Let x1, x2, ........., xn be n observations, and let \bar{x} be their arithmetic mean and \sigma2 be their variance. Statement 1: Variance of 2x1, 2x2, ...., 2xn is 4\sigma2. Statement 2: Arithmetic mean of 2x1, 2x2, ...., 2xn is 4\bar{x}.

Engineering

Mathematics

Measure of Dispersion

Question

Let x₁, x₂, ........., xn be n observations, and let $\bar{x}$ be their arithmetic mean and $σ$ ² be their variance.

Statement 1: Variance of 2x₁, 2x₂, ...., 2x_n is 4 $σ$ ².

Statement 2: Arithmetic mean of 2x₁, 2x₂, ...., 2x_n is 4 $\bar{x}$ .

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Statement 1 is false, Statement 2 is true.

Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

Statement 1 is true, Statement 2 is false.

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

S‑2 : Arithmetic Mean $= \frac{2 x_{1} + 2 x_{2} + . ....... + 2 x_{n}}{n} = 2 (\frac{x_{1} + x_{2} + . ...... + x_{n}}{n}) = 2 \bar{x}$

$∴$ Statement‑2 is false

Statement‑1

We know variance of x₁, x₂, x₃, ....... x_n.

$σ = \frac{\sum x_{i}^{2}}{n} - {(\frac{\sum x_{i}}{n})}^{2}$

variance of 2x₁ ; 2x₂ ; ............ ; 2x_n

_{$σ^{2} = \frac{4 \sum x_{i}^{2}}{n} - 4 (\frac{\sum x_{i}}{n})$}

_{$= 4 (\frac{\sum x_{i}^{2}}{n} - \frac{\sum x_{i}}{n}) = 4 σ^{2}$}