1. Given the sample data taken from the brewery industry in ml: 5, 6, 7, 8, 9,
find the coefficient of variation

The **coefficient of variation (CV)** is a statistical measure of the relative variability of data. It is calculated as the ratio of the standard deviation to the mean, typically expressed as a percentage. The formula is:

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

### Step 1: Calculate the Mean
First, we calculate the mean of the sample data.

\[
\text{Mean} = \frac{5 + 6 + 7 + 8 + 9}{5} = \frac{35}{5} = 7
\]

### Step 2: Calculate the Standard Deviation
Next, we calculate the standard deviation. The formula for the sample standard deviation \(s\) is:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

where:
- \(x_i\) is each data point,
- \(\bar{x}\) is the mean of the data,
- \(n\) is the number of data points.

\[
s = \sqrt{\frac{(5-7)^2 + (6-7)^2 + (7-7)^2 + (8-7)^2 + (9-7)^2}{5-1}} 
\]
\[
s = \sqrt{\frac{(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2}{4}} 
\]
\[
s = \sqrt{\frac{4 + 1 + 0 + 1 + 4}{4}} 
\]
\[
s = \sqrt{\frac{10}{4}} = \sqrt{2.5} \approx 1.58
\]

### Step 3: Calculate the Coefficient of Variation
Finally, we calculate the coefficient of variation.

\[
\text{CV} = \left(\frac{1.58}{7}\right) \times 100\% \approx 22.57\%
\]

### Final Answer:
The coefficient of variation is approximately **22.57%**.

Do you want more details or have any questions?

Here are 5 related questions:
1. How is the coefficient of variation used in comparing datasets?
2. What does a higher coefficient of variation indicate about a dataset?
3. How would the CV change if the dataset had a different mean but the same standard deviation?
4. Can the coefficient of variation be used for datasets with negative values?
5. What are some limitations of using the coefficient of variation?

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

Learn how to calculate the coefficient of variation (CV) for a sample data set from the brewery industry. Understand the statistical measure of relative variability using the formula and step-by-step calculation. Suitable for advanced high school students and beginners in statistics.

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