To calculate the coefficient of variation (CV) for the data set $[5, 10, 15, 20, 25]$, we need the following steps:

Step 1: Calculate the Mean

The mean ($\mu$) is the average of the data set.

$\mu = \frac{5 + 10 + 15 + 20 + 25}{5} = \frac{75}{5} = 15$

Step 2: Calculate the Standard Deviation

The standard deviation ($\sigma$) measures the spread of the data. We can calculate it using the formula:

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

Where:

• $x_i$ are the individual data points,
• $\mu$ is the mean,
• $N$ is the number of data points.

For the given data:

$\text{For } x_1 = 5, (x_1 - \mu)^2 = (5 - 15)^2 = (-10)^2 = 100$ $\text{For } x_2 = 10, (x_2 - \mu)^2 = (10 - 15)^2 = (-5)^2 = 25$ $\text{For } x_3 = 15, (x_3 - \mu)^2 = (15 - 15)^2 = 0^2 = 0$ $\text{For } x_4 = 20, (x_4 - \mu)^2 = (20 - 15)^2 = 5^2 = 25$ $\text{For } x_5 = 25, (x_5 - \mu)^2 = (25 - 15)^2 = 10^2 = 100$

Sum of squared deviations: $100 + 25 + 0 + 25 + 100 = 250$

Now, divide by the number of data points $N = 5$:

$\frac{250}{5} = 50$

Take the square root to find the standard deviation:

$\sigma = \sqrt{50} \approx 7.071$

Step 3: Calculate the Coefficient of Variation

The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage:

$CV = \frac{\sigma}{\mu} \times 100 = \frac{7.071}{15} \times 100 \approx 47.05\%$

Final Answer:

The coefficient of variation for the data set $[5, 10, 15, 20, 25]$ is approximately 47.05%.

---

Would you like more details on any of the steps, or do you have further questions?

Here are some related questions:

1. How does the coefficient of variation help in comparing different data sets?
2. What happens to the coefficient of variation if the mean of a data set increases?
3. How do you interpret a high coefficient of variation in a data set?
4. How can you calculate the standard deviation for a sample instead of a population?
5. What other measures of dispersion are used alongside standard deviation and coefficient of variation?

Tip: The coefficient of variation is especially useful for comparing the relative variability between datasets with different units or scales!