To find the coefficient of variation (CV) of a data set, we first need to calculate the mean ($\mu$) and the standard deviation ($\sigma$) of the data set. The formula for the coefficient of variation is given by:

$\text{CV} = \frac{\sigma}{\mu} \times 100$

Step 1: Calculate the Mean ($\mu$)

The mean $\mu$ of a data set is given by:

$\mu = \frac{\sum x_i}{N}$

where $x_i$ are the data points and $N$ is the number of data points.

Given data set: $253, 323, 350, 373, 402, 461, 461, 510, 559, 598$

$\mu = \frac{253 + 323 + 350 + 373 + 402 + 461 + 461 + 510 + 559 + 598}{10}$

Step 2: Calculate the Variance ($\sigma^2$) and Standard Deviation ($\sigma$)

The variance is given by:

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

The standard deviation $\sigma$ is the square root of the variance:

$\sigma = \sqrt{\sigma^2}$

Step 3: Calculate the Coefficient of Variation (CV)

$\text{CV} = \frac{\sigma}{\mu} \times 100$

Let's calculate these values step-by-step.### Results:

1. Mean ($\mu$): $429.0$
2. Standard Deviation ($\sigma$): $103.14455875129816$
3. Coefficient of Variation (CV): $24.043020687948292\%$

The coefficient of variation for the given data set is approximately $24.043020687948292\%$.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the importance of the coefficient of variation in statistics?
2. How does the coefficient of variation differ from standard deviation?
3. How would the coefficient of variation change if the dataset values were doubled?
4. Can the coefficient of variation be negative, and what does it imply if so?
5. How does one interpret a high or low coefficient of variation in practical scenarios?

Tip: The coefficient of variation is particularly useful when comparing the relative variability of data sets that have different units or widely different means.