## Tables of Higher Order Direct Relationships

For a direct linear relationship, the ratio between the values of each variable should always be the same.  Well, for a direct square relationship, the ratio between the value of one variable and the square of the other variable should always be the same.  So say we had this table of data:

We could check whether a linear relationship existed by finding the  or  ratio for each pair of values:

 x 2 3 4 y 8 18 32 y/x 4 6 8

These ratios aren’t the same, so it’s not a linear relationship (so  does not describe this data).  We can check whether  describes the data by finding the ratios of y/x2:

 x 2 3 4 y 8 18 32 y/x2 2 2 2

Bingo!  The ratios are all the same.  This means that y is proportional to the square of x.  Note that you could also have checked the inverse ratios – so instead of checking y/x2, we could have checked x2/y.  The actual value of the ratio would have been different to ‘2’, but they would still all be the same:

 x 2 3 4 y 8 18 32 x2/y 0.5 0.5 0.5