A group of data elements has a mean M and a standard deviation S.
a) Find the new mean and the new standard deviation if a constant number C is added to each data element in the group? Explain thoroughly.
b) Find the new mean and the new standard deviation if each data element in the group is multiplied by the same constant number C? Explain thoroughly.
a) To find the new mean and standard deviation when a constant number C is added to each data element in the group, follow these steps:
1. Mean Calculation: Adding a constant C to each data element will result in shifting the entire data set up by C units. Since the mean represents the average value of the data set, the new mean (M') can be obtained by simply adding C to the original mean (M).
M' = M + C
For example, if the original mean is 5 and C is 2, then the new mean would be 5 + 2 = 7.
2. Standard Deviation Calculation: The standard deviation measures the spread or dispersion of the data set. When a constant C is added to each data element, the spread or dispersion of the data does not change. Therefore, the standard deviation remains the same.
Thus, the new standard deviation (S') would be equal to the original standard deviation (S).
S' = S
For example, if the original standard deviation is 3, then the new standard deviation would also be 3.
b) To find the new mean and standard deviation when each data element in the group is multiplied by the same constant number C, follow these steps:
1. Mean Calculation: Multiplying each data element by C will result in scaling the entire data set by a factor of C. Since the mean represents the average value of the data set, the new mean (M') can be obtained by multiplying the original mean (M) by C.
M' = M * C
For example, if the original mean is 5 and C is 2, then the new mean would be 5 * 2 = 10.
2. Standard Deviation Calculation: The standard deviation measures the spread or dispersion of the data set. When each data element is multiplied by C, the spread or dispersion of the data is also scaled by a factor of C. Therefore, the new standard deviation (S') is equal to the original standard deviation (S) multiplied by absolute value of C.
S' = |C| * S
Note: Taking the absolute value of C is important since the standard deviation cannot be negative.
For example, if the original standard deviation is 3 and C is -2, then the new standard deviation would be |-2| * 3 = 6.
In summary, when a constant is added to each data element, only the mean changes while the standard deviation remains the same. When each data element is multiplied by a constant, both the mean and the standard deviation are affected. The mean is scaled by the constant, and the standard deviation is scaled by the absolute value of the constant.