Question:
Let \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) be observations with mean \(m\) and standard deviation \(s\). Then, which of the following statements are correct:
A. Mean of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(m+5k\)
B. Mean of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(km\).
C. The standard deviation of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(s\).
D. The standard deviation of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(k^{5}s\).
Choose the correct answer from the options given below :
Let \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) be observations with mean \(m\) and standard deviation \(s\). Then, which of the following statements are correct:
A. Mean of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(m+5k\)
B. Mean of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(km\).
C. The standard deviation of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(s\).
D. The standard deviation of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(k^{5}s\).
Choose the correct answer from the options given below :
A. Mean of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(m+5k\)
B. Mean of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(km\).
C. The standard deviation of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(s\).
D. The standard deviation of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(k^{5}s\).
Choose the correct answer from the options given below :
Show Hint
Adding a constant shifts the mean but not the spread; multiplying by \(k\) scales the mean by \(k\) and the standard deviation by \(|k|\).
Updated On: Nov 12, 2026
- A, B, C, D
- A, C only
- B, D only
- B, C only
Show Solution
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The Correct Option is D
Solution and Explanation
A: new mean \(=\frac{\sum (x_i+k)}{5}=m+k\), not \(m+5k\). So A is false.
B: new mean \(=\frac{\sum kx_i}{5}=km\). True.
C: deviations from the mean are unchanged by adding \(k\), so the standard deviation stays \(s\). True.
D: the standard deviation becomes \(|k|s\), not \(k^5 s\). False.
Hence only B and C are correct.
Answer: (4) B, C only
B: new mean \(=\frac{\sum kx_i}{5}=km\). True.
C: deviations from the mean are unchanged by adding \(k\), so the standard deviation stays \(s\). True.
D: the standard deviation becomes \(|k|s\), not \(k^5 s\). False.
Hence only B and C are correct.
Answer: (4) B, C only
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