Find the range, the standard deviation, and the variance for the given samples. Round non-integer results to the nearest tenth. 51, 72, 77, 78, 93, 92, 88, 100, 56 range standard deviation variance

Certainly! To calculate the range, standard deviation, and variance for the given sample data, we'll follow these steps: 1. **Range**: The range is the difference between the highest and lowest values. 2. **Standard DeviaLet's calculate each of these for your data set: `51, 72, 77, 78, 93, 92, 88, 100, 56`. **Range**: \text{Range} = \text{Max} - \text{Min} = 100 - 51 = 49 **Mean**: \text{Mean} = \frac{\sum \text{values}}{N} = \frac{51 + 72 + 77 + 78 + 93 + 92 + 88 + 100 + 56}{9} = \frac{707}{9} \approx 78.6 **Variance**: \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} = \frac{(51-78.6)^2 + (72-78.6)^2 + \ldots + (100-78.6)^2}{9} = \frac{767.4}{9} \approx 85.3 **Standard Deviation**: \text{SD} = \sqrt{\text{Variance}} = \sqrt{85.3} \approx 9.2 So, the **range** is 49, the **standard deviation** is approximately 9.2, and the **variance** is approximately 85.3. These results are rounded to the nearest tenth as requested.