1. **Problem Statement:** Calculate the arithmetic mean and geometric mean of pollutant concentrations 20, 30, 50, and 120 micrograms per cubic meter. 2. **Arithmetic Mean Formula:** The arithmetic mean $\bar{x}$ of $n$ observations $x_1, x_2, ..., x_n$ is given by: $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$ It is the sum of all values divided by the number of values. 3. **Calculate Arithmetic Mean:** $$\bar{x} = \frac{20 + 30 + 50 + 120}{4} = \frac{220}{4} = 55$$ So, the arithmetic mean concentration is 55 micrograms per cubic meter. 4. **Geometric Mean Formula:** The geometric mean $\bar{x}_g$ of $n$ observations $x_1, x_2, ..., x_n$ is: $$\bar{x}_g = \sqrt[n]{x_1 \times x_2 \times ... \times x_n}$$ It is the $n$th root of the product of all values. 5. **Calculate Geometric Mean:** Calculate the product: $$20 \times 30 \times 50 \times 120 = 3,600,000$$ Then take the 4th root: $$\bar{x}_g = \sqrt[4]{3,600,000} = (3,600,000)^{\frac{1}{4}}$$ Calculate: $$\bar{x}_g \approx 43.45$$ So, the geometric mean concentration is approximately 43.45 micrograms per cubic meter. 6. **Comparison and Justification:** The arithmetic mean (55) is higher than the geometric mean (43.45) because the arithmetic mean is influenced more by the large value 120. The geometric mean reduces the impact of very high or low values and provides a more balanced estimate when data are multiplicative or skewed. **Final answers:** - Arithmetic mean = 55 - Geometric mean $\approx$ 43.45 - The geometric mean provides a more balanced estimate of pollutant levels across locations because it lessens the effect of extreme values and better represents multiplicative relationships.