Understanding Consistency in Batting Performance

When we want to compare the consistency of different batsmen, especially when their average scores are different, using just the standard deviation can be misleading. A batsman with a higher average might naturally have a higher standard deviation of scores just because the numbers are larger. The coefficient of variation (CV) is a better measure for comparison in such cases because it is a relative measure of variability. It expresses the standard deviation as a percentage of the mean.

Calculating the Coefficient of Variation (CV)

The formula for the coefficient of variation is:

\( \text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\% \)

A lower coefficient of variation indicates less variability relative to the mean, meaning the performance is more consistent. A higher coefficient of variation indicates greater variability, meaning the performance is less consistent.

Calculating CV for Each Batsman

Let's calculate the coefficient of variation for each batsman A, B, and C using the given mean and standard deviation of their runs scored over 9 consecutive matches.

Batsman A:

• Mean (\(\bar{x}_A\)): 40
• Standard Deviation (\(s_A\)): 4

CV for Batsman A:

\( \text{CV}_A = \left( \frac{s_A}{\bar{x}_A} \right) \times 100\% = \left( \frac{4}{40} \right) \times 100\% \)

\( \text{CV}_A = (0.1) \times 100\% = 10\% \)

Batsman B:

• Mean (\(\bar{x}_B\)): 64
• Standard Deviation (\(s_B\)): 8

CV for Batsman B:

\( \text{CV}_B = \left( \frac{s_B}{\bar{x}_B} \right) \times 100\% = \left( \frac{8}{64} \right) \times 100\% \)

\( \text{CV}_B = (0.125) \times 100\% = 12.5\% \)

Batsman C:

• Mean (\(\bar{x}_C\)): 72
• Standard Deviation (\(s_C\)): 12

CV for Batsman C:

\( \text{CV}_C = \left( \frac{s_C}{\bar{x}_C} \right) \times 100\% = \left( \frac{12}{72} \right) \times 100\% \)

\( \text{CV}_C = \left( \frac{1}{6} \right) \times 100\% \approx 0.1667 \times 100\% \approx 16.67\% \)

Comparing the Coefficients of Variation

Let's put the calculated CV values in a table:

To determine the most consistent batsman using the coefficient of variation, we look for the lowest CV value.

• CV(A) = 10%
• CV(B) = 12.5%
• CV(C) \(\approx\) 16.67%

Comparing these values, we see that 10% is the lowest value among 10%, 12.5%, and 16.67%. This means Batsman A has the lowest coefficient of variation.

Identifying the Most Consistent Batsman

Since the coefficient of variation is a measure of relative variability, and a lower value indicates higher consistency, Batsman A is the most consistent among the three.

Evaluating the Options

Let's examine each given option based on our calculations:

1. A is the most consistent: Our calculations show that CV(A) is the lowest (10%), making A the most consistent. This statement is correct.
2. C is the most consistent: Our calculations show that CV(C) is the highest (\(\approx\) 16.67%), making C the least consistent. This statement is incorrect.
3. B is more consistent than A: Our calculations show that CV(B) (12.5%) is greater than CV(A) (10%). This means A is more consistent than B. This statement is incorrect.
4. Both (2) and (3): Since options (2) and (3) are both incorrect, this combined option is also incorrect.

Based on the analysis, the correct statement is that A is the most consistent.

Revision Table: Comparing Batsman Consistency

Additional Information: Measures of Variability

Standard deviation and coefficient of variation are both measures of variability or dispersion, but they provide different types of information.

• Standard Deviation: This is an absolute measure of dispersion. It tells you the average distance of each data point from the mean in the original units of the data. A higher standard deviation means the data points are more spread out. However, comparing standard deviations directly for datasets with very different means can be misleading. For example, a standard deviation of 10 might indicate high variability for data with a mean of 20 but relatively low variability for data with a mean of 1000.
• Coefficient of Variation (CV): This is a relative measure of dispersion. It expresses the standard deviation as a percentage of the mean. Because it's relative to the mean, it is useful for comparing the variability of two or more datasets that have different units of measurement or significantly different means. A lower CV indicates that the data points are clustered more tightly around the mean relative to the size of the mean itself.

In the context of comparing batsmen's consistency, where their average scores (means) are different, the coefficient of variation is the appropriate tool as specified in the question, providing a standardized way to assess which batsman's performance is less variable relative to their average score.