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The standard deviation is a statistical concept that provides insight into the amount of variation or spread in a set of data values. It tells us how much the individual data points differ from the mean (average) of the data set. A smaller standard deviation indicates that the data points are closer to the mean, which means less variability. Conversely, a larger standard deviation signifies greater variation among the data points.

To calculate standard deviation, we use the formula: \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\). Here, \(x_i\) represents each individual data point, \(\bar{x}\) is the mean of the data, and \(n\) is the number of data points.

From the exercise, the quarter-mile times have a standard deviation of 0.056 minutes. This shows that these times are tightly clustered around the average time of 0.966 minutes. Meanwhile, the mile times have a standard deviation of 0.120 minutes, indicating slightly more spread or variability around their mean of 4.534 minutes.

The coefficient of variation (CV) is a useful statistical tool for comparing the relative variability of different data sets. It is defined as the standard deviation divided by the mean, expressed as a percentage. By using the CV, we can determine which data set has more variability relative to its mean.

The CV is calculated using the formula: \( \text{CV} = \left(\frac{s}{\bar{x}}\right) \times 100\% \), where \(s\) is the standard deviation and \(\bar{x}\) is the mean. This ratio tells us how much variability exists relative to the average value of the data set.

In our data, the quarter-mile times have a CV of 5.8%, while the mile times have a CV of 2.6%. This indicates that, in relative terms, the mile times demonstrate less variability compared to the quarter-mile times. Even though the absolute variability is larger for the mile times, when considering their average value, they show greater consistency.

Data variability refers to how much the data points in a set differ from each other and from the mean. It is an important aspect to understand because it affects how we interpret data.

When assessing variability, the standard deviation reveals the overall spread of the data, providing a straightforward way to see differences within a single data set. However, the coefficient of variation adjusts this spread for the scale or magnitude of the data, offering a normalized measure that allows for comparisons across different data sets or units.

In the context of the running times in the exercise, the variability is assessed using both standard deviation and the coefficient of variation. While the standard deviation gives us a sense of absolute differences, the CV helps us understand variability in relation to the mean. By employing multiple variability measures, we gain a more comprehensive view of the data, which is crucial for making informed conclusions like whether quarter-mile or mile times are genuinely more consistent.