Q.63 Which of the following assertions are CORRECT?
P: Adding 7 to each entry in a list adds 7 to the mean of the list
Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list
R: Doubling each entry in a list doubles the mean of the list
S: Doubling each entry in a list leaves the standard deviation of the list unchanged
(A) P, Q (B) Q, R (C) P, R (D) R, S
Adding a constant to every data point shifts the mean but not the standard deviation,
while scaling doubles the mean and also scales the standard deviation.
The correct assertions are P and R.
✅ Correct Answer: (C) P, R
These two assertions hold true based on fundamental statistical properties of mean and standard deviation transformations.
🔍 Detailed Assertion Analysis
P: Adding 7 Adds 7 to Mean ✅ CORRECT
x₁, x₂, ..., xₙ with mean x̄ = (1/n)ΣxᵢNew list:
xᵢ + 7, new mean x̄' = (1/n)Σ(xᵢ + 7) = x̄ + 7The mean shifts exactly by the added constant.
Q: Adding 7 Adds 7 to SD ❌ INCORRECT
σ = √[(1/n)Σ(xᵢ - x̄)²]New deviations:
(xᵢ + 7) - (x̄ + 7) = xᵢ - x̄, so σ' = σ (unchanged)Standard deviation measures spread around the mean, which remains identical.
R: Doubling Doubles the Mean ✅ CORRECT
2xᵢ, new mean x̄' = (1/n)Σ(2xᵢ) = 2x̄The mean scales linearly with the scaling factor.
S: Doubling Leaves SD Unchanged ❌ INCORRECT
σ' = √[(1/n)Σ(2xᵢ - 2x̄)²] = √[4 × (1/n)Σ(xᵢ - x̄)²] = 2σStandard deviation also doubles because deviations are scaled by 2.
🎯 Why P and R Matter in Practice
These properties underpin data standardization in biotechnology and statistics. For instance, in microbial growth kinetics, shifting measurements (like adding a baseline OD value) adjusts means predictably without altering variability, aiding precise model fitting in fermentation processes and enzyme kinetics studies.
