Comprehension
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:
\[ r_1 = \text{Number of runs scored in completed innings} \] \[ n_1 = \text{Number of completed innings} \] \[ r_2 = \text{Number of runs scored in incomplete innings} \] \[ n_2 = \text{Number of incomplete innings} \]
\[ BA = \frac{r_1 + r_2}{n_1} \]
To better assess a batsman’s accomplishments, the ICC is considering two other measures MBA1 and MBA2, defined as follows:
\[ MBA_1 = \frac{r_1}{n_1} + \frac{n_2}{n_1} \max \left( 0, \left( \frac{r_2}{n_2} - \frac{r_1}{n_1} \right) \right) \]
\[ MBA_2 = \frac{r_1 + r_2}{n_1 + n_2} \]
Question: 1
Based on the given definitions of BA, MBA$_1$, and MBA$_2$, which of the following is true?
Based on the given definitions of BA, MBA$_1$, and MBA$_2$, which of the following is true?
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When comparing averages with different denominators, remember: increasing the denominator with same numerator decreases the value.
Updated On: Aug 4, 2025
- MBA$_1 \le$ BA $\le$ MBA$_2$
- BA $\le$ MBA$_2 \le$ MBA$_1$
- MBA$_2 \le$ BA $\le$ MBA$_1$
- None of these
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The Correct Option is B
Solution and Explanation
Given:
$\text{BA} = \frac{r_1 + r_2}{n_1}$,
$\text{MBA}_1 = \frac{r_1}{n_1} + \frac{n_2}{n_1} \max\left[0, \frac{r_2}{n_2} - \frac{r_1}{n_1}\right]$,
$\text{MBA}_2 = \frac{r_1 + r_2}{n_1 + n_2}$.
From structure:
- BA uses all runs divided by completed innings only, hence BA $\ge$ MBA$_2$ (since MBA$_2$ divides by total innings, which is larger).
- MBA$_1$ is constructed to be at least as large as BA, since the adjustment term is non-negative.
Therefore BA $\le$ MBA$_2 \le$ MBA$_1$ holds.
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Question: 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:
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:
Show Hint
Incomplete innings increase BA because denominator stays the same, but effect on other metrics requires more data.
Updated On: Aug 4, 2025
- BA and MBA$_1$ will both increase
- BA will increase and MBA$_1$ will decrease
- BA will increase and not enough data is available to assess change in MBA$_1$ and MBA$_2$
- None of these
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The Correct Option is C
Solution and Explanation
Initially no incomplete innings: BA = $\frac{r_1}{n_1} = 50$. Adding an incomplete innings with $r_2 = 45$, BA formula $\frac{r_1 + r_2}{n_1}$ increases numerator without changing $n_1$, so BA increases. However, MBA$_1$ depends on comparison between $\frac{r_2}{n_2}$ and $\frac{r_1}{n_1}$, which needs $n_2$ and prior $r_2$ values. MBA$_2$ also depends on $n_2$ and may increase or decrease. Hence, we cannot conclude changes in MBA$_1$ or MBA$_2$.
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