1 Table $2.1$ gives the number of deaths by age for the state of Minnesota in $1987 .$

If a data set is to be grouped to form a frequency distribution, difficulties should be recognized, and an efficient strategy is needed for better communication. First, there is no clear-cut rule on the number of intervals or classes. With too many intervals, the data are not summarized enough for a clear visualization of how they are distributed. On the other hand, too few intervals are undesirable because the data are oversummarized, and some of the details of the distribution may be lost. In general, between 5 and 15 intervals are acceptable; of course, this also depends on the number of observations, we can and should use more intervals for larger data sets.

The widths of the intervals must also be decided. Example $2.1$ shows the special case of mortality data, where it is traditional to show infant deaths (deaths of persons who are born live but die before living one year). Without such specific reasons, intervals generally should be of the same width. This common width $w$ may be determined by dividing the range $R$ by $k$, the number of intervals:
$$w=\frac{R}{k}$$
where the range $R$ is the difference between the smallest and largest in the data set. In addition, a width should be chosen so that it is convenient to use or easy to recognize, such as a multiple of 5 (or 1 , for example, if the data set has a narrow range). Similar considerations apply to the choice of the beginning of the first interval; it is a convenient number that is low enough for the first interval to include the smallest observation. Finally, care should be taken in deciding in which interval to place an observation falling on one of the interval boundaries. For example, a consistent rule could be made so as to place such an observation in the interval of which the observation in question is the lower limit.

## 统计代写|生物统计学作业代写Biostatistics代考|Histogram and the Frequency Polygon

A convenient way of displaying a frequency table is by means of a histogram and/or a frequency polygon. A histogram is a diagram in which:

1. The horizontal scale represents the value of the variable marked at interval boundaries.
1. The vertical scale represents the frequency or relative frequency in each interval (see the exceptions below).

A histogram presents us with a graphic picture of the distribution of measurements. This picture consists of rectangular bars joining each other, one for each interval, as shown in Figure $2.2$ for the data set of Example 2.2. If disjoint intervals are used, such as in Table $2.2$, the horizontal axis is marked with true boundaries. A true boundary is the average of the upper limit of one interval and the lower limit of the next-higher interval. For example, $19.5$ serves as the true upper boundary of the first interval and the true lower boundary for the second interval. In cases where we need to compare the shapes of the histograms representing different data sets, or if intervals are of unequal widths, the height of each rectangular bar should represent the density of the interval, where the interval density is defined by
$$\text { density }=\frac{\text { relative frequency }(\%)}{\text { interval width }}$$
The unit for density is percent per unit (of measurement): for example, percent per year. If we graph densities on the vertical axis, the relative frequency is represented by the area of the rectangular bar, and the total area under the histogram is $100 \%$. It may always be a good practice to graph densities on the vertical axis with or without having equal class width; when class widths are

equal, the shape of the histogram looks similar to the graph, with relative frequencies on the vertical axis.

To draw a frequency polygon, we first place a dot at the midpoint of the upper base of each rectangular bar. The points are connected with straight lines. At the ends, the points are connected to the midpoints of the previous and succeeding intervals (these are make-up intervals with zero frequency, where widths are the widths of the first and last intervals, respectively). A frequency polygon so constructed is another way to portray graphically the distribution of a data set (Figure 2.3). The frequency polygon can also be shown without the histogram on the same graph.

The frequency table and its graphical relatives, the histogram and the frequency polygon, have a number of applications, as explained below; the first leads to a research question and the second leads to a new analysis strategy.

1. When data are homogeneous, the table and graphs usually show a unimodal pattern with one peak in the middle part. A bimodal pattern might indicate the possible influence or effect of a certain hidden factor or factors.

## 统计代写|生物统计学作业代写Biostatistics代考|Cumulative Frequency Graph and Percentiles

Cumulative relative frequency, or cumulative percentage, gives the percentage of persons having a measurement less than or equal to the upper boundary of the class interval. Data from Table $2.2$ for the distribution of weights of 57 children are reproduced and supplemented with a column for cumulative relative frequency in Table 2.6. This last column is easy to form; you do it by successively accumulating the relative frequencies of each of the various intervals. In the table the cumulative percentage for the first three intervals is
$$8.8+33.3+17.5=59.6$$
and we can say that $59.6 \%$ of the children in the data set have a weight of $39.5$ $\mathrm{lb}$ or less. Or, as another example, $96.4 \%$ of children weigh $69.5 \mathrm{lb}$ or less, and so on.

The cumulative relative frequency can be presented graphically as in Figure 2.6. This type of curve is called a cumulative frequency graph. To construct such a graph, we place a point with a horizontal axis marked at the upper class boundary and a vertical axis marked at the corresponding cumulative frequency. Each point represents the cumulative relative frequency and the points are connected with straight lines. At the left end it is connected to the lower boundary of the first interval. If disjoint intervals such as
\begin{aligned} &10-19 \ &20-29 \ &\text { etc. } . \end{aligned}
are used, points are placed at the true boundaries.
The cumulative percentages and their graphical representation, the cumulative frequency graph, have a number of applications. When two cumulative frequency graphs, representing two different data sets, are placed on the same

graph, they provide a rapid visual comparison without any need to compare individual intervals. Figure $2.7$ gives such a comparison of family incomes using data of Example $2.5$.

The cumulative frequency graph provides a class of important statistics known as percentiles or percentile scores. The 90 th percentile, for example, is the numerical value that exceeds $90 \%$ of the values in the data set and is exceeded by only $10 \%$ of them. Or, as another example, the 80 th percentile is that numerical value that exceeds $80 \%$ of the values contained in the data set and is exceeded by $20 \%$ of them, and so on. The 50 th percentile is commonly

called the median. In Figure $2.7$ the median family income in 1983 for nonwhites was about $\$ 17,500$, compared to a median of about$\$22,000$ for white families. To get the median, we start at the $50 \%$ point on the vertical axis and go horizontally until meeting the cumulative frequency graph; the projection of this intersection on the horizontal axis is the median. Other percentiles are obtained similarly.

The cumulative frequency graph also provides an important application in the formation of health norms (see Figure $2.8$ ) for the monitoring of physical progress (weight and height) of infants and children. Here, the same percentiles, say the 90 th, of weight or height of groups of different ages are joined by a curve.

## 统计代写|生物统计学作业代写Biostatistics代考|Histogram and the Frequency Polygon

1. 水平刻度表示在区间边界处标记的变量的值。
2. 垂直刻度表示每个间隔中的频率或相对频率（请参阅下面的例外情况）。

密度 = 相对频率 (%) 间隔宽度

1. 当数据是同质的时，表格和图表通常显示一个单峰模式，中间有一个峰。双峰模式可能表明某个或多个隐藏因素的可能影响或效果。

## 统计代写|生物统计学作业代写Biostatistics代考|Cumulative Frequency Graph and Percentiles

8.8+33.3+17.5=59.6

10−19 20−29  等等 .

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## MATLAB代写

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