Calculating Variance of Cricket Scores from Coefficient of Variation and Mean

The question asks us to find the variance of cricket scores given the coefficient of variation and the mean score. We are provided with the following information:

• Coefficient of Variation (CV) = 16
• Mean ($\bar{x}$) = 25

We need to find the variance ($\sigma^2$).

Understanding Coefficient of Variation, Mean, and Variance

Let's quickly define the terms involved:

• Mean ($\bar{x}$): This is the average score of the players.
• Standard Deviation ($\sigma$): This measures the dispersion or spread of the scores around the mean. A higher standard deviation means the scores are more spread out.
• Variance ($\sigma^2$): This is the square of the standard deviation. It also measures dispersion but is in squared units of the original data.
• Coefficient of Variation (CV): This is a standardized measure of dispersion. It expresses the standard deviation as a percentage of the mean. It is useful for comparing the degree of variation between data sets, even if their means are drastically different. The formula is commonly given as:
  $$ \text{CV} = \frac{\sigma}{\bar{x}} \times 100\% $$ Sometimes, the CV is given as a decimal or fraction, in which case the formula used is simply:
  $$ \text{CV} = \frac{\sigma}{\bar{x}} $$ Given the options are numerical values derived from standard deviation squared, it is most likely that the CV given as '16' in the question represents 16%, or the formula used is $16 = \frac{\sigma}{25} \times 100$. Let's use the latter interpretation as it fits the options.

Step-by-Step Calculation of Variance

We will use the formula for the coefficient of variation to first find the standard deviation ($\sigma$), and then square it to find the variance ($\sigma^2$).

Given:

• CV = 16
• Mean ($\bar{x}$) = 25

Using the formula $\text{CV} = \frac{\sigma}{\bar{x}} \times 100$:

Substitute the given values into the formula:

$$ 16 = \frac{\sigma}{25} \times 100 $$

Now, we need to solve for $\sigma$. Divide both sides by 100:

$$ \frac{16}{100} = \frac{\sigma}{25} $$ $$ 0.16 = \frac{\sigma}{25} $$

Multiply both sides by 25 to isolate $\sigma$:

$$ \sigma = 0.16 \times 25 $$ $$ \sigma = 4 $$

So, the standard deviation ($\sigma$) is 4.

Now, we need to find the variance ($\sigma^2$). Variance is the square of the standard deviation:

$$ \text{Variance} = \sigma^2 $$ $$ \text{Variance} = 4^2 $$ $$ \text{Variance} = 16 $$

Thus, the variance of the cricket scores is 16.

Summary of Results

The variance of the cricket scores is 16.

Revision Table: Cricket Scores Statistics

Additional Information on Variability and Dispersion in Cricket Scores

In statistics, measures of dispersion tell us how spread out a set of data is. For cricket scores, this could indicate how consistent the players' scores are.

• Standard Deviation and Variance: These are absolute measures of dispersion. A standard deviation of 4 for scores with a mean of 25 tells us the typical deviation from the average score is 4 runs. Variance is useful in many statistical calculations but is less intuitive than standard deviation because it's in squared units.
• Coefficient of Variation: This is a relative measure. A CV of 16% (assuming 16 represents 16%) means the standard deviation is 16% of the mean. This allows comparison. For example, a player with an average of 50 and a standard deviation of 8 would have a CV of $(\frac{8}{50}) \times 100\% = 16\%$. This player has the same relative variability as the one in the question, even though their average score is higher. A lower CV generally indicates more consistency relative to the mean.
• These statistical measures are used by analysts and coaches to assess player performance, consistency, and compare different players or teams.