Percentiles and Standard Scores

Graham Tall   research@grahamtall.com   September 203 

I.    Standardised Scores

The formula to create standardised scores is reasonably straight forward and uses the two most widely known statistics, the mean and the standard deviation (if uncertain of these look up basic statistics). The basic formula used in research is simply:

i.e. subtract each raw score from the mean and divide by the standard deviation. With standardised (z) scores, therefore, the mean is 0 and the standard deviation is 1.

However, when standardised scores are used in schools, it is common practice to transform z scores to scores which have a mean of 50 and a standard deviation of between 14 and 16. Creating a more user-friendly number. The formula therefore becomes:

The reason schools use such scores is to ensure that subjects are given the same weight when banding or streaming pupils. Consider Table 1, where the scores of 10 pupils are given for English, mathematics and subject X. In English the scores given range from 50 to 90, in Mathematics from 0 to 90 and in subject X from 31 to 40. The result of this is that even when the scores of the three subjects are combined the final position is virtually the same as if the maths scores alone had been used; this is because only the Mathematics markers used the whole mark range when marking their examination.

 Top of Page

Table 1 Raw Scores in Three Subjects Used to Band Pupils

When the scores in each subject are standardised, the situation changes dramatically.   In table 2, the mean and standard deviation of each subject is transformed to 50 and 16 respectively.  As a result the order of the students changes dramatically; with the child who scored highest in English  now becoming third rather than last, whilst the child who scored highest in mathematics drops to the eighth position.

Table 2 Standardised Scores in Three Subjects Used to Band Pupils

II    Percentiles

Percentiles are normally calculated only when large numbers of people are involved.  As a result, parents and teachers will only meet them if they are informed of, say, the results their child's/pupil's results on a psychological test. What does information that a child/pupil is on the fifth or the fifteenth percentile mean?

Converting examination/test marks or other numbers to percentages is done because percentages are easier to consider than marks out of arbitrary total(s).  Percentiles are based on the same idea, but are linked to the order or position of the number concerned. The only point of possible confusion is that whilst position is directly linked to percentile rank, the calculation means that a mark that earns position 1 has a percentile rank of 100 (see Table 1).

III.   Illustration of Marks, Percentages, Percentiles and Standard Scores
          (note these marks are artificial)

Consider the results of a test taken by 1000 pupils, where the maximum possible mark (total) is, say, 50, the mean is 25 and the standard deviation is 8.   For information on mean, median, and standard deviation see basic statistics.

In table 3, although a 1000 pupils were theoretically involved, the marks of just 21 are indicated.  The first column indicates marks which fit a normal distribution, the second column transforms these marks into percentages.  The third column states the position of the pupils used out of the 1000 who 'took' the exam.  The fourth identifies the percentile rank (the percentile commonly referred to);  although two percentile ranks are given as decimals, this is NOT normal practice - the 97.5 percentile would normally be described as the 98th.  The percentile point column states the mark that the pupils attaining a particular percentile attained.  With much smaller numbers of individuals rounding up would occur so that the top person is on the 100th percentile.  The final column gives the standard score for this particular set of results; for information a z score of ±1.96 is the normal distribution probability score that is used for significance at the 5% level (z = ±2.33 and  ±3.08 define significance at the 1% and 0.1% levels) in two-tailed tests, a subject discussed in statistical testing.

Table 3 Percentages, Percentiles and Standard Scores (z)

 The formula for calculating percentile scores =  (Number  of individuals - Position of individual)  * 100
                                                                                                  Number  of individuals

e.g.       the 250th person of the 1000, percentile score = (1000 - 250)   * 100  = 75th percentile
                                                                                            1000

Ferguson, G.A. (1981) Statistical Analysis in Psychology and Education. £17.95 McGraw Hill: London. Excellent statistics book formulae and worked examples of a wide range of tests- Strength is in range of different Analyses of Variance More up-to-date edition now available

Guilford, J.P. (1965) Fundamental Statistics in Psychology and Education. A sound general statistical textbook                                                                                                                              Top of Page

Home Page    Research Introduction  Quantitative Advice   Index   Statistical Tests    Basic Statistics    
Research and Statistics Courses