Variance And Standard Deviation

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