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.