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:
|
x |
2 |
3 |
4 |
|
y |
8 |
18 |
32 |
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 |