Standard tests
In the previous section, Z-scores and stanines, we looked at a single set of class marks in isolation. In the current section, we ask how these marks compare to a larger population — for example, the rest of the country.
In New Zealand, various standard tests have been written and are available for any school to use. These serve several purposes:
- They are well-written and effectively assess student ability.
- They allow student performance to be compared to a national distribution of marks. Guidelines for interpretation of marks are often provided.
- The results from these tests provide educational researchers with information about how student ability is changing from year to year.
We concentrate on the second of these points. The national distribution of marks for a standard test provides a guideline for interpreting an individual student or class.
School Entry Assessment
One standard assessment package is the School Entry Assessment kit which is used by many new entrant teachers in New Zealand. Student summary sheets are returned to the Ministry of Education and the diagram below shows the national distribution of marks for one part of the assessment ('Checkout') from about 30,000 students between 1997 and 2000.
The numbers of students getting each mark are shown on the axis on the left. The axis on the right shows the proportions and is of more interest. Drag over the bar chart to read off the proportion of students with marks less than any value — the sum of the bar heights to the left of the value.
No need to smooth
For a small set of class marks, a normal distribution may be an adequate description of the distribution since there is not enough data to accurately assess the shape of the distribution.
However, when there are many thousands of marks, a simple distribution such as a normal distribution is unlikely to be adequate. Since a bar chart or histogram of the data should be reasonably smooth, there is usually no need for further smoothing.
Warning
The students whose marks are compiled to form a 'national distribution' may not form a typical cross-section of the country. If only high-decile schools conduct a particular test, the 'national distribution' of marks will be centred on a higher mark than would be typical from the country as a whole.
For example, many of the better students in New Zealand sit the exam for the Australian Mathematics Competition. Since this is a fairly selective group of students, an 'average mark' in this exam should not be taken as an indication that the student is only an 'average student'.
National distributions should therefore be interpreted with caution unless they represent a genuine cross-section of students.