## 统计代写|金融统计代写Financial Statistics代考|Percentiles, Quartiles, and Interquartile Range

One useful way of describing the relative standing of a value in a set of data is through the use of percentiles. Percentiles give valuable information about the rank of an observation. Most of you are familiar with percentiles from taking standardized college admissions tests such as the SAT or ACT. These tests assign each student not only a raw score but also a percentile to indicate his or her relative performance. For example, a student scoring in the 85 th percentile scored higher than $85 \%$ of the students who took the test and lower than $(100-85)=15 \%$ of those who took it.

Let $x_1, x_2, \ldots$, be a set of measurements arranged in ascending (or descending) order. The $P$ th percentile is a number $x$ such that $P$ percent of the measurement fall below the Pth percentile and $(100-P)$ percent fall above it.

Quartiles are merely particular percentiles that divide the data into quarters. The 25th percentile is known as the first quartile $\left(Q_1\right)$, the 50th percentile is the second $\left(Q_2\right)$, and the 75 th percentile is the third $\left(Q_3\right)$.

To approximate the quartiles from a population containing $N$ observations, the following positioning point formulas are used:
$Q_1=$ value corresponding to the $\frac{N+1}{4}$ ordered observation
$Q_2=$ median, the value corresponding to the $\frac{2(N+1)}{4}=\frac{N+1}{2}$ ordered observation
$Q_3=$ value corresponding to the $\frac{3(N+1)}{4}$ ordered observation
The formulas given for $Q_1$ and $Q_3$ sometimes are defined as the $(N+1) / 4$ th and $(3 N+1) / 4$ th observations, respectively. If $Q_1, Q_2$, or $Q_3$ is not an integer, then the interpolation method can he used to estimate the value of the corresponding observation.

Interquartiles range (IQR), a measure commonly used in conjunction with quartiles, can be defined as
$$\mathrm{IQR}=Q_3-Q_1$$

## 统计代写|金融统计代写Financial Statistics代考|Box and Whisker Plots

A box and whisker plot is a graphical representation of a set of sample data that illustrates the lowest data value $(L)$, the first quartile $\left(Q_1\right)$, the median $\left(Q_2\right.$, Md), the third quartile $\left(Q_3\right)$, the interquartile range (IQR), and the highest data value $(H)$.
In the last section, the following values were determined for the aptitude test scores in Table 4.6: $L=20, Q_1=41+.25(42-41)=41.25, Q_2=\mathrm{Md}=55.5$, $Q_3=75+.75(78-75)=77.25, I Q R=36$, and $H=100$.

A box and whisker plot of these values is shown in Fig. 4.5. The ends of the box are located at the first and third quartiles, and a vertical bar is inserted at the median. Consequently, the length of the box is the interquartile range. The dotted lines are the whiskers; they connect the highest and lowest data values to the end of the box. This means that approximately $25 \%$ of the data values will lie in each whisker and in each portion of the box. If the data are symmetric, the median bar should be located at the center of the box. Consequently, the location of the bar informs us about any skewness of the data; if the bar is located in the left (or right) half of the box, the data are skewed right (or left), as defined in the next section.

In Fig. 4.5, the distribution of the data is skewed to the right because the median bar is located in the left. A box and whisker plot using MINITAB is shown in Fig. 4.6. In this figure, a rectangle (the box) is drawn with the ends (the hinges) drawn at the first and third quartiles $\left(Q_1\right.$ and $\left.Q_3\right)$. The median of the data is shown in the box by the symbol $+$. There are two boxes in Fig. 4.6. The only difference is that the second specifies the starting value at $15 .$

Example $4.13$ Using MINITAB to Compute Some Important Statistics of 40 Aptitude Test Scores. The MINITAB/PC input and printout are presented in Fig. 4.7. This printout presents mean, median, standard deviation, $L$ (MIN), $Q_1, Q_3$, and $H$ (MAX), which we have calculated and analyzed before. Note that the MINITAB/ PC can calculate this information very effectively. In Fig. 4.7, 40 aptitude test scores are first entered into the PC. Then ten statistics will automatically print if the command “MTB $>$ describe Cl” is entered. Two of those statistics. TRMEAN and SEMEAN, are not discussed in this book.

## 统计代写|金融统计代写Financial Statistics代考|Percentiles, Quartiles, and Interquartile Range

$Q_1=$ 对应的值 $\frac{N+1}{4}$ 有序观察
$Q_2=$ 中位数，对应的值 $\frac{2(N+1)}{4}=\frac{N+1}{2}$ 有序观察
$Q_3=$ 对应的值 $\frac{3(N+1)}{4}$ 有序观察

$$\mathrm{IQR}=Q_3-Q_1$$

## 统计代写|金融统计代写Financial Statistics代考|Box and Whisker Plots

\begin{aligned} &L=20, Q_1=41+.25(42-41)=41.25, Q_2=\mathrm{Md}=55.5, \ &Q_3=75+.75(78-75)=77.25, I Q R=36 ， \quad \text { 和 } H=100 . \end{aligned}

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