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jane123456 · 2024年08月06日

请问知道标准差,是否可以直接算MAD

NO.PZ2021061603000025

问题如下:

Annual returns and summary statistics for three funds are listed in the following exhibit:

The fund with the highest absolute dispersion is:

选项:

A.Fund PQR if the measure of dispersion is the range B.Fund XYZ if the measure of dispersion is the variance C.Fund ABC if the measure of dispersion is the mean absolute deviation

解释:

C is correct. The mean absolute deviation (MAD) of Fund ABC's returns is greater than the MAD of both of the other funds.

MAD=i=1nXiXˉnMAD = \frac{{\sum\limits_{i = 1}^n {\left| {{X_i} - \bar X} \right|} }}{n}, where Xˉ{\bar X} is the arithmetic mean of the series.

MAD for Fund ABC =

20(4)+23(4)+14(4)+5(4)+14(4)5=14.4%\frac{{\left| { - 20 - ( - 4)} \right| + \left| {23 - ( - 4)} \right| + \left| { - 14 - ( - 4)} \right| + \left| {5 - ( - 4)} \right| + \left| { - 14 - ( - 4)} \right|}}{5} = 14.4\%

MAD for Fund XYZ=

33(10.8)+12(10.8)+12(10.8)+8(10.8)+11(10.8)5=9.8%\frac{{\left| { - 33 - ( - 10.8)} \right| + \left| { - 12 - ( - 10.8)} \right| + \left| { - 12 - ( - 10.8)} \right| + \left| { - 8 - ( - 10.8)} \right| + \left| {11 - ( - 10.8)} \right|}}{5} = 9.8\%

MAD for Fund PQR=

14(5)+18(5)+6(5)+2(5)+3(5)5=8.8%\frac{{\left| { - 14 - ( - 5)} \right| + \left| { - 18 - ( - 5)} \right| + \left| {6 - ( - 5)} \right| + \left| { - 2 - ( - 5)} \right| + \left| {3 - ( - 5)} \right|}}{5} =8.8\%

A and B are incorrect because the range and variance of the three funds are as follows:


The numbers shown for variance are understood to be in "percent squared" terms so that when taking the square root, the result is standard deviation in percentage terms. Alternatively, by expressing standard deviation and variance in decimal form, one can avoid the issue of units. In decimal form, the variances for Fund ABC, Fund XYZ, and Fund PQR are 0.0317, 0.0243, and 0.0110, respectively.

不考虑实际意义的话,题目告诉3个数据集的标准差,要求3个数据集的MAD的大小,那么是否根据标准差和MAD的公式,直接通过n*SD^2=(MAD*n)^2即MAD=(SD^2/n)^1/2?如此,还可以推导出标准差更大的数据集其实MAD也更大?

1 个答案

品职助教_七七 · 2024年08月07日

嗨,爱思考的PZer你好:


无法通过标准差来计算MAD,提问中的公式关系不成立。

MAD的每一项是绝对值,不是平方项。如将MAD整体平方,平方后会产生交叉项,并不等于标准差的平方。

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