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When working with normal distributions, z-scores play an essential role. A z-score is a measure of how many standard deviations an element is away from the mean of the distribution. You can think of it as a measure of distance within the distribution.

Calculating z-scores involves using the formula:
\[z = \frac{X - \mu}{\sigma}\]

• \(X\) is the value in question from the data set.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For example, if you want to find the z-score for \(X = 110\) with a mean \(\mu = 100\) and standard deviation \(\sigma = 15\), it becomes:
\[z = \frac{110 - 100}{15} = 0.67\]
This z-score tells us that 110 is 0.67 standard deviations above the mean.

Once z-scores are determined, they help in finding probabilities within a normal distribution—a critical concept in statistics. Probability reflects how likely an event is to occur, and for continuous data like a normal distribution, we look at areas under the curve.

Utilizing a z-score, you can use a z-table or software to find the probability of a score being less than the given z-score, which represents the area to the left of that z-score in the standard normal distribution. By assessing these areas, we derive probabilities for ranges.

For instance, if \(z = 0.67\), a z-table indicates that \(P(Z \leq 0.67) = 0.7486\). This number shows there's a 74.86% chance a randomly chosen score from the distribution will be 110 or lower.

Standard deviation is a vital statistic measuring how spread out numbers are in a data set. In a normal distribution, it directly impacts the shape of the curve.

Here's precisely what it represents:

• A small standard deviation means data points are close to the mean, creating a narrow and steep curve.
• A larger standard deviation indicates a wider spread of data points, resulting in a flatter and wider curve.

Knowing the standard deviation is crucial when calculating z-scores and probabilities, as it determines how data values relate to the mean.

For this exercise, \(\sigma = 15\) reflects how much individual test scores vary around the average score of 100. Calculating z-scores with this standard deviation helps in quantifying these variances within the dataset.