10 Question MCQ (including Assertion)

Assertion (A): For the dataset \(\{2, 4, 6, 8, 10\}\), the Mean Deviation about mean is \(2.4\).
Reason (R): For any non-constant dataset, the Mean Deviation about mean is strictly greater than the Standard Deviation.

Two batsmen A and B have identical mean scores (\(\bar{x} = 56\)). The dot plot below displays the score distribution for Batsman B across 8 matches: Calculate the Standard Deviation (\(\sigma\)) of Batsman B's scores (\(50, 52, 54, 56, 56, 58, 60, 62\)).

Investment Fund P has average return \(\bar{x}_P = 12\%\) with \(\sigma_P = 2\%\). Fund Q has average return \(\bar{x}_Q = 15\%\) with \(\sigma_Q = 5\%\). Calculate the Coefficients of Variation \(C.V._P\) and \(C.V._Q\).

Assertion (A): If asset return \(X\) has variance \(\sigma_X^2 = 5\), then the daily portfolio return \(Y = 4X + 7\) has variance \(\sigma_Y^2 = 80\).
Reason (R): Under linear transformation \(Y = aX + b\), variance transforms according to the formula \(\sigma_Y^2 = a \cdot \sigma_X^2 + b\).

Assertion (A): Range is considered the most reliable and scientifically sound measure of dispersion for highly scattered datasets with extreme outliers.
Reason (R): Range is calculated strictly as \(Range = \text{Maximum Value} - \text{Minimum Value}\), ignoring all intermediate values.

Find the mean deviation about the mean for the dataset: \(6, 7, 10, 12, 13, 4, 8, 12\).

Assertion (A): Adding an extreme outlier \(x = 100\) to the dataset \(\{10, 12, 14, 16, 18\}\) leaves its variance virtually unaffected.
Reason (R): Variance is highly sensitive to outliers because deviations from the mean are squared in its computation.

A dataset contains \(15\) identical numbers, all equal to \(7\). Calculate its Range, Mean Deviation about mean, and Standard Deviation.

The variance of \(20\) observations is \(5\). If each observation is multiplied by \(2\), calculate the variance of the resulting observations.