A normal score for an 8th grader on standardized tests in the United States typically falls within the range of the mean score, which is often around 500. However, this can vary depending on the specific test and the year it was taken. For instance, the National Assessment of Educational Progress (NAEP) has a mean score of 300, with a standard deviation of about 100. This means that a score of 500 would be slightly above average, while a score of 400 would be at the median. The normal range can be considered to be scores that are within one standard deviation of the mean, which would be between 200 and 600.

The reason for this range is rooted in the statistical properties of standardized tests. Standardized tests are designed to have a normal distribution, also known as a bell curve, where the majority of scores cluster around the mean. This distribution allows for a clear understanding of where a student's score stands in relation to their peers. The mean score serves as a benchmark for average performance, while the standard deviation indicates the spread of scores around the mean. By considering scores within one standard deviation of the mean, we account for the natural variability in test performance and provide a more nuanced view of what is considered "normal" for 8th graders.

Data from the NAEP, which is considered a reliable indicator of student achievement, supports this analysis. In 2022, the mean score for 8th graders in mathematics was 285, with a standard deviation of 100. This means that a score of 385 would be at the 95th percentile, while a score of 195 would be at the 5th percentile. Similarly, in reading, the mean score was 266 with a standard deviation of 100, suggesting that a score of 366 would be at the 95th percentile and a score of 166 would be at the 5th percentile.

It's important to note that these scores can vary significantly depending on the specific test and the subject area. For example, the Scholastic Assessment Test (SAT) has a different scoring scale and distribution than the NAEP. On the SAT, a score of 500 in each section is considered average, with a normal range of scores between 200 and 800. This reflects the fact that the SAT is designed to assess a broader range of skills and knowledge compared to the NAEP.

In conclusion, a normal score for an 8th grader in the United States can be estimated to be within one standard deviation of the mean score on a standardized test, which is typically around 500. This range accounts for the natural variability in test performance and provides a clear reference point for understanding a student's achievement in comparison to their peers.