How can I find correlation?

Question

How can I find the correlation for X: 19, 31, 34, 35, 39, 39, 43 and Y: 410, 580, 590, 570, 640, 680, 660?

If you don't mind, can you tell me how you got it. I need to learn.

Answer 1

ŷ=a+bx is the regression equation based on (x,y) data pairs, and ŷ is an estimated output value for some input value x. The hat symbol is used to distinguish estimated from actual. In a dataset, the y values are actually observed values.

So how are a and b calculated? Here are the formulas:

Slope, gradient or rate of change, b=(n∑x_iy_i-∑x_i∑y_i)/(n∑x_i²-(∑x_i)²). Intercept a=(∑y_i-b∑x_i)/n for 1≤i≤n, where n is the number of data pairs, which is 7 in this problem.

The table below shows what you need to calculate from the given data:

x	y	x2	xy
19	410	361	7790
31	580	961	17980
34	590	1156	20060
35	570	1225	19950
39	640	1521	24960
39	680	1521	26520
43	660	1849	28380
∑xi=240	∑yi=4130	∑xi2=8594	∑xiyi=145640

Putting all these figures together into the formulas above, where n=7 we get a=210.9539, b=11.0555 (approx).

Take x=34, for example, then ŷ=210.9539+11.0555×34=587 approx, compared with 590 actually. A good fit.

The correlation coefficient, r=0.9606 approx, which indicates a strong correlation between x and y.