To determine which of the two distributions is more consistent, we need to compare their coefficients of variation (CV). The coefficient of variation is a measure of relative variability and is given by the ratio of the standard deviation (SD) to the arithmetic mean (AM).
Here are the steps to find the coefficient of variation for each distribution and determine which one is more consistent:
1. Calculate the coefficient of variation for Distribution X:
Given:
- Arithmetic Mean (AM_X): 100
- Standard Deviation (SD_X): 10
The formula for the coefficient of variation (CV) is:
[tex]\[
\text{CV}_X = \frac{\text{SD}_X}{\text{AM}_X}
\][/tex]
Substituting the given values:
[tex]\[
\text{CV}_X = \frac{10}{100} = 0.1
\][/tex]
2. Calculate the coefficient of variation for Distribution Y:
Given:
- Arithmetic Mean (AM_Y): 90
- Standard Deviation (SD_Y): 18
Using the same formula for the coefficient of variation:
[tex]\[
\text{CV}_Y = \frac{\text{SD}_Y}{\text{AM}_Y}
\][/tex]
Substituting the given values:
[tex]\[
\text{CV}_Y = \frac{18}{90} = 0.2
\][/tex]
3. Compare the coefficients of variation:
- For Distribution X, [tex]\(\text{CV}_X = 0.1\)[/tex]
- For Distribution Y, [tex]\(\text{CV}_Y = 0.2\)[/tex]
The lower the coefficient of variation, the more consistent the distribution is.
4. Conclusion:
Since [tex]\(\text{CV}_X = 0.1\)[/tex] and [tex]\(\text{CV}_Y = 0.2\)[/tex], Distribution X has a smaller coefficient of variation. Therefore, Distribution X is more consistent than Distribution Y.
So, the more consistent distribution is Distribution X.
