Students are subject to all kinds of state and national tests now to determine how much they have learned compared to other students in the same state or country. Educators, and politicians, need some way to compare the scores to know how well the students are doing. How well the students score determines education funds for the state, the school district and the individual schools. Standard scores and z-scores are used to determine how those funds are distributed.
Standard Scores Explained
Standardized tests all have their own scoring systems. The SAT is scored on an 800 point scale for each section, whereas the ACT is scored on a 36 point scale for each test section. When a student receives her scores back from the testing agencies, she will only know how well she scored for that test. This score is the raw, or standard, score. The student will not know how her score compares to others nationally or statewide without having other information available to her.
When all standard scores are gathered and graphed on a histogram by how many times that particular score occurred, a bell shape tends to emerge on the graph. This bell shape is called the standard distribution. All scores can be plotted and found on this bell curve. The top of the curve is where the mean, median and mode of the scores usually lie. That is the average, the middle score, and the score that occurred the most, respectively.
Standard Normal Distribution
A standard normal distribution is similar to the standard distribution described above. However, in the case of a standard normal distribution, the mean, median, and mode are all zero (0). The standard normal distribution will have a standard deviation, that is the average distance from the mean, of 1. This means that most of the scores will be found within one deviation from the mean. In fact, 68 percent of all scores on the distribution will be within 1 standard deviation of the mean, 95 percent within 2 standard deviations and 99.7 percent within 3 standard deviations.
A z-score is a score that falls somewhere within the standard normal distribution. Its counterpart on the standard distribution is the standard score. The standard normal distribution and z-score allow all the standardized scores to be weighed equally through a few calculations. The z-score can then be used to determine what percentile rank the score falls into by finding the z-score on a z-distribution chart.
Standard Scores to Z-Scores
As mentioned above, a simple calculation can transform the standard score to a z-score. The z-score can be found by subtracting the mean of the standard scores from the standard score being evaluated and then dividing that difference by the standard deviation of the standard distribution. The formula is: z=(X-M)/s.d.
This z-score is how many standard deviations from the mean that the standard score falls. If the z-score is positive, then the standard score is above the mean. If the z-score is negative, then the standard score is below the mean.
Other Standard Scores
Standard scores are used any time there is any kind of testing or evaluation involving some kind of scoring system. Standard and z-score scores are also used when testing claims that manufacturers make regarding their products' durability, strength or other measurable feature.