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To calculate the mean, also known as the average, you need to add up all the values of a dataset and then divide by the number of values. This provides a central point that represents all the data. For the given dataset, which includes the salt content percentages for 10 frankfurters, we calculate the mean by:

• First, summing all the salt content percentages: 2.26 + 2.11 + 1.64 + 1.17 + 1.64 + 2.36 + 1.70 + 2.10 + 2.19 + 2.40.
• Next, divide this sum by the total number of frankfurters, which in this case is 10.

You will have: \[\mu = \frac{2.26 + 2.11 + 1.64 + 1.17 + 1.64 + 2.36 + 1.70 + 2.10 + 2.19 + 2.40}{10}\]This results in the mean salt content for the frankfurters. The mean gives a good central value that summarizes the dataset, providing a snapshot of the typical salt content for the frankfurters.

Variance measures how much the numbers in a dataset differ from the mean. It gives us an idea of the spread or dispersion of the data. To calculate variance, follow these steps:

• Subtract the mean from each data point. This gives you how far each point is from the mean.
• Square each of these differences. This ensures all differences are positive and larger differences weigh more than smaller ones.
• Sum all the squared differences.
• Divide the total by the number of data points, which is 10 in this example.

The formula looks like this:\[\sigma^{2} = \frac{\sum (x_{i} - \mu)^{2}}{n}\]Where \(x_{i}\) represents each data point and \(\mu\) is the mean. The result will give you the variance of the salt content. A higher variance means more variability, while a lower variance indicates the data points are closer to the mean.

The standard deviation is a measure of how spread out the numbers in your dataset are. It is simply the square root of the variance, making it a very useful tool. Since variance gives squared deviations, taking the square root brings it back to the original units of measurement, which are the same as the data.To compute the standard deviation:

• Calculate the variance as explained previously.
• Take the square root of the variance.

The formula is:\[\sigma = \sqrt{\sigma^{2}}\]Now, about unbiasedness: Estimations can be biased, especially in small samples like our 10 frankfurters. Typically, to reduce bias in a small sample, the variance formula divides by \(n-1\) instead of \(n\) due to Bessel's correction. This is not applied in our current calculation, meaning the estimator we used might intentionally underestimate the population standard deviation slightly. As your dataset grows larger, this bias becomes negligible.