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To solve the given tasks, I'll first extract the relevant data from the table and proceed with the required calculations.

### Step 1: Extract Data
- **Volume of water delivered (mL):** 50 mL for all trials.
- **Mass of beaker + water delivered (g):**
  - Trial 1: 96.190 g
  - Trial 2: 93.964 g
  - Trial 3: 92.850 g
- **Mass of water delivered (g):**
  - Trial 1: 48.445 g
  - Trial 2: 46.219 g
  - Trial 3: 45.105 g
- **Density of water (g/mL):**
  - Trial 1: 0.9689 g/mL
  - Trial 2: 0.9244 g/mL
  - Trial 3: 0.9021 g/mL

### Step 2: Calculating the Average Density
To find the average density:
\[
\text{Average density} = \frac{0.9689 + 0.9244 + 0.9021}{3} = \frac{2.7954}{3} = 0.9318 \text{ g/mL}
\]

### Step 3: Calculating the Standard Deviation
The standard deviation (σ) of the densities is calculated as:
\[
\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (x_i - \bar{x})^2}
\]
Where \(x_i\) are the individual densities, \( \bar{x} \) is the mean density, and \( N \) is the number of trials.

Plugging in the values:
\[
\sigma = \sqrt{\frac{1}{3-1} \left[(0.9689 - 0.9318)^2 + (0.9244 - 0.9318)^2 + (0.9021 - 0.9318)^2\right]}
\]
\[
\sigma = \sqrt{\frac{1}{2} \left[0.001372 + 0.000055 + 0.000876\right]}
\]
\[
\sigma = \sqrt{\frac{1}{2} \times 0.002303} = \sqrt{0.0011515} \approx 0.0339
\]

### Step 4: Calculating the 90% Confidence Limit for the Average Density
The 90% confidence interval can be calculated using:
\[
\text{Confidence limit} = \bar{x} \pm t \frac{\sigma}{\sqrt{N}}
\]
For a 90% confidence level and \( N = 3 \), the t-value (from the t-distribution table) is approximately 1.886.

Thus, the confidence limit is:
\[
0.9318 \pm 1.886 \times \frac{0.0339}{\sqrt{3}}
\]
\[
0.9318 \pm 1.886 \times 0.0196 = 0.9318 \pm 0.0370
\]
So, the 90% confidence interval is approximately \(0.8948\) to \(0.9688\) g/mL.

### Step 5: Calculating the Relative Error
The relative error can be calculated as:
\[
\text{Relative Error} = \frac{\text{Standard deviation}}{\text{Average density}} \times 100
\]
\[
\text{Relative Error} = \frac{0.0339}{0.9318} \times 100 \approx 3.64\%
\]

### Summary
1. **Average Density:** 0.9318 g/mL
2. **Standard Deviation:** 0.0339 g/mL
3. **90% Confidence Interval:** 0.8948 to 0.9688 g/mL
4. **Relative Error:** 3.64%

Would you like further details or explanations on any of these steps?

### Related Questions:
1. How does increasing the number of trials affect the accuracy of the average density?
2. What is the significance of using a t-value in calculating the confidence interval?
3. How would the results change if the volume of water delivered was different for each trial?
4. Why is it important to use the standard deviation when analyzing experimental data?
5. How can outliers affect the calculation of standard deviation and mean in this experiment?

### Tip:
When calculating confidence intervals, always ensure you have the correct t-value based on your sample size and desired confidence level.

The user is asked to calculate the average density, standard deviation of density, 90% confidence limit, and relative error based on trials of water density measurements.

This problem involves calculating the average density, standard deviation, 90% confidence limit, and relative error based on three trials of water measurements. Detailed calculations for each step, including key formulas like the standard deviation and confidence interval, are provided.

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