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Standard deviation is a crucial concept in statistics that measures the extent of variation or dispersion in a set of values. In this exercise, the standard deviation of gasoline prices is given as \( \$0.06 \) per gallon. This tells us how much the price of gasoline can deviate from the mean price, on average. A smaller standard deviation means that the prices are closely packed around the mean, while a larger standard deviation indicates prices are spread out over a wider range.
To better grasp this, imagine we have a mean price of \( \$3.06 \) per gallon. If the prices at different stations deviate significantly from this value, the standard deviation tells us how typical or atypical these deviations are. The formula for calculating the standard deviation \(\sigma\) is:
\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \] where \( N \) is the number of observations, \( x_i \) represents each data point, and \( \mu \) is the mean.
Understanding standard deviation helps us determine the spread of gasoline prices around the mean price.

The mean price, or average price, is another fundamental concept in this exercise. The mean is calculated by summing up all individual prices and dividing it by the number of prices. In this problem, the mean price of gasoline was given as \( \$3.06 \) per gallon. This represents the central point around which the prices at different gasoline stations are distributed.
The formula for calculating the mean \( \mu \) is straightforward:
\[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \] where \( N \) is the total number of observations, and \( x_i \) is each individual observation.
Knowing the mean price is useful as it serves as the reference point for calculating the standard deviation and for applying Chebyshev's Theorem. For instance, when we say a gas station's price is within 2.5 standard deviations of the mean, we are measuring how far that price is from the average price of \( \$3.06 \).
Thus, mean price gives us a central figure, making it easier to understand and compare distributions of different prices.

To determine the minimum percentage of gasoline stations with prices within a certain range, we use Chebyshev's Theorem. This theorem provides a way to estimate the spread of data points in any distribution, regardless of its shape. According to Chebyshev's Theorem, at least \( \left(1 - \frac{1}{k^2}\right) \times 100\% \) of data values lie within \( k \) standard deviations from the mean.
For example, in step 2 of our solution, we calculate the minimum percentage of gasoline stations with prices within 3 standard deviations of the mean: \[ \left(1 - \frac{1}{3^2}\right) \times 100\% = 88.89\% \] This means at least 88.89% of gasoline stations had prices within 3 standard deviations of \( \$3.06 \).
Similarly, for gasoline prices between \( \$2.94 \) and \( \$3.18 \), which are within 2 standard deviations of the mean, Chebyshev's Theorem states that at least 75% of data values fall within this range: \[ \left(1 - \frac{1}{2^2}\right) \times 100\% = 75\% \] Thus, knowing how to apply Chebyshev's Theorem not only helps us understand the spread and concentration of data but also provides a useful tool for making informed decisions and predictions.