Please help Thank you A constant is added to each observation in a set of data (
ID: 3273239 • Letter: P
Question
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A constant is added to each observation in a set of data (relocation). Describe the effect on the median, lower quartile, range, interquartile range, minimum, mean, variance, and standard deviation. What is the effect on these statistics if each observation is multiplied by a constant (rescaling)? Relocation and rescaling, called linear transformations, are frequently used: for example, converting from degree C to degree F, defined by degree F = 1.8 times degree C + 32. What is the rescaling constant? Give two more examples of rescaling and relocation. An example of nonlinear transformation is going from the radius of a circle to its area: A = pi r^2. Give two more examples of nonlinear transformations.Explanation / Answer
Part (a)
When a constant is added to each observation, the same constant gets added to the following:
Median, lower quartile, minimum, and mean.
The addition of a constant to each observation does not have any effect on the following:
Range, inter-quartile range, variance and standard deviation. ANSWER
[additional input: In general, when a constant is added to each observation, the same constant gets added to each of the Measures of Central Tendency, while the addition of a constant to each observation does not have any effect on any of the Measures of Dispersion]
Part (b)
When each observation is multiplied by a constant, following get multiplied by the same constant:
Median, lower quartile, minimum, mean, range, inter-quartile range, and standard deviation.
However, variance gets multiplied by the square of that constant. ANSWER
[additional input: In general, when each observation is multiplied by a constant, all Measures of Central Tendency and all Measures of Dispersion, EXCEPT variance, get multiplied by the same constant ]
Part (c)
The rescaling constant in temperature units is 1.8 ANSWER
[additional input: In general, if the original observations are xi, and yi = a + bxi is the rescaling and relocating transformation, then b and a are respectively the rescaling and relocating constants.]
Part (d)
Examples of rescaling and relocating transformation
If the test scores, x, are out of 50, each student is given a grace mark of 5, and the resultant score is expressed in percentage, the new score, y = 2(x + 5) is a rescaling and relocating transformation.
In a corporate scenario, the salary of an employee consists of a basic pay which changes from employee to employee and the dearness allowance which is constant for all employees. Now, given the basic pay, xi, of each employee as the original observation, and d as the common darkness allowance, the annual gross salaries are given by: yi = 12(xi + d) is yet another example of rescaling and relocating transformation. ANSWER [Additional inputs: Classical terminology for rescaling and relocating transformation is 'change of scale and origin']