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To understand the concept of a z-score, think of it as a way to measure how far a particular data point is from the mean of a dataset, expressed in terms of standard deviations. This is particularly useful when dealing with data that follows a normal distribution, often represented as a bell curve.

To calculate a z-score, the formula used is:

• The z-score formula is: \[z = \frac{x - \mu}{\sigma}\] where:
  • \(x\) is the value you are examining,
  • \(\mu\) is the mean of the dataset, and
  • \(\sigma\) is the standard deviation.

For example, in our gasoline price scenario, to find the z-score for a price of \(2.20\), you would subtract the mean price \(2.30\) from \(2.20\) and then divide by the standard deviation \(0.10\). This tells us that \(2.20\) is one standard deviation below the mean. Similarly, a price of \(2.40\) results in a z-score of \(1.0\), meaning it is one standard deviation above the mean. Understanding z-scores helps in determining the percentage of data that falls within a certain range.

Standard deviation is a crucial concept when dealing with statistics and normal distributions. It tells us how spread out the values in a dataset are around the mean. A smaller standard deviation indicates that the values are closer to the mean, while a larger standard deviation means there is more variability in the dataset.

Standard deviation is symbolized by \(\sigma\) and is calculated using the formula:

• \[\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}\]

Where:

• \(x_i\) is each value in the dataset,
• \(\mu\) is the mean of the dataset, and
• \(N\) is the number of values in the dataset.

In our scenario, the standard deviation of \(0.10\) indicates how much individual gasoline prices vary from the average price of \(2.30\). Understanding standard deviation helps us determine the reliability and consistency of the data presented, which in turn assists with calculating z-scores and interpreting the normal distribution.

The mean, often referred to as the average, is the sum of all values in a dataset divided by the number of values. It is a common measure of central tendency, which provides insight into the overall level of the data you are looking at.

Calculated using the formula:

• \[\mu = \frac{\sum x_i}{N}\]

Where:

• \(\mu\) is the mean,
• \(x_i\) represents each individual data point, and
• \(N\) is the total number of data points.

In this gasoline price example, the mean price \(\mu = 2.30\) serves as a reference point. It allows us to calculate z-scores and standard deviations to understand how often prices deviate from this central point. The mean is a foundational aspect of understanding datasets, providing a straightforward summary of where values generally fall on the spectrum of variability. It is crucial for interpreting and predicting trends within a normal distribution.