Exam 5: The Normal Curve and Standard Scores

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Standard (z) scores, also known as standard scores or z-values, are a statistical measure that describe the position of a raw score in terms of its distance from the mean, measured in units of standard deviation. This allows for comparison between scores from different data sets or distributions by converting them into a common scale.

The formula to calculate a z-score for a given data point is:

z = (X - μ) / σ

Where:
- z is the z-score,
- X is the value of the data point,
- μ is the mean of the data set,
- σ is the standard deviation of the data set.

A z-score indicates how many standard deviations an element is from the mean. A positive z-score means the data point is above the mean, while a negative z-score indicates it is below the mean. A z-score of 0 means the data point is exactly at the mean.

Z-scores are particularly useful in statistics for identifying outliers, standardizing scores for comparison, and for calculating probabilities using the standard normal distribution. In the context of the standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1), z-scores can be directly associated with percentiles, which tell us the percentage of data that falls below the given z-score.

For example, if a test score has a z-score of 2, it means that the score is 2 standard deviations above the mean of the test scores. If the distribution of test scores is normal, we can infer that this score is higher than approximately 97.72% of all the other scores (since about 97.72% of the data in a normal distribution lies within two standard deviations below the mean).