Analyzing Wage Variability
In statistics, understanding the distribution of data is crucial for comparing different groups. When we look at two firms, simply comparing the average wage is often insufficient. We must look at the variability (or dispersion) of the wages to understand how consistent or 'uniform' the pay is across workers.
The Metric: Coefficient of Variation (CV)
To determine which firm has a more uniform distribution (or less variability), we use the Coefficient of Variation (CV). The CV is a normalized measure of dispersion:
$$CV = \frac{\sigma}{\bar{x}} \times 100$$
Where:
- $\sigma$ is the standard deviation (the square root of the variance).
- $\bar{x}$ is the mean (average wage).
A lower CV indicates a more uniform (consistent) distribution of data relative to its mean.
Step-by-Step Solution
Given Data:
- Firm A: $\bar{x}_A = 586$, $Variance_A = 81$
- Firm B: $\bar{x}_B = 575$, $Variance_B = 100$
Step 1: Calculate Standard Deviation ($\sigma$):
- $\sigma_A = \sqrt{81} = 9$
- $\sigma_B = \sqrt{100} = 10$
Step 2: Calculate CV for each firm:
- $CV_A = (9 / 586) \times 100 \approx 1.536\%$
- $CV_B = (10 / 575) \times 100 \approx 1.739\%$
Conclusion
Since $CV_A < CV_B$, Firm A has less relative variability, meaning its wage distribution is more uniform compared to Firm B.