Issues

Standardized Scores

by Michael T. Martin

Standardized scores provide important information about student achievement. They were developed because raw scores often need complicated interpretations to make sense. By using standardized scores, the average person can quickly get a comprehensive idea of how a particular score ranks compared to other scores.

The key is the word "ranks" because standardized scores are specifically intended to display a ranking of achievement rather than a level of achievement. The standardized score displays how an individual score compares to all the other scores from that test, but it does not display any information about the achievement level of the score. In fact, as explained in Martin's Paradox, it can grossly distort the level of achievement. To see why standardized scores are splendid tools for reporting individual scores, but ridiculous for groups, consider a hypothetical case of using a raw score.

Suppose your child came home with a score of 47 on a test. That might bend your nose out of shape a little, but that doesn't really tell you anything. It’s just a number. Suppose your child then announced that the average score on the test was a 40. That tells you your child scored above average. A raw score has no meaning at all without knowing what the average score was. Therefore, to have any meaning, test scores normally would require reporting at least two scores. Even then, knowing your child was above average doesn't really tell you how much above average.

For that, you need the average difference between all the scores and the average, the Standard Deviation. In other words, we don't really know if anybody actually scored at the average. Most students probably did not score at the average. The question, then, is how much did each of the scores of the students differ from the average. Did everybody score right at the average, except for your child, or were the scores really widespread? The standard deviation tells you, on average, how far away the other scores were from the average.

A normal bell curve is six standard deviations wide, three on each side of the mean. About two-thirds (68%) of all scores will be within one standard deviation of the average. That leaves about one-third (32%) either above or below this middle two-thirds, with one-sixth (16%) above this range and one-sixth below this range. Notice there are two standard deviations in each of those 16% ranges, plus the two in the middle for six standard deviations. Technically, that doesn't include everything but it includes 997 out of every thousand.

So to tell how well your child did on the test you really need three scores: your child's score, the average score, and the standard deviation. Then you can see where your child scored in the bell curve of scores.

For example, if the class mean was 40 and the Standard Deviation was 10, this would indicate that your child's score of 47 was well within the first standard deviation above the mean. When the median and the mean are about the same,this is the point where 50% of the scores are on either side and called the 50th percentile. One standard deviation above this equals the 84th percentile (50% + half of 68%).

However, if the Standard Deviation was 5, this would indicate that your child did very well, because 84th percentile is the score of 45 and your child scored a 47. If the Standard Deviation was 3, then your child did better than about 98% of the other scores. Same score, same average, but a huge difference in the significance of the score with a different standard deviation.

The fact that the same score can have such a radical difference in importance is why scores are generally "standardized" before they are released. A standardized score always uses some predetermined score as the average. In the college SAT test (and the AIMS tests) the predetermined average is 500.

The standardized score then uses the Standard Deviation as a measuring stick to arrange the actual scores around this predetermined average. In the SAT (and the AIMS tests) the Standard Deviation is set to 100. Thus your child's score of 47 would be standardized above 500 (the predetermined average) but exactly where would depend on the Standard Deviation.

If the Standard Deviation on your child's test was 10, then the standardized score for 47 would be 500 for the 40 mean plus 70 (500 plus 7 tenths of 100) for a score of 570.

If the Standard Deviation was 5, then the standardized score for 47 would be 640 (500 plus 7 fifths of 100). If the Standard Deviation was 3, then the standardized score for 47 would be 733 (500 plus 7 thirds of 100).

The advantage of standardized scores is that they preserve the relationship between each actual score with the other actual scores in the group that determined the average and Standard Deviation, but you no longer have to know the average and the standard deviation because they are predetermined.

Standardized scores are extremely helpful in showing how an individual score fits into a group of scores. Knowing that your child scored a 47 was meaningless, but knowing that your child scored a standardized 570, or 640, or 733 tells you quite a lot (assuming you know that the lowest score possible is 200 and the highest score possible is 800).

Standardized scores are very useful in comparing one student to other students in the group, because they tell you how an individual score compared to the other scores in the group. But what it tells you applies only to the group that determined the average and Standard Deviation. Where standardized scores become useless is in trying to compare groups.

If you are comparing students within a group, it does not matter what the shape of the distribution of scores looks like. It can look like a bell curve, it can look like a human nose, it can look like a 1947 Buick. It doesn't matter because standardized scores will preserve the relationship between individuals within that distribution. The relationship between a 570 or 640 or 733 will be the same whether the shape of the distribution resembles a 1947 Buick or a bell curve.

Standardized scores were not developed to compare groups. They were developed to rank individuals within a group. Standardized scores provide a very effective way of showing how individuals rank in comparison to other individuals who take the same test.

But as national testing expert Dr. Carl Popham frequently points out, you can't measure a child's temperature using a spoon, you need the right tool for the task. Standardized scores cannot be used to measure a level of achievement or to compare one group to another. They only measure relative rankings of individual scores on the same test.