Understanding Standard Deviations and Their Impact on Test Scores

Data analysis often involves understanding the distribution of scores, particularly through the use of standard deviations. When interpreting test scores, it is common to wonder how a score that is 2 standard deviations below the mean compares to the rest of the scores in a given distribution.

Standard Deviations and Their Interpretation

When someone scores 2 standard deviations below the mean, it does not mean that they performed 2 times worse than the average. The relationship between standard deviations and test scores can be more accurately explained using a different perspective. Let's explore the concept and why the traditional interpretation is misleading.

The Non-Direct Relationship Between Scores and Standard Deviations

Assume a test score is out of 100. If the mean score is 50, then a score of 50 indicates that the test-taker performed in the average range. A score of 25, being 2 standard deviations below the mean, doesn't necessarily mean the test-taker performed 2 times worse. In fact, it might be more appropriate to say they performed half as well as someone scoring close to the mean.

An Example with Specific Scores

For a standardized test with a range of 1 to 100, a score of 50 would be considered average. If a test-taker scores 25, a score which is two standard deviations below the mean, they would be performing much lower than the average. In this example, it is more accurate to say the test-taker performed half as well as someone who scored 50, rather than twice as poorly.

Understanding Percentile Scores

A more meaningful representation of a score's position within a distribution is through percentile scores. Percentile scores indicate the percentage of the test-takers who scored below a specific score.

Example of Percentile Scores

Let's consider a test-taker who scored 50 out of 100. If 50% of the test-takers scored below or at this level, then 50 is their percentile score. The test-taker who scored 25 would fall into the lower 10% of the distribution. Hence, they performed much lower in comparison to the majority of the test-takers.

Normal Distribution and Standard Deviations

When test scores are normally distributed, standard deviations become particularly useful. In a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. For a test with a mean of 100 and a standard deviation of 15, a score of 70 (two standard deviations below the mean) would place the test-taker in the lower 2.275% of the distribution.

Example with IQ Scores

In the context of IQ scores, a common mean and standard deviation are 100 and 15, respectively. Hence, a score of 70 would be considered significantly lower than the mean, placing the individual in the lowest 2.275% of the population based on the standard normal deviate.

Why Standard Deviations Don't Always Mean Much

While standard deviations are useful, they do not always provide a complete picture of where a test-taker stands. Percentile scores offer a clearer and more direct comparison within a distribution. For example, if a test-taker is at the 25th percentile, then they have outperformed 25% of the test-takers but fallen short of 75%.

Conclusion

When interpreting test scores, it is important to consider the context and the distribution of the scores. While standard deviations can provide a rough indication of where a score falls within the distribution, it is often more meaningful to use percentile scores. Percentile scores provide a clearer picture of a test-taker's performance relative to their peers.

Keywords

Standard deviations, test scores, percentile scores