Other aspects of dimensional analysis do carry over to the social sciences , however , and in ways which may not be immediately obvious . Population density ( of interest within social geography , for example ) is defined as population per unit of area , but as area is measured in the square of the units of length , such as square metres ( m ²) or square kilometres ( km ²), in dimensional analysis this would be represented as the S-formula P . L ¯ ².
The minus sign in the above formula may need a moment of explanation for readers who do not possess a natural affinity for mathematics . Dividing by a number is equal to multiplying by the reciprocal of that number , and the reciprocal of any quantity is simply that quantity raised to the power of minus one , so ½ is numerically equal to 2¯ ¹, for example . Dividing by an area with the units L ², therefore , is the same as multiplying by L¯ ².
The units of time may be similarly modified using negative exponents . Speed , for example , as [ distance ] / [ time ] has , as shown above , the units [ length ]/[ time ], which becomes [ length ] [ time ] ¯ ¹ and so in S-theory terms , L T ¯ ¹. The rate of population growth also uses the concept of ‘ per unit of time ’, and so would be P T ¯ ¹.
Dodd ’ s concept went far beyond simple dimensional analysis however . He was interested in representing the complete ‘ outer nature ’ of a data set – any data set – and so he devised methods for symbolising the presence of qualitative data ( not the data itself ), for the existence of data from different ‘ classes . Examples included separate regions on a map . Dodd ’ s theory thus formulated how data might be : ( a ) expressed in class-intervals , for instance as a series of equallyspaced periods of time ; and ( b ) presented as cases , for instance as individual dates ( Dodd 1942 , 904 ). Crucially , as with qualitative data , his interest was not in the classes , class-intervals or dates themselves , but the fact of their inclusion as part of the data .
Dodd also included several new ‘ operators ’ in S-theory in addition to the familiar operations of +, -, x and ÷, such as ‘:’ for ‘ each with a corresponding ’, ‘::’ for ‘ cross-classified with ’ and ‘•’ for ‘ is correlated with ’ and ‘;’ for the intriguing concept of ‘ unspecified operator ’. Other symbols referred to the nature of a given quantity , such as underlining to denote indefiniteness , overlining to imply a geometric interpretation and the use of primes , such as ‘ ‘’ and ‘’’ to represent individually identified items , such as named persons or a specific time or date .