(a) Range:

The range is calculated as the difference between the largest and smallest values in the dataset.

Given values:
84.38, 85.74, 85.07, 85.92, 85.74, 84.56, 85.95, 85.77, 85.01

• Maximum value: 85.95
• Minimum value: 84.38

So, the range = 85.95 - 84.38 = 1.57

(b) Standard Deviation:

The standard deviation for a sample is given by the formula:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

Where:

• $x_i$ are the data points,
• $\bar{x}$ is the sample mean,
• $n$ is the sample size.

Step 1: Calculate the mean ($\bar{x}$).

$\bar{x} = \frac{84.38 + 85.74 + 85.07 + 85.92 + 85.74 + 84.56 + 85.95 + 85.77 + 85.01}{9} = \frac{769.14}{9} = 85.46$

Step 2: Calculate the squared deviations from the mean.

$(x_i - \bar{x})^2$

• (84.38 - 85.46)² = 1.1681
• (85.74 - 85.46)² = 0.0774
• (85.07 - 85.46)² = 0.1521
• (85.92 - 85.46)² = 0.2116
• (85.74 - 85.46)² = 0.0774