统计代写|金融统计代写Financial Statistics代考|QMS230

统计代写|金融统计代写Financial Statistics代考|The Mean

Sometimes data are in grouped form, so calculating the mean from raw data is impossible. Recall from Chap. 3 that raw data are sometimes grouped into classes. For example, a teacher might group exam scores into A’s (90-100), B’s (80-89), and so on. In cases such as this, we can estimate the mean from the grouped data by multiplying the midpoint of each class by the number of observations and dividing by the total number of observations $(N)$ :
$$\mu=\sum_{i=1}^k f_i m_i / N$$
where $f_i=$ the frequency or number of observations in the $i$ th group, $m_i=$ the midpoint of the ith group, and $k=$ the number of groups. Note that
$$\sum_{i=1}^k f_i=N$$
Although this is the formula for estimating the population mean from grouped data, we estimate the sample mean in the same manner by substituting $n$ for $N$.
Example $4.14$ Finding the Mean of Market Rates of Return in Terms of Grouped Data. Table $4.8$ presents a frequency distribution for the rate of return on the $\mathrm{S} \& \mathrm{P}$ composite stock index from 1990 to 2009 ; it has eight classes or groups. Suppose we do not have access to the raw data that underlie this frequency distribution (which, however, are shown in Table 4.9). Will our calculated group mean be reasonably close to the true mean? In this example,
$$\sum_{i=1}^7 f_i=19$$

统计代写|金融统计代写Financial Statistics代考|The Median

Although a median can also be calculated for grouped data, it may be impossible to determine the exact value of the median if individual data values are not available. However, we can approximate the median value by first assuming that data fall equally throughout the median class. For example, suppose we have five students who scored between 90 and 100 on the exam. By assuming that the observations are equally spaced, we can hazard an educated guess of the five students’ scores. Because the width of this class is 10 and because there are five students in the class, an assumption of equal spacing means each pair of “adjacent” scores should be separated by 2 points. Making this assumption for all the classes enables us to find what is approximately the median or middle score.

To obtain the median for grouped data, find the class in which the median observation appears. Then apply the following formula to estimate the median:
$$m=L+\frac{(N / 2-F)}{f}(U-L)$$
where $L$ and $U$ are the lower and upper boundaries, respectively, of the class that contains the median; $f$ is the frequency in this class; and $F$ is the cumulative frequency of the observations in the classes prior to this class.

Example $4.15$ Finding the Median of Stock Rates of Return in Terms of Grouped Data. Referring to the grouped data in Example 4.14, we know that the median is the 11th observation. Thus, we know that the median is in the .110 to $.200$ class. $F$ is equal to 9 because nine observations occurred before the class; $f$ is equal to 8 because there are eight observations in the class; and the lower $(\mathrm{L})$ and upper (U) boundaries of the class are 110 and .200, respectively. Hence, the median estimate is
$$m=.110+\frac{(21 / 2-9)}{8}(.200-.110)=.1269$$
The median for the raw data is $.1240$, which is similar to that calculated from grouped data.

统计代写|金融统计代写Financial Statistics代考|The Mean

$$\mu=\sum_{i=1}^k f_i m_i / N$$

$$\sum_{i=1}^k f_i=N$$

$$\sum_{i=1}^7 f_i=19$$

统计代写|金融统计代写Financial Statistics代考|The Median

$$m=L+\frac{(N / 2-F)}{f}(U-L)$$

$$m=.110+\frac{(21 / 2-9)}{8}(.200-.110)=.1269$$

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