Statistical Analysis for Education and Psychology Researchers

position. If two schools have the same pass rate they are both assigned a rank, but this
rank is calculated as the average of the ranks that each of the schools would have
occupied.
To use a simplified example, consider the following list of numbers of A-level passes
achieved at one sitting for 10 schools:
School A B C D E F G H I J
No. of A-level passes 29 18 0 56 52 60 18 52 85 52

To make these data more meaningful they are first displayed in a table in order of
magnitude with the highest rank placing given to the school with the greatest number of
A-level passes.

Table 2.1: Ranking of A-level passes by school

School Number of A-level passes School rank

I 85 1

F 60 2

D 56 3

E 52 5

H 52 5

J 52 5

A 29 7

B 18 8.5

G 18 8.5

C 0 10

The three schools with 52 A-level passes in Table 2.1 would have been assigned ranks 4,
5 and 6 because they came after rank score 3 and are the next three possible ranked
places. It would be unfair to give either school E, H or J a higher ranked position because
they all obtained 52 A-level passes. These schools are therefore given the average of the
three ranks, i.e., 5. When scores are equal they are referred to as tied scores. Similarly,
the two schools with 18 A-level passes are given a rank of 8.5, the average of the two
ranks 8 and 9.

Quantitative Variables

Up to now the use of numbers for categories and ranks only has been considered.
Numbers, however, can also be used quantitatively to measure amount. Technically,
measurement is when numbers are assigned to data according to a set of logical rules.
Numbers when used for measurement make use of the logical properties of the number
system and therefore can be manipulated mathematically.
The number system is entirely logical and numbers are unique, in the sense that no
two numbers are exactly identical. Numbers also have the properties of order, additivity
and can be subtracted, multiplied or divided. Whenever numbers are manipulated they
result in new unique numbers. For example, the subtraction of one particular number
from another particular number always yields a unique number. The usefulness of

Measurement issues 21