Correlation | Linear Regression | Variance and Standard Deviation | Normal Distribution

Calculator Correlation and Significance

Calculates the correlation coefficient for two characteristics and the significance of the data. Correlation is the statistical relationship. It tells how much two quantifiable characteristics have to do with each other. The correlation coefficient indicates the degree of this relationship, it is between -1 and 1. 1 means perfect relationship, -1 means perfect inverse relationship and 0 is no relationship. Correlation says nothing about causality. If two things are related, it does not necessarily mean that one thing causes the other.
Significance tells how certain a measured relationship actually is. There can always be errors and coincidences with measured values. This can only be ruled out with a certain probability, 95% and 99% are common as significance levels. A t-test determines the significance.

Please enter the values ​​of the two characteristics separately. For each characteristic, the values ​​must be separated from one another with a blank or a line break. The number of values ​​per characteristic must be the same. The n-th value of the first feature belongs to the n-th value of the second feature. Example calculates with the size (in 1000 km²) and population (in millions) of some European countries.

Characteristic x

Characteristic y

Result

Number of value pairs:
Covariance x and y:
Standard deviation x:
Standard deviation y:
Correlation coefficient:
Test variable:
Degrees of freedom:
95% significance:
99% significance:

The formulas are:

n: number of value pairs, Σ: sum i=1 to n
x_m: mean of all x_i, y_m: mean of all y_i

Covariance x und y: s_xy = 1/n * Σ (x_i−x_m)(y_i−y_m)

Standard deviation x: s_x = √ 1/n * Σ (x_i−x_m)²

Standard deviation y: s_y = √ 1/n * Σ (y_i−y_m)²

Correlation coefficient: r_xy = s_xy / (s_x*s_y)

Test variable: t=√(n−2)/(1−r_xy²)

Degrees of freedom: df=n-2

t-distribution

df	95%	99%		df	95%	99%
1	6.314	63.656		19	1.729	2.861
2	2.92	9.925		20	1.725	2.845
3	2.353	5.841		21	1.721	2.831
4	2.132	4.604		22	1.717	2.819
5	2.015	4.032		23	1.714	2.807
6	1.943	3.707		24	1.711	2.797
7	1.895	3.499		25	1.708	2.787
8	1.86	3.355		30	1.697	2.75
9	1.833	3.25		40	1.684	2.704
10	1.812	3.169		50	1.676	2.678
11	1.796	3.106		60	1.671	2.66
12	1.782	3.055		70	1.667	2.648
13	1.771	3.012		80	1.664	2.639
14	1.761	2.977		100	1.66	2.626
15	1.753	2.947		150	1.655	2.609
16	1.746	2.921		200	1.653	2.601
17	1.74	2.898		500	1.648	2.586
18	1.734	2.878		1000	1.646	2.581

Correlation is rarely used in everyday language. This is unfortunate, because it is an important information that helps us better understand and correctly classify many things. Far too often, statistical correlations and cause and effect are lumped together.

Last updated on 06/23/2025. Author: Jürgen Kummer

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