The terms “mean” and “average” are similar. The distinction is that average is found in everyday conversation such as the average person or the average time to do something. The result is that customary language implications have changed even though both the mean and average mathematically are identical. Mean is utilized in statistical analysis and is defined as the sum of a group of numbers divided by the number of items in the group. Average, which is the same mathematical calculation as a mean, implies a general description that is not usually associated with a mathematical calculation.

The median is the exact middle of the group where one-half of the items in the group are above and one-half of the items in the group are below. However, the term “median” has found little acceptance in traditional data reporting by the media and, as a result, average is often thought to imply the exact middle. The range is defined as the distance between the minimum and maximum amounts in a set of data.All too often in both the media and in school board meetings, only the average is presented for the data being discussed. Without additional information the average is mistakenly viewed as a single absolute fact about the group. For example, this single absolute fact is seen when the media reports SAT or PSSA scores.

In statistical usage, the mean is one of several terms that are referred to as descriptive statistics. Descriptive statistics are used to present a picture of a group of data or data set.

The following example shows the problem with relying on one simple statistical value such as the mean for total test scores. As shown in Figure 1 – Example Test Scores, the mean or average test score is 1621. If we limit our discussion to only the mean, we fail to develop a full understanding of the data being presented. By focusing only on the mean, the discussion becomes one of, “How do we increase the average test score?”

By expanding the descriptive statistics, a clearer picture of the data starts to emerge. Adding the median (exact middle) expands the picture of our data set. The median for the example test data is 1670. We now have two descriptors of the test data, the average of 1621 and the median of 1670. This tells us that the exact middle is larger than the average test score which indicates that there is a “skew” in the data that was used to calculate the mean. A skew in data as used in this example tells us that the distance between the median and the bottom score is greater than the distance between the median and the top score.
 
Figure 1: Example Test Scores Table

To improve our understanding of the example data, the addition of the range of test scores allows us to visualize the difference between the extremes of the test. In our example the range is 1210 to 1880.

By comparing the range to the median of 1670 the number of points difference with the lowest score is 460, the difference with the highest score is 210. The differences between the mean and the lowest or minimum and the highest or maximum scores are 411 and 259 respectively.

By expanding the view of the data it is now easier to refocus the discussion. “How do we increase the average test score?” can become “How can we improve the scores for those individuals below the median?”However, the discussion of the example test scores is comprised of two parts, math and reading. Therefore, a set of descriptive statistics needs to be developed for both parts. 

Descriptive Statistics for Example Test Scores

The descriptive statistics for our example data shows that the reading scores tend to have a slightly narrower distance between the minimum and maximum scores. The average and median are greater for reading than math. Now we can begin to address the question of either “How do we increase the average test score?” or “How can we improve the scores for those individuals below the median?” The results of the data indicate that in this scenario the students need additional work in math to bring the level of scores up to the level of the reading scores. This analysis indicates that improving math generally and focusing on low end reading through a remedial program should result in improved test scores.

Another critical issue associated with standardized test scores remains with the simple fact that each year there is a different group of students than the prior year. The difference between groups of students usually produces different results on standardized tests. In some instances, the results can be significant and are not limited to only increases but also declines in scores on standardized tests.

Certain standardized tests are designed specifically to measure improvements of students over time. Unfortunately, high stakes tests such as the PSSA and SAT are not among the tests structured to measure improvement over time. Descriptive statistics, when used properly, will not measure or evaluate change over time but only describe the performance of the specific students taking the test.

For questions or comments, contact David W. Davare, PhD, at (717) 774-2331 ext. 3372 or (800) 932-0588.