Consider the data sample:
10; 12; 15; 18; 18; 18; 20; 20; 24.
What is the coefficient of variation for this sample?

To calculate the coefficient of variation (CV) for the given data sample, follow these steps:

1. **Find the Mean (μ):**
   \[
   \text{Mean} = \frac{\sum \text{data points}}{\text{number of data points}}
   \]
   \[
   \text{Mean} = \frac{10 + 12 + 15 + 18 + 18 + 18 + 20 + 20 + 24}{9} = \frac{155}{9} \approx 17.22
   \]

2. **Calculate the Standard Deviation (σ):**
   \[
   \text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}
   \]
   where \( x_i \) represents each data point, \( \mu \) is the mean, and \( n \) is the number of data points.

   Let's compute the deviations:

   \[
   (10 - 17.22)^2 \approx 52.13
   \]
   \[
   (12 - 17.22)^2 \approx 27.22
   \]
   \[
   (15 - 17.22)^2 \approx 4.93
   \]
   \[
   (18 - 17.22)^2 \approx 0.61 \quad \text{(this is repeated three times since we have three 18s)}
   \]
   \[
   (20 - 17.22)^2 \approx 7.71 \quad \text{(this is repeated twice since we have two 20s)}
   \]
   \[
   (24 - 17.22)^2 \approx 45.85
   \]

   Summing these deviations:
   \[
   \sum (x_i - \mu)^2 \approx 52.13 + 27.22 + 4.93 + 3(0.61) + 2(7.71) + 45.85 \approx 148.98
   \]

   Now, divide by the number of data points (9):
   \[
   \text{Variance} = \frac{148.98}{9} \approx 16.55
   \]
   \[
   \text{Standard Deviation} = \sqrt{16.55} \approx 4.07
   \]

3. **Compute the Coefficient of Variation (CV):**
   \[
   \text{CV} = \frac{\sigma}{\mu} \times 100\%
   \]
   \[
   \text{CV} = \frac{4.07}{17.22} \times 100\% \approx 23.64\%
   \]

So, the coefficient of variation for the sample is approximately **23.64%**.

Would you like further details or have any questions?

Here are some related questions you might consider:
1. What is the significance of the coefficient of variation in statistical analysis?
2. How does the CV compare to the standard deviation in terms of relative variability?
3. Can the coefficient of variation be used for comparing datasets with different units?
4. How would the CV change if all the data points were doubled?
5. Is a higher or lower CV generally preferable in real-world data analysis?
6. What are the limitations of using the CV for data with a mean close to zero?
7. How would adding an outlier to this data sample affect the CV?
8. What other measures of relative dispersion are available besides the CV?

**Tip:** The coefficient of variation is particularly useful when comparing the variability of two datasets with different units or different means.

Learn how to calculate the coefficient of variation (CV) for a given data sample step-by-step. Includes detailed calculations of mean, standard deviation, and their implications. Suitable for advanced high school students and anyone interested in statistics.

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