Understanding Variability in Employee Salaries

Variability in salary refers to how much the individual salaries differ from the average salary within a group of employees. A higher variability means salaries are more spread out, while lower variability means salaries are clustered closer to the average.

Two common measures of variability are the standard deviation and the coefficient of variation.

• Standard Deviation: Measures the absolute spread of data points around the mean. It's in the same units as the data (in this case, dollars). A higher standard deviation indicates greater absolute variability.
• Coefficient of Variation (CV): Measures the relative spread of data points around the mean. It is the ratio of the standard deviation to the mean, usually expressed as a percentage. It is unitless and is particularly useful for comparing the variability of datasets that have different means or are measured in different units.

Analyzing Salary Data for Factory A and Factory B

Let's look at the data provided for both factories:

Comparing Absolute Variability using Standard Deviation

Comparing the standard deviations directly:

• Standard Deviation for Factory A = $5
• Standard Deviation for Factory B = $4.5

Since $5 > $4.5, Factory A has a greater standard deviation than Factory B. This means the absolute spread of salaries is larger in Factory A.

Comparing Relative Variability using Coefficient of Variation

To compare variability between groups with different average values (like Factory A and Factory B, which have different average salaries), the Coefficient of Variation (CV) is often a more appropriate measure. The formula for CV is:

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

Let's calculate the CV for each factory:

For Factory A:

\( \text{CV}_A = \left( \frac{\$5}{\$34.5} \right) \times 100\% \)

\( \text{CV}_A = \left( \frac{5}{34.5} \right) \times 100\% \)

\( \text{CV}_A \approx 0.1449 \times 100\% \)

\( \text{CV}_A \approx 14.49\% \)

For Factory B:

\( \text{CV}_B = \left( \frac{\$4.5}{\$28.5} \right) \times 100\% \)

\( \text{CV}_B = \left( \frac{4.5}{28.5} \right) \times 100\% \)

\( \text{CV}_B = \left( \frac{45}{285} \right) \times 100\% \)

\( \text{CV}_B = \left( \frac{9}{57} \right) \times 100\% \)

\( \text{CV}_B = \left( \frac{3}{19} \right) \times 100\% \)

\( \text{CV}_B \approx 0.1579 \times 100\% \)

\( \text{CV}_B \approx 15.79\% \)

Concluding Which Factory Has Greater Variability

Comparing the coefficients of variation:

• CV for Factory A \(\approx\) 14.49%
• CV for Factory B \(\approx\) 15.79%

Since \( \text{CV}_B \approx 15.79\% \) is greater than \( \text{CV}_A \approx 14.49\% \), Factory B has greater relative variability in paying individual salary compared to its average salary.

When comparing variability between two datasets with significantly different means, the coefficient of variation provides a better perspective on how spread out the data is relative to its average value. In this case, although Factory A has a higher absolute standard deviation, Factory B has a higher standard deviation relative to its lower average salary.

Therefore, Factory B has greater variability in paying individual salary when considering relative variability.

Revision Table: Key Statistical Concepts

Additional Information on Comparing Variability

Choosing between standard deviation and coefficient of variation depends on the context. If the goal is to understand the absolute range of deviation from the mean, standard deviation is suitable. However, if the goal is to compare the degree of variability relative to the average size of the values, especially when the averages are different, the coefficient of variation is the preferred measure. For instance, a standard deviation of $10 might be high for salaries averaging $100 but low for salaries averaging $10,000. The CV helps standardize this comparison.