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The standard deviation is a crucial statistical measure that helps us understand how data points are spread out from the mean in a dataset. It is denoted by the symbol \(s\) and is calculated as the square root of the variance. In simple terms, if the standard deviation is small, the data points tend to be very close to the mean, indicating less variability. Conversely, a large standard deviation suggests that the data points are spread out over a wider range.
In our exercise, the standard deviation is given as 12, which means, on average, the data points in the sample stray from the mean, or central value of 74, by about 12 units. This measure is key in determining the intervals around the mean that encompass a given percentage of observations, especially when using Chebyshev's theorem.

The mean is often referred to as the average and is a fundamental measure of central tendency in statistics. It is calculated by summing up all data points in a dataset and then dividing by the number of data points.
In our exercise, the mean of the sample of observations is given as 74. This value serves as the center point for our intervals when applying Chebyshev's theorem. Understanding the mean is essential because it provides a baseline from which the standard deviation and statistical intervals are measured.

Statistical intervals are ranges designed to contain a specific portion of data points from a dataset. In the context of Chebyshev's theorem, these intervals are centered around the mean and extend a certain number of standard deviations on either side.
For example, the interval \(\bar{x} \pm 2s\) signifies that data is captured within two standard deviations from the mean. In our exercise, with \(\bar{x} = 74\) and \(s = 12\), this interval translates to \([74 - 2(12), 74 + 2(12)]\), which is [50, 98]. Statistical intervals are very effective in estimating how spread out the data is around the mean without knowing the exact distribution of the data.

Chebyshev's theorem provides a way to estimate the minimum percentage of observations that lie within given intervals around the mean, regardless of the distribution of data. The theorem states that for any dataset, at least \(1 - \frac{1}{k^2}\) of the data points must lie within \(k\) standard deviations from the mean.
This is powerful because it holds for any data distribution, whether it's a normal distribution or not. In our exercise, we calculated the minimum percentage of data points within the intervals of \(\pm 2s\), \(\pm 2.5s\), and \(\pm 3s\), yielding minimum percentages of 75%, 84%, and 89%, respectively. Understanding this theorem allows analysts to make broad predictions about data distribution without detailed insights into actual data movements.