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The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It's a critical concept in statistics, especially when dealing with a normal distribution. The formula for calculating the z-score is \[ z = \frac{x - \mu}{\sigma} \]where:

• \( x \) is the value from the sample set,
• \( \mu \) is the mean of the data set,
• \( \sigma \) is the standard deviation.

A z-score tells us how many standard deviations an element is from the mean. A z-score can be positive or negative, indicating that the value is either above or below the mean, respectively. For example, Dolores's z-score calculation in the exercise shows us whether the price she paid is more or less than what is considered "average" for the Rhino car.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It gives insight into how much individual measurements typically deviate from the mean. In essence, it provides a summary of how data points spread out over a distribution and is calculated using the formula:\[ \sigma = \sqrt{\frac{1}{n} \sum (x_i - \mu)^2} \]where:

• \( x_i \) is each individual value,
• \( \mu \) is the mean of all values,
• \( n \) is the total number of values.

In the context of the exercise, a standard deviation of \( \\(350 \) for the price of Rhino cars shows how much customers typically pay above or below the mean price of \( \\)19,800 \). A smaller standard deviation would mean prices are closely clustered around the mean, whereas a larger standard deviation indicates prices vary more widely.

Percentiles are statistical measures that represent the rank of a particular value within a given data set. In the context of a normal distribution, percentiles help us understand what proportion of the data falls below a certain point. When you calculate the z-score, you can use it to find the percentile from a standard normal distribution table, which helps to interpret the overall standing of a specific value. For Dolores, once her z-score is determined, the corresponding percentile indicates what percentage of customers paid less than she did for their car. Thus, if her score falls at the 25th percentile, it means 25% of other customers paid less than she did. For Cuthbert, calculating percentiles enables us to determine what percentage of customers paid more by subtracting his percentile from 100%. This process provides insight into how their payments compare within the population of all customers.

The mean is a fundamental statistical measure that acts as a "central" point in a data set. For data that fits a normal distribution, the mean is particularly informative since it represents the most common value, around which all other data points are symmetrically distributed. It is calculated using the following formula:\[ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i \]Where:

• \( x_i \) are each of the values in the set,
• \( n \) is the number of values.

In the car dealership example, the mean price of \( \$19,800 \) indicates the typical selling price of the Rhino car model. The mean acts as the benchmark against which individual sales are compared, as seen through the calculation of z-scores. Understanding the concept of mean helps contextualize how individual sales, like those of Dolores and Cuthbert, fit into the larger picture of overall sales trends.