Correct option is A
The coefficient of variation (CV) is a standardized measure of dispersion of a dataset, expressed as a percentage. It provides a way to compare variability across datasets with different units or scales.
Derivation of CV Formula:
- σ\sigmaσ: Standard deviation, which measures the spread of data points around the mean.
- Xˉ\bar{X}X̄: Mean of the dataset.
The CV formula is:
The result is expressed as a percentage to make it unitless, allowing cross-comparison of datasets with different scales. For example, comparing the variability of income (measured in dollars) to variability in exam scores (measured in percentages).
The CV is particularly useful in financial analysis to assess the risk-return tradeoff, where lower CV indicates more consistency in returns.
Why Other Options Are Incorrect:
- (b): Standard deviation is always non-negative. Since it is derived as the square root of variance (σ2\sigma^2σ2), it cannot take a negative value.
- (c): Multiplying data by σ\sigmaσ would multiply the variance by the square of σ\sigmaσ, not 36 unless specifically defined.
- (d): Variance equals the square of the standard deviation (σ2\sigma^2σ2), not the other way around.
Information Booster:
The coefficient of variation (CV) plays an essential role in comparing variability or relative dispersion in data. Key points to understand about CV:
- Comparison Across Datasets: CV allows comparisons across datasets with different means and units, unlike standard deviation, which is absolute.
- Fields of Application: It is used in fields like finance, quality control, and operational research. For example, in finance, CV evaluates risk in relation to return.
- Interpretation: A lower CV indicates greater stability and consistency, whereas a higher CV shows more variability and risk.
Example of Application:
If dataset A has a mean of 50 and a standard deviation of 5 (CV=10%CV = 10\%CV=10%) while dataset B has a mean of 100 and a standard deviation of 20 (CV=20%CV = 20\%CV=20%), dataset A is more consistent than dataset B.
Additional Knowledge:
Option (b): The standard deviation (σ\sigmaσ) measures the spread of values from the mean. Even if all values in the dataset are negative, the deviations from the mean are squared during variance calculation, ensuring σ\sigmaσ is non-negative.
Option (c): When all values in a dataset are multiplied by a constant kkk, the variance is multiplied by k2k^2k2. For variance to be multiplied by 36, k=6k = 6k=6. However, multiplying by σ\sigmaσ itself does not necessarily result in 36.
Option (d): Variance (σ2\sigma^2σ2) is the square of the standard deviation. The reverse is incorrect. For example, if σ=4\sigma = 4σ=4, variance = 42=164^2 = 1642=16, not the other way around.
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