Mean, CV, and Standard Deviation Relationship

This problem requires us to calculate the standard deviation for four different datasets (labeled A, B, C, and D). We are provided with the Mean ($M$) and the Coefficient of Variation ($CV$) for each dataset. Our final task is to arrange these datasets in ascending order based on their calculated standard deviations.

Standard Deviation Calculation Formula

The Coefficient of Variation ($CV$) is a measure of relative variability. It is defined as the ratio of the standard deviation ($SD$) to the mean ($M$), typically expressed as a percentage. The formula is:

$CV = \frac{SD}{M} \times 100\%$

To find the standard deviation ($SD$), we can rearrange this formula algebraically:

$SD = \frac{CV \times M}{100}$

We will use this rearranged formula to calculate the standard deviation for each dataset.

Calculate Standard Deviation for Each Dataset

Dataset A

For Dataset A, the Mean ($M$) is 60 and the Coefficient of Variation ($CV$) is $\frac{70}{3}\%$.

Using the formula $SD = \frac{CV \times M}{100}$, the Standard Deviation for Dataset A ($SD_A$) is calculated as:

$SD_A = \frac{(\frac{70}{3}) \times 60}{100} = \frac{70 \times 20}{100} = \frac{1400}{100} = 14$

Dataset B

For Dataset B, the Mean ($M$) is 80 and the Coefficient of Variation ($CV$) is $6.25\%$.

Using the formula, the Standard Deviation for Dataset B ($SD_B$) is calculated as:

$SD_B = \frac{6.25 \times 80}{100} = \frac{500}{100} = 5$

Dataset C

For Dataset C, the Mean ($M$) is 70 and the Coefficient of Variation ($CV$) is $\frac{160}{7}\%$.

Using the formula, the Standard Deviation for Dataset C ($SD_C$) is calculated as:

$SD_C = \frac{(\frac{160}{7}) \times 70}{100} = \frac{160 \times 10}{100} = \frac{1600}{100} = 16$

Dataset D

For Dataset D, the Mean ($M$) is 90 and the Coefficient of Variation ($CV$) is $\frac{40}{9}\%$.

Using the formula, the Standard Deviation for Dataset D ($SD_D$) is calculated as:

$SD_D = \frac{(\frac{40}{9}) \times 90}{100} = \frac{40 \times 10}{100} = \frac{400}{100} = 4$

Arrange Datasets by Standard Deviation

Let's summarize the calculated standard deviations for each dataset:

Now, we need to arrange these datasets in ascending order based on their standard deviation values ($SD_D=4$, $SD_B=5$, $SD_A=14$, $SD_C=16$).

Comparing the values: $4 < 5 < 14 < 16$.

This order corresponds to the datasets D, B, A, and C.

Therefore, the correct arrangement of the datasets in ascending order of standard deviation is D, B, A, C.