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A standardized score, also known widely as a z-score in statistics, is a way of describing a specific data point in terms of its distance from the mean of a dataset, measured in standard deviations. This concept is especially helpful when comparing scores from different datasets or understanding how a score fits into a particular distribution.

• The mean is the average value for the dataset.
• The standard deviation shows how much the values deviate from the mean on average.
• The z-score indicates how far and in what direction a particular score deviates from the mean.

For example, a z-score of \( 0 \) represents a score that is exactly at the mean, while a z-score of \( 2 \) would mean the score is two standard deviations above the mean. Conversely, a negative z-score, like \( -0.5 \), indicates that the score is below the mean. Knowing a score’s z-score helps you quickly understand how typical or atypical it is compared to a set benchmark.

The z-score formula offers a mathematical way to find the relative position of an individual score within the context of a group of scores. The formula is: \[ z = \frac{x - \mu}{\sigma} \]Where:

• \( z \) is the z-score.
• \( x \) is the actual score.
• \( \mu \) is the mean of the dataset.
• \( \sigma \) is the standard deviation of the dataset.

With this formula, you can rearrange to solve for different components of the equation. For example, to find the actual score \( x \), you would rearrange the formula to: \[ x = z \cdot \sigma + \mu \]This alteration can assist when you know the z-score, mean, and standard deviation but need to determine what the actual score is. This method was used in the problem to find that Wayne's actual ACT score is 15, given his z-score of \( -0.5 \), a mean of 18, and a standard deviation of 6.

The ACT (American College Testing) exam scores are a crucial component of college admissions in the United States. Like many standardized tests, ACT scores are often normally distributed, forming a bell-shaped curve. This distribution indicates that most students score around the mean, with fewer students achieving extremely high or low scores.

• The mean score on the ACT is typically used to gauge the average performance.
• The standard deviation helps identify how scores are spread out around the mean.
• Wayne's score being below the mean and having a z-score of \( -0.5 \) suggests he scored lower than the average student.

Understanding standardized test scores through concepts like the mean, standard deviation, and z-score allows students and educators to make informed decisions about performance relative to peers. This helps in evaluating whether students are on track with their goals, particularly when comparing them to typical college admission standards.