Question

Let $a$, $b$, $c$, and $d$ be the observations with mean $m$ and standard deviation $S$. The standard deviation of the observations $a + k, b + k, c + k, d + k$ is:

• A. $kS$
• B. $S + k$
• C. $\frac{S}{k}$
• D. $S$

Answer 1

The Correct Option is D

1. Understand the problem:

We are given five observations: a, b, c, d, e with mean m and standard deviation S. We need to find the standard deviation of the transformed observations: a + k, b + k, c + k, d + k, e + k.

2. Recall properties of standard deviation:

Standard deviation is a measure of dispersion and is unaffected by adding a constant to all data points. Only the mean changes by the constant k, while the spread (standard deviation) remains the same.

3. Verify mathematically:

Original mean (m) = (a + b + c + d + e)/5

New mean = [(a+k) + (b+k) + (c+k) + (d+k) + (e+k)]/5 = m + k

Original variance (S²) = Σ(xᵢ - m)²/5

New variance = Σ[(xᵢ + k) - (m + k)]²/5 = Σ(xᵢ - m)²/5 = S²

Thus, standard deviation remains S.

Correct Answer: (D) S

Answer 2

Step 1: Understand the transformation

If you add a constant \( k \) to each observation, it shifts the mean but does not affect the spread (the variation or standard deviation) of the data. The standard deviation only depends on the spread of the data and not on the mean.

Step 2: Effect on standard deviation

Adding a constant to each observation does not change the standard deviation. The spread remains the same.

Step 3: Conclusion

Since the standard deviation of the original observations is \( S \), and adding a constant \( k \) to each observation does not affect the standard deviation, the standard deviation of the new observations is also \( S \).