Understanding the Arithmetic Mean
The arithmetic mean of n observations (X1, X2, X3, ..., Xn) is calculated as follows:
- Arithmetic Mean (M) = (X1 + X2 + X3 + ... + Xn) / n
This represents the central tendency of the dataset.
Transformation of Variables
When we apply a transformation to the variable X, such as Y = 5X ± 2, we need to analyze how this affects the mean.
Mean of the Transformed Variable Y
1. Substituting the mean of X into the transformation:
- If M is the mean of X, then the mean of Y can be expressed as:
- Mean of Y = 5 * (Mean of X) ± 2
2. Therefore, we can express it as:
- Mean of Y = 5M ± 2
This means that the new mean Y will scale the original mean M by a factor of 5 and then add or subtract 2 depending on the sign in the transformation.
Conclusion
To summarize:
- The mean of Y = 5X ± 2 is derived from the mean of X by scaling and shifting.
- The final expression for the mean of Y is:
- Mean of Y = 5M ± 2
This transformation provides insight into how changes in the original data affect the average of the new variable, Y. Understanding this relationship is essential for data analysis in various contexts.