1. **Problem Statement:** Calculate the arithmetic mean and geometric mean of pollutant concentrations 20, 30, 50, and 120 micrograms per cubic meter, then determine which mean provides a more balanced estimate. 2. **Arithmetic Mean Formula:** $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$ This is the sum of all observations divided by the number of observations. 3. **Geometric Mean Formula:** $$\bar{x}_g = \left( \prod_{i=1}^n x_i \right)^{\frac{1}{n}}$$ This is the nth root of the product of all observations. 4. **Calculate Arithmetic Mean:** $$\bar{x} = \frac{20 + 30 + 50 + 120}{4} = \frac{220}{4} = 55$$ 5. **Calculate Geometric Mean:** First, calculate the product: $$20 \times 30 \times 50 \times 120 = 3,600,000$$ Then take the 4th root: $$\bar{x}_g = (3,600,000)^{\frac{1}{4}} = \sqrt{\sqrt{3,600,000}}$$ Calculate stepwise: $$\sqrt{3,600,000} \approx 1897.37$$ $$\sqrt{1897.37} \approx 43.56$$ So, $$\bar{x}_g \approx 43.56$$ 6. **Interpretation:** The arithmetic mean is 55, while the geometric mean is approximately 43.56. 7. **Which Mean is More Balanced?** The geometric mean is more balanced when data involves rates, ratios, or skewed distributions because it reduces the impact of very high or low values. Here, the high value 120 skews the arithmetic mean upward, while the geometric mean provides a central tendency less influenced by this outlier. **Final answers:** - Arithmetic mean = 55 - Geometric mean ≈ 43.56 - Geometric mean provides a more balanced estimate for pollutant levels across locations.