The problem involves determining statistical values (mean, variance, and standard deviation) for a series of tape measurements.

Data Provided:

The following measurements were made:

• 1000.58 m
• 1000.46 m
• 1000.38 m
• 1000.40 m

The tasks are:

1. Mean of the measurements ($\mu$)
2. Variance of the measurements ($\sigma^2$)
3. Standard deviation of the measurements ($\sigma$)

Let's calculate the required values:

1. Mean ($\mu$)

The mean is given by: $\mu = \frac{1}{N} \sum_{i=1}^N x_i$ Where:

• $N$ is the number of measurements (here $N = 4$)
• $x_i$ are the individual measurements.

2. Variance ($\sigma^2$)

The variance is given by: $\sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2$

3. Standard deviation ($\sigma$)

The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^2}$

I will now compute these values.Here are the results for the given measurements:

1. Mean ($\mu$): 1000.455 m
2. Variance ($\sigma^2$): 0.006075 $m^2$
3. Standard deviation ($\sigma$): 0.07794 m

Would you like a detailed explanation of these results or any specific part of the process?

Here are 5 related questions for further exploration:

1. How would the results change if another measurement was added to the data set?
2. What is the significance of the standard deviation in these measurements?
3. How can we interpret the variance in terms of measurement consistency?
4. What happens to the variance if all measurements were exactly the same?
5. How would you calculate the confidence interval for this data set?

Tip: The standard deviation provides insight into the spread of the data. A smaller standard deviation means the measurements are closer to the mean, indicating higher precision.