If the standard deviation of the observation −5, −4, −3, −2, −1, 0, 1, 2, 3,…
2019
If the standard deviation of the observation −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 is √10, then the standard deviation of the observation 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 will be:
- A.
√10
- B.
√10 + 10
- C.
√10 + 20
- D.
None of the options
Attempted by 56 students.
Show answer & explanation
Correct answer: A
Key idea: adding the same constant to every observation does not change the standard deviation.
Given set: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 has SD = √10 (given).
New set: 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 is each original value plus 20 (i.e., y_i = x_i + 20).
If y_i = x_i + 20 then mean(y) = mean(x) + 20, so y_i − mean(y) = x_i − mean(x).
Therefore each squared deviation (y_i − mean(y))^2 equals (x_i − mean(x))^2, so variance and standard deviation remain unchanged.
Conclusion: the standard deviation of the new observations is √10.