First example. I want to study the link between the DA (A: air) distance in a crow line from Genoa to other places in Northern Italy and the DS (S: street) distance along the road.  I collect data relating to the minimum road distances from Genoa of the provincial capitals of Piedmont, Lombardy and Veneto and those relating to the distances in the crow flies: 55,   70, 155, 160, 155, 115, 105, 165, 110, 115,   85, 165, 110, 155, 105 85, 115, 205, 230, 185, 185, 145, 240, 140, 155, 125, 290, 170, 195, 135 I put (0,0) as "fixed point".  I get (rounding):  y = 1.42 x,  correlation coefficient = 0.913.

 

How to graphically represent what was obtained:  see here.

Second example. A sample of "ideal gas". y: approximate values (with some units of uncertainty) of the temperature in degrees Celsius;  x: corresponding pressure values (in mm of mercury). x:   65, 75, 85, 95, 105 y:  -21, 18, 43, 95, 127 I get (rounding):  y = 3.73 x − 264.65   (264.65 is close to −273),  correl. coeffic. = 0.995.

 

How to graphically represent what was obtained:  see here.

Third example.
In an experiment on the growth of wheat during the winter, the average temperature (in °C) of the soil at a depth of 8 cm (x) and the days necessary for sprouting (y) are recorded.  Is there a relationship between soil temperature and sprouting time? Can we mathematize this relationship?
x:    5, 5.5, 6, 6.5,  7, 7.5,  8, 8.5
y:  40, 36, 32, 27, 23, 19, 19, 20
We can roughly express the relationship with the function:  y = −6.4·x + 70  (if x ≤ 8).

 

How to graphically represent what was obtained:  see here.

To associate the linear correlation with a measure of accuracy it is convenient to use WoframAlpha:

Result:  -0.955778    p-value:  0.000209086     What is p-value?
It is the probability of obtaining (with a given amount of data) a given correlation coefficient under the assumption that the two variables are totally uncorrelated (ie that the correlation coefficient was actually 0).  In our case we got 0.0209086%:  there is a very low probability that the data is uncorrelated.  A p-value no greater than 5% is often assumed as the conventional limit for the possible existence of a correlation.

The p-value (also used in many other situations) does not, in itself, establish probabilities of hypotheses; rather, it is a tool for deciding whether to reject the tested model. For further information see en.wikipedia. If I have less data (see the graph on the right), I can obtain a greater correlation coefficient but at the same time have a much greater p-value: the probability that there is no statistical link between the data has greatly increased: correlation test [(5.5,6.5,7,8),(36,27,23,19)]   correlation: -0.979561   p-value: 0.020439 = 2.0439%   In the case shown on the left   p-value  is almost 100%: correlation test [(10,15, 25,40,35,20),(25,15,10,20,35,35)] p-value:   0.833509 = 83.3509%

If you want the initial graph with WolframAlpha: plot (5,40);(5.5,36);(6,32);(6.5,27);(7,23);(7.5,19);(8,19);(8.5,20) color brown

For further development it is better to use R (see).

Fourth example.
See.