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When you hear about the mean in statistics, think of it as the average value of a dataset. Here, the mean gross sales for companies in the city is $2.3 million. It's the central value around which all the company sales hover. This helps us understand where the center of the data lies.

The standard deviation is a measure of how spread out the numbers in a data set are. In this exercise, the standard deviation is $0.6 million, which tells us how much the individual sales figures tend to deviate from the mean. A smaller standard deviation indicates that the data points are close to the mean, while a larger one points to a wide variation around the mean.

Together, mean and standard deviation are crucial for understanding the data's overall behavior. They help us quantify how typical or unusual a particular data point is compared to the mean.

A probability distribution provides a statistical function that describes all the possible values and probabilities that a random variable can take within a given range. It paints the picture of how data is distributed across different possible outcomes.

Chebyshev's Theorem, which is used in this exercise, isn't associated with a normal distribution but gives us insight into how data is spread out regardless of the distribution shape. Chebyshev's Theorem specifically tells us that, for any dataset, the proportion of values that fall within a certain number of standard deviations from the mean will be at least a certain amount. This makes it a handy tool when the distribution is unknown or not perfectly normal.

In this context, it helps us find out the percentage of companies whose sales fall within specified ranges by calculating how far those ranges are from the mean in terms of standard deviations.

To use Chebyshev's Theorem effectively, you need to determine how many standard deviations away a given range is from the mean. This is done by calculating the number of standard deviations each boundary value of the range is from the mean.

For example, to find the percentage of companies with sales between \(1.1 and \)3.5 million, you calculate how many standard deviations these values are from the mean of \(2.3 million. Once you have these values (let's say they are 2 standard deviations from the mean), you can apply Chebyshev's Theorem. This theorem states that at least \(1 - \frac{1}{k^2}\) of the data falls within \(k\) standard deviations from the mean.

This allows us to state confidently, without assuming a normal distribution, what percentage of companies fall within each specified sales range. For the range \)1.1 to \(3.5 million, at least 75% of companies fall within this range; for \)0.8 to \(3.8 million, it's at least 84%; and for \)0.5 to $4.1 million, at least 89% of companies have sales in this range.