Summarizing Assessment Results: Comparing Test Scores to a Larger Population

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Instructor: Melissa Hurst

Melissa has a Masters in Education and a PhD in Educational Psychology. She has worked as an instructional designer at UVA SOM.

Assessment results can yield valuable information about the individual test-taker and the larger population of test-takers. This lesson will describe how to compare test scores to a larger population by explaining standard score, stanines, z-score, percentile rank and cumulative percentage.

Standard Score, Stanines and Z-Score

Okay, you explained how to use a normal distribution to understand test scores. Now I still need to compare individual test scores to a larger population. Can you help me understand how to do that?

A common method to transform raw scores (the score based solely on the number of correctly answered items on an assessment) in order to make them more comparable to a larger population is to use a standard score. A standard score is the score that indicates how far a student's performance is from the mean with respect to standard deviation units.

In another lesson, we learned that standard deviation measures the average deviation from the mean in standard units. Deviation is defined as the amount an assessment score differs from a fixed value. The standard score is calculated by subtracting the mean from the raw score and dividing by standard deviation.

Example of a standard deviation graph
Standard Deviation Graph

In education, we frequently use two types of standard scores: stanine and Z-score.

Stanines are used to represent standardized test results by ranking student performance based on an equal interval scale of 1-9. A ranking of 5 is average, 6 is slightly above average and 4 is slightly below average. Stanines have a mean of 5 and a standard deviation of 2.

Z-scores are used frequently by statisticians and have a mean of 0 and a standard deviation of 1. A Z-score tells us how many standard deviations someone is above or below the mean.

To calculate a Z-score, subtract the mean from the raw score and divide by the standard deviation. For example, if we have a raw score of 85, a mean of 50 and a standard deviation of 10, we will calculate a Z-score of 3.5.

Cumulative Percentage and Percentile Rank

Another method to convert a raw score into a meaningful comparison is through percentile ranks and cumulative percentages.

Percentile rank scores indicate the percentage of peers in the norm group with raw scores less than or equal to a specific student's raw score. In this lesson, 'norm group' is defined as a reference group that is used to compare one score against similar others' scores.

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