To calculate the coefficient of variation (CV), we need to use the formula:

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

You have provided a data set:

71, 218, 72, 223, 77, 220, 67, 207, 71, 189, 80, 180, 80, 181, 72, 207, 77, 176, 68, 177, 69, 206, 80, 212

Let's go through the steps:

Step 1: Calculate the Mean (μ)

The mean is the average of all the data points. We add up all the values and divide by the number of data points.

$\text{Mean} = \frac{\sum \text{Data Points}}{n}$

Where $n$ is the number of data points. First, let me calculate this.

Step 2: Calculate the Standard Deviation (σ)

The standard deviation measures the spread of data around the mean. The formula for standard deviation is:

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

Where:

• $x_i$ is each data point
• $\mu$ is the mean
• $n$ is the number of data points

I'll calculate both the mean and the standard deviation, and then use the CV formula. numpy as np

Given data set

data = [71, 218, 72, 223, 77, 220, 67, 207, 71, 189, 80, 180, 80, 181, 72, 207, 77, 176, 68, 177, 69, 206, 80, 212]

Calculate mean and standard deviation

mean = np.mean(data) std_dev = np.std(data)

Calculate the Coefficient of Variation (CV)

CV = (std_dev / mean) * 100 mean, std_dev, CVThe calculations give us the following:

• Mean (μ): 136.67 (rounded to two decimal places)
• Standard Deviation (σ): 64.24 (rounded to two decimal places)
• Coefficient of Variation (CV): 47.00%

Conclusion:

The coefficient of variation for this data set is 47.00%. This indicates that the standard deviation is 47% of the mean, suggesting a relatively high variation in the data compared to the mean. Since CV is unitless, you can compare this data set to others with different units of measurement.

Would you like to dive deeper into the concept of CV or how to interpret it further? Feel free to ask!

Here are some related questions:

1. How can the coefficient of variation help in comparing different data sets?
2. What are the limitations of using the coefficient of variation?
3. How does the coefficient of variation behave in data with a small or large standard deviation?
4. Can the coefficient of variation be used for all types of distributions?
5. How does changing the mean of a data set affect the coefficient of variation?

Tip: The CV is particularly useful in comparing data sets that are on different scales or have different units of measurement, as it standardizes the variation relative to the mean.