Understanding the Relationship Between x and y
The equation given is 3x + 4y = 20. This linear equation allows us to express y in terms of x, which helps in understanding how changes in x affect y.
Finding y in terms of x
- Rearranging the equation:
- 4y = 20 - 3x
- y = (20 - 3x) / 4
This indicates that y is a linear function of x.
Quartile Deviation of x
- The quartile deviation (QD) of x is given as 12.
- QD can be calculated as (Q3 - Q1) / 2, which indicates the spread of the middle 50% of data.
Calculating Quartile Deviation of y
To find the quartile deviation of y, we need to understand how the transformation from x to y affects the dispersion:
- The linear transformation of y = (20 - 3x) / 4 implies that y is influenced by the coefficients of x.
- The transformation has a multiplication factor of -3/4, which affects the spread of the data.
Effect of Transformation on Quartile Deviation
- Since transformations of the form y = mx + c scale the quartile deviation by the absolute value of the coefficient:
- QD(y) = (1/4) * | -3 | * QD(x)
- QD(y) = (3/4) * 12
Calculating QD(y)
- QD(y) = 9
Conclusion
In summary, the quartile deviation of y, derived from the quartile deviation of x through the linear transformation, is 9. This demonstrates how linear relationships impact data spread.