The table you've provided includes information for calculating the average density, standard deviation, 90% confidence limit, and relative error based on three trials. Let's break down the required calculations.

Data from the Table:

• Trial 1:

  • Volume of water: 12.00 mL
  • Mass of flask + water: 104.772 g
  • Mass of water delivered: 10.719 g
  • Density: 0.893 g/mL

• Trial 2:

  • Volume of water: 12.00 mL
  • Mass of flask + water: 116.886 g
  • Mass of water delivered: 22.833 g
  • Density: 1.903 g/mL

• Trial 3:

  • Volume of water: 12.00 mL
  • Mass of flask + water: 129.658 g
  • Mass of water delivered: 35.605 g
  • Density: 2.967 g/mL

1. Average Density (g/mL):

To calculate the average density, sum the densities from each trial and divide by the number of trials. $\text{Average Density} = \frac{(0.893 + 1.903 + 2.967)}{3}$ $\text{Average Density} = \frac{5.763}{3} = 1.921 \text{ g/mL}$

2. Standard Deviation of Density:

The standard deviation ($\sigma$) is calculated using: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n - 1}}$ Where $x_i$ are the individual densities, $\mu$ is the mean, and $n$ is the number of trials.

First, calculate the squared differences:

• $(0.893 - 1.921)^2 = 1.058^2 = 1.119364$
• $(1.903 - 1.921)^2 = 0.018^2 = 0.000324$
• $(2.967 - 1.921)^2 = 1.046^2 = 1.094116$

Now sum the squared differences: $\sum = 1.119364 + 0.000324 + 1.094116 = 2.213804$ Finally, divide by $n-1$ and take the square root: $\sigma = \sqrt{\frac{2.213804}{2}} = \sqrt{1.106902} = 1.052 \text{ g/mL}$

3. 90% Confidence Limit for the Average Density:

To calculate the 90% confidence interval, use: [ \text{Confidence Interval} = \mu \pm t_{\alpha/2}