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Calculating the mean is the first step in many statistical analyses, providing the average value of a dataset. To find the mean of a set of numbers, add them all together and divide by the number of observations. For example, in our dataset about oxidation-induction times, we sum up each value provided and then divide this sum by the number of observations, which in this case is 19. The equation looks like this:\[\bar{x} = \frac{1}{19}(87 + 103 + 130 + \ldots + 129)\]Calculating this gives us a mean (\bar{x}) of 133.42 minutes. This mean represents the central tendency of our sample data and will be used in calculating both the variance and standard deviation.

Data reexpression involves transforming your dataset using a consistent mathematical operation. In our case, we need to reexpress the observation times from minutes to hours. Since there are 60 minutes in an hour, we simply divide each observation by 60. It’s important to understand how this affects calculations for variance and standard deviation. When reexpressing data, one must adjust the calculations accordingly. For a variance, which is a squared value, if each original observation is divided by a factor, i.e., 60, then the variance is divided by 60 squared. For standard deviation, which is not squared, we simply divide by 60. This understanding of data transformation allows us to apply results in different units without recalculating everything anew.

Sample variance describes how data points in a sample differ from the mean. It indicates the degree of spread in our data. Calculating it involves several steps:

• First, compute the deviation of each observation from the mean. Subtract the mean calculated earlier from each data point.
• Next, square each deviation. This eliminates negative values and emphasizes larger differences.
• Finally, sum these squared deviations and divide by \(n-1\), where \(n\) is the number of observations. The formula to calculate sample variance is:\[s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\]For our dataset, this computation results in a sample variance of 1304.58 minutes squared.

Variance is crucial for determining how varied or similar the dataset is to the mean, providing foundational understanding before calculating standard deviation.

Sample standard deviation provides insight into the amount of variation or dispersion in a dataset. A low standard deviation means data points are close to the mean, while a high standard deviation indicates more spread out data.To find the standard deviation, take the square root of the sample variance. Using our example's variance of 1304.58, the standard deviation is:\[s = \sqrt{1304.58} \approx 36.12\]This tells us that, on average, our data points deviate from the mean by about 36.12 minutes. This measure helps us understand data volatility and is often used to compare different datasets or to set expectations for future observations. When data is reexpressed, remember that standard deviation changes linearly with the transformation factor, in contrast to variance that changes quadratically.