1. **Problem:** Calculate the mean and median of the corn production data (in millions of bushels) for seven states: 27, 692, 700, 342, 191, 148, 311. 2. **Formula and rules:** - Mean is the sum of all data points divided by the number of points: $$\text{Mean} = \frac{\sum x_i}{n}$$ - Median is the middle value when data is sorted. If $n$ is odd, median is the middle number; if even, average of two middle numbers. 3. **Calculate mean:** $$\sum x_i = 27 + 692 + 700 + 342 + 191 + 148 + 311 = 2411$$ Number of data points $n=7$ $$\text{Mean} = \frac{2411}{7} \approx 344.43$$ 4. **Calculate median:** Sort data: 27, 148, 191, 311, 342, 692, 700 Middle value (4th) is 311 $$\text{Median} = 311$$ 5. **Interpretation:** The mean is $344.43$ and the median is $311$. 6. **Better summary measure:** Since the data has extreme values (692, 700) much larger than others, the median is a better measure of central tendency because it is less affected by outliers. 7. **Calculate variance, standard deviation, and coefficient of variation:** - Variance formula: $$s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$$ - Standard deviation: $$s = \sqrt{s^2}$$ - Coefficient of variation: $$CV = \frac{s}{\bar{x}} \times 100\%$$ Calculate squared deviations: $(27 - 344.43)^2 = 101,011.18$ $(692 - 344.43)^2 = 120,011.18$ $(700 - 344.43)^2 = 126,916.18$ $(342 - 344.43)^2 = 5.90$ $(191 - 344.43)^2 = 23,580.18$ $(148 - 344.43)^2 = 38,702.18$ $(311 - 344.43)^2 = 1,113.18$ Sum of squared deviations = 411,340.96 Variance: $$s^2 = \frac{411,340.96}{6} \approx 68,556.83$$ Standard deviation: $$s = \sqrt{68,556.83} \approx 261.87$$ Coefficient of variation: $$CV = \frac{261.87}{344.43} \times 100 \approx 76.03\%$$ **Final answers:** - Mean = $344.43$ - Median = $311$ - Variance = $68,556.83$ - Standard deviation = $261.87$ - Coefficient of variation = $76.03\%$