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The Medical College Admission Test (MCAT) is an essential standardized exam required for admission to almost all medical schools in the United States. The test assesses a candidate's problem solving, critical thinking, and knowledge of natural, behavioral, and social sciences. The total score for the MCAT ranges between 472 and 528, encompassing four distinct sections.

In the spring of 2019, the mean of MCAT scores was recorded at 500.9 with a standard deviation of 10.6. These metrics are critical as they provide a foundation for understanding the distribution of scores across examinees. Since MCAT scores follow a normal distribution, the mean score also represents the median score. This property comes in handy when estimating quartiles, intervals, or other statistical measures from the scores.

Quartiles are statistical values used to divide data sets into four equal parts, each representing 25% of the data. In a normally distributed set of data like MCAT scores, quartiles help determine where an individual score falls within the spectrum of all test scores.

For MCAT scores, we calculate the first quartile, also known as Q1 or the 25th percentile, using the formula: \[ Q_1 = ext{mean} + z_{0.25} \times ext{standard deviation} \]Given that the z-score for the first quartile is approximately -0.675, Q1 is computed as \[ 493.8 \].

The third quartile, or Q3, is synonymous with the 75th percentile and can be found using a similar formula but with the z-score of 0.675 - resulting in Q3 being \[ 507.9 \]. These points demarcate the boundaries of the interquartile range.

The central 80% interval of a data set encompasses the middle 80% of scores and is found between the 10th and the 90th percentiles. In practical terms, this interval helps highlight the range within which the bulk of MCAT scores lie.

To identify this interval for MCAT scores, we use z-scores for the 10th and 90th percentiles, which are approximately -1.28 and 1.28, respectively. Applying these to the MCAT scores, using the formula: \[ \text{Percentile Score} = ext{mean} + z \times ext{standard deviation} \]- The 10th percentile is calculated as approximately \( 487.27 \) - The 90th percentile as \( 514.53 \).

Thus, the scores that fall in the central 80% of the distribution range between roughly 487.27 and 514.53.

The Interquartile Range (IQR) is a measure of statistical dispersion and indicates the range within which the middle 50% of the data lies. To compute the IQR of MCAT scores, calculate the difference between the third quartile (Q3) and the first quartile (Q1), which are 507.9 and 493.8, respectively.

The formula for IQR is: \[ \text{IQR} = Q_3 - Q_1 \]Substituting the quartile values gives an IQR of \( 14.1 \).

This value represents the spread of the middle half of scores in the MCAT distribution, offering insights into the consistency and variability of test performance across the examined population. A smaller IQR value could indicate that scores are closely bunched together, whereas a larger IQR would suggest more variability in test performances.