This question asks us to find the variance of a frequency distribution given its mean and coefficient of variation. Let's break down how to solve this using the definitions and formulas related to these statistical measures.

We are given:

• Mean (\(\bar{x}\)) = 100
• Coefficient of Variation (CV) = 45%

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

Understanding Coefficient of Variation in Statistics

The Coefficient of Variation (CV) is a measure of relative variability. It expresses the standard deviation as a percentage of the mean. This is useful for comparing the degree of variation between data sets, even if their means are drastically different. The formula for the Coefficient of Variation is:

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

We can represent this using symbols:

\(\text{CV} = \left( \frac{\sigma}{\bar{x}} \right) \times 100\%\)

where \(\sigma\) is the standard deviation and \(\bar{x}\) is the mean.

Calculating Standard Deviation from CV and Mean

We are given CV = 45% and \(\bar{x}\) = 100. We can plug these values into the formula and solve for the standard deviation (\(\sigma\)).

\(45\% = \left( \frac{\sigma}{100} \right) \times 100\%\)

To solve for \(\sigma\), first convert the percentage to a decimal:

\(0.45 = \frac{\sigma}{100}\)

Now, multiply both sides by 100:

\(0.45 \times 100 = \sigma\)

\(45 = \sigma\)

So, the standard deviation is 45.

Calculating Variance from Standard Deviation

The variance (\(\sigma^2\)) is the square of the standard deviation (\(\sigma\)). It measures how spread out the data is from the mean. The relationship is straightforward:

\(\text{Variance} = (\text{Standard Deviation})^2\)

Using symbols:

\(\sigma^2 = \sigma^2\)

Since we found that the standard deviation (\(\sigma\)) is 45, we can now calculate the variance:

\(\sigma^2 = 45^2\)

\(\sigma^2 = 45 \times 45\)

Let's calculate \(45 \times 45\):

\(45 \times 45 = (40 + 5) \times (40 + 5) = 40 \times 40 + 40 \times 5 + 5 \times 40 + 5 \times 5\)

\(= 1600 + 200 + 200 + 25\)

\(= 1600 + 400 + 25\)

\(= 2025\)

Therefore, the variance is 2025.

Summary of Calculation Steps

1. Identify the given values: Mean (\(\bar{x}\)) = 100, Coefficient of Variation (CV) = 45%.
2. Use the CV formula: \(\text{CV} = (\sigma / \bar{x}) \times 100\%\) to find the Standard Deviation (\(\sigma\)).
3. Substitute values: \(45\% = (\sigma / 100) \times 100\%\).
4. Solve for \(\sigma\): \(0.45 = \sigma / 100 \implies \sigma = 45\).
5. Use the relationship between Variance and Standard Deviation: Variance (\(\sigma^2\)) = \(\sigma^2\).
6. Calculate Variance: \(\sigma^2 = 45^2 = 2025\).

The value of the variance is 2025.

Revision Table: Key Statistical Measures

Additional Information on Variability Measures

Standard deviation and variance are both measures of the dispersion or spread of a data set. A higher standard deviation or variance indicates that the data points are more spread out from the mean, while a lower value indicates that they are clustered closer to the mean.

• Standard Deviation is often preferred because it is in the same units as the original data, making it easier to interpret.
• Variance is important in many statistical tests and theories, including analysis of variance (ANOVA) and regression analysis.
• Coefficient of Variation is particularly useful when comparing variability between data sets that have different scales or units, or significantly different means. For instance, comparing the variability in heights of infants versus adults.

Understanding the relationships between these statistical measures is crucial for analyzing frequency distributions and interpreting data variability.