Question

A study of bone density on 5 random women at a hospital produced the following results. Age 41 53 57 61 69 Bone Density 360 355 350 345 315 Step 1 of 3 : Calculate the correlation coefficient, r. Round your answer to six decimal places.

Answer 1

Objective: To compute the correlation coefficient r for the variables Age (X, say) and Bone densities (say, Y).

By definition of Pearson's correlation coefficient,

• r measures the strength and direction of the linear relationship that exists between two continuous variables.
• It ranges from -1 to 1, negative and positive values indicating a negative and positive linear relationship respectively.
• Values close to unity, depicts a strong linear relationship and those close to zero implies weak or no linear relationship.

For two variables X and Y, the correlation coefficient r can be calculated using the formula:

Substituting the values:

Age (X)	Bone Density (Y)	XY	X2	Y2
41	360	14760	1681	129600
53	355	18815	2809	126025
57	350	19950	3249	122500
61	345	21045	3721	119025
69	315	21735	4761	99225
281	1725	96305	16221	596375

= -0.874173

We find that r is negative and close to unity. We may say that the variables Age and Bonsity exhibit a strong negative linear relationship i.e. Age and Bone density are negatively correlated - As age increases, bone density is found to decrease and vice versa.