To understand the characteristics of the two random variables, X and Y, we can start by analyzing their means and standard deviations.
Understanding the Means
The mean is a measure of central tendency, indicating where the center of the data lies. For random variable X, the mean is 120, suggesting that the values of X are expected to cluster around this number. Conversely, random variable Y has a mean of 100, showing that Y’s values are centered around this lower figure.
Examining the Standard Deviations
The standard deviation quantifies the amount of variation or dispersion of a set of values. A smaller standard deviation means that the data points tend to be closer to the mean, whereas a larger standard deviation indicates more spread out data.
For X, the standard deviation is 15, meaning the values of X will generally vary by 15 units above or below the mean of 120. Therefore, most of the values for X are likely to fall between 105 (120 – 15) and 135 (120 + 15).
On the other hand, Y has a standard deviation of 9. This indicates that its values are more tightly clustered around the mean of 100, typically ranging between 91 (100 – 9) and 109 (100 + 9).
Comparative Insights
When comparing these two random variables:
- Central Tendency: X tends to be higher on the scale compared to Y.
- Variability: X exhibits more variability (15) than Y (9), meaning that a wider range of values is expected for X.
- Distribution Shape: If we assume both distributions are normal (which is common for random variables, though not always true), X may have a flatter and broader bell curve compared to the narrower bell curve of Y.
Conclusion
In summary, random variable X has a higher average with more variability, while Y is lower in average and more consistent. Understanding these properties aids in predicting the behavior of these random variables in various scenarios, such as in statistical calculations or probability assessments.