Batting Average Measures

CAT 2001 Slot 1 · QA · Medium · Statistics

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Passage / data set

The batting average (BA) of a Test batsman is computed from runs scored and innings played — completed innings and incomplete innings (not out) in the following manner: r1=r_1 = Number of runs scored in completed innings n1=n_1 = Number of completed innings r2=r_2 = Number of runs scored in incomplete innings n2=n_2 = Number of incomplete innings BA=r1+r2n1\text{BA} = \frac{r_1 + r_2}{n_1} To better assess a batsman's accomplishments, the ICC is considering two other measures MBA1\text{MBA}_1 and MBA2\text{MBA}_2 defined as follows: MBA1=r1n1+n2n1max⁡[0,(r2n2−r1n1)]\text{MBA}_1 = \frac{r_1}{n_1} + \frac{n_2}{n_1} \max \left[ 0, \left( \frac{r_2}{n_2} - \frac{r_1}{n_1} \right) \right] MBA2=r1+r2n1+n2\text{MBA}_2 = \frac{r_1 + r_2}{n_1 + n_2}

Question 1 of 2

Based on the above information which of the following is true?

  1. A.

    MBA1≤BA≤MBA2\text{MBA}_1 \le \text{BA} \le \text{MBA}_2

  2. B.

    BA≤MBA2≤MBA1\text{BA} \le \text{MBA}_2 \le \text{MBA}_1

  3. C.

    MBA2≤BA≤MBA1\text{MBA}_2 \le \text{BA} \le \text{MBA}_1

  4. D.

    None of these

Answer

D

Explanation

Since BA=r1+r2n1\text{BA} = \frac{r_1 + r_2}{n_1} and MBA2=r1+r2n1+n2\text{MBA}_2 = \frac{r_1 + r_2}{n_1 + n_2}, clearly BA≥MBA2\text{BA} \ge \text{MBA}_2 because n1+n2≥n1n_1 + n_2 \ge n_1. Also, comparing BA\text{BA} and MBA1\text{MBA}_1, if r2n2<r1n1\frac{r_2}{n_2} < \frac{r_1}{n_1}, MBA1=r1n1\text{MBA}_1 = \frac{r_1}{n_1}, while BA=r1+r2n1\text{BA} = \frac{r_1+r_2}{n_1}. Depending on values, BA\text{BA} can be greater or smaller than MBA1\text{MBA}_1, so BA≥MBA1≥MBA2\text{BA} \ge \text{MBA}_1 \ge \text{MBA}_2 holds generally. Thus, 'None of these' is the correct choice.

Question 2 of 2

An experienced cricketer with no incomplete innings has BA of 50. The next time he bats, the innings is incomplete and he scores 45 runs. It can be inferred that:

  1. A.

    BA and MBA1\text{MBA}_1 will both increase

  2. B.

    BA will increase and MBA2\text{MBA}_2 will decrease

  3. C.

    BA will increase and not enough data is available to assess change in MBA1\text{MBA}_1 and MBA2\text{MBA}_2

  4. D.

    None of these

Answer

B

Explanation

Initial: n2=0n_2 = 0, so BA=r1n1=50\text{BA} = \frac{r_1}{n_1} = 50. Next innings: r2=45r_2 = 45, n2=1n_2 = 1. New BA=50n1+45n1=50+45n1>50\text{BA} = \frac{50n_1 + 45}{n_1} = 50 + \frac{45}{n_1} > 50, so BA\text{BA} increases. New MBA2=50n1+45n1+1=50−5n1+1<50\text{MBA}_2 = \frac{50n_1 + 45}{n_1 + 1} = 50 - \frac{5}{n_1 + 1} < 50, so MBA2\text{MBA}_2 decreases.

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