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The concept of a 'z-score' is fundamental in statistics, particularly when working with normal distribution. Simply put, a z-score is a measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. When you calculate the z-score of a data point, you are finding out how many standard deviations away from the mean it is.

Mathematically, it is represented by the equation: \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. A z-score can be positive or negative, indicating whether the score is above or below the mean, respectively. In the example of the midterm exam, by calculating the z-score, we can determine whether a given exam score falls within the top 15% of all scores, a necessary step to find out if a student earned an A grade.

Standard deviation, denoted as \( \sigma \), measures the amount of variation or dispersion of a set of values. A low standard deviation means that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Understanding standard deviation is critical when interpreting z-scores since it forms the denominator of the z-score formula. It gives context to the z-score value, helping to understand how extreme a data point is in relation to the overall distribution. For instance, in the context of the midterm exam, a standard deviation of 7 points tells us that most students scored within 7 points above or below the average score of 78.

The percentile-to-z-score conversion is a powerful tool when working with the standard normal distribution. A percentile rank is a value below which a given percentage of observations fall. To find the z-score associated with a certain percentile, one often uses a z-score table or statistical software.

In our exercise, finding the z-score for the top 15% of scores (or the 85th percentile) allows us to set a cutoff point for grading. Typically, the 85th percentile z-score is around 1.04, meaning that a score 1.04 standard deviations above the mean marks the lower bound of the top 15%. Transforming a percentile to a z-score is an essential step when setting benchmarks such as passing grades or qualification limits.

Normal distribution is a key concept in statistics with remarkable properties, making it invaluable in many fields. It has a bell-shaped curve which is symmetrical about the mean. Here are a few crucial properties:

• The mean, median, and mode of a normal distribution are all equal.
• Approximately 68% of the data within a normal distribution falls within one standard deviation of the mean. Similarly, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.
• The tails of the distribution curve extend infinitely, theoretically covering every possible value of x.

For example, when we apply this understanding to the midterm exam scores, we anticipate that most students' scores will cluster around the mean, with few extreme high or low scores.