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.

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.**Cumulative percentages** determine placement among a group of scores.

Cumulative percentages do not determine how much greater one score is than another or how much less it is than another. Cumulative percentages are ranked on an ordinal scale and are used to determine order or rank only. Specifically, this means that the highest scores in the group will be the top score no matter what that score is.For example, let’s take a test score of 85, the raw score. If 85 were the highest grade on this test, the cumulative percentage would be 100%. Since the student scored at the 100th percentile, she did better than or the same as everyone else in the class. That would mean that everyone else made either an 85 or lower on the test.

Cumulative percentages and percentiles are ranked on a scale of 0%-100%. Changing raw scores to cumulative percentages is one way to standardize raw scores within a certain population.

## Lesson Summary

So you can see there are a few ways to understand and summarize assessment results.

Let’s recap what we discussed, and hopefully you will be able to apply these concepts to your classrooms.Test scores fall along a normal distribution, which we learned about in a previous lesson. A normal distribution shows that the majority of scores fall in the middle of the curve, with a few falling along the upper or lower range. This distribution shows us the spread of scores and the average of a set of scores.The normal distribution enables us to find the **standard deviation** of test scores, which measures the average deviation from the mean in standard units.**Standard scores** indicate how far a student’s performance is from the mean with respect to standard deviation, and there are a few types of standard scores used in education, including **stanines** and **Z-scores**.Finally, we also discussed ways to represent scores by **percentile** and **cumulative percentage** rankings, which indicate the percentage of peers in the comparison group with raw scores less than or equal to the specified top score.