Long page
descriptions

Chapter 4   Comparison of groups

4.1   Z-scores and stanines

4.1.1   Normal parameters

The parameters of the normal distribution are its mean and standard deviation. The best fit for a set of marks is obtained by setting these to the mean and standard deviation of the data.

4.1.2   Standard normal distribution

All normal distributions have the same 'shape' but different centre and spread. Any normal distribution can therefore be translated into a standard normal distribution by a simple scaling.

4.1.3   70-95-100 rule

All normal distributions have the same proportions of values within 1, 2, and 3 standard deviations of the mean. This corresponds to standardised values between ±1, ±2, and ±3. This is a rule-of-thumb that also applies approximately to other distributions.

4.1.4   Standardising data

Data can be standardised in the same way as a normal distribution -- subtract the mean and divide by the standard deviation. These are called z-scores.

4.1.5   Stanines

Stanines are z-scores, translated into an integer scale betweeen 1 and 9.

4.2   Reference populations

4.2.1   National distributions

For some standard tests, a national distribution of marks is available for use as a reference population.

4.2.2   Percentiles from national distns

Individual marks in a class can be assessed against a reference population. Each mark can be translated into a percentile in the reference population.

4.2.3   Stanines from national distns

The reference population can also be used to translate individual marks into stanines.

4.3   Scaling marks

4.3.1   Linear scaling

A simple linear scaling of a set of marks can change the mean and standard deviation to any desired values.

4.3.2   Piecewise linear scaling

Simple linear scaling can change the top marks too much. A piecewise linear scaling changes the extreme marks less.

4.3.3   Doing it in Excel

Piecewise linear scaling can be done easily in Excel.