Calculating correlation co-efficients

The correlation co-efficient for a set of pairs of data provides a measure of the strength and direction (positive or negative) of the linear relationship between two variables. The most commonly used is the Pearson correlation co-efficient, which uses a least squares method of calculating the dispersion of the data pairs from a theoretical straight-line (linear) relationship.

The correlation co-efficient, r, for a set of (x,y) data pairs is calculated as follows:


The following steps can be followed:

1. Calculate the average values for both x (x*) & y (y*);
2. For each row, calculate (x – x*) and (y – y*);
3. For each row, calculate (x-x*)², (y – y*)², and (x-x*)(y-y*);
4. Add up the values in each column, and store the totals
5. For both x & y values, calculate the standard deviation, s_x and s_y, using the totals for (x-x*)² and (y-y*)² dividing each by the number of rows, and taking the square root of the results.
6. Calculate r using the total for the column of (x-x*)(y-y*) and dividing by (n*s_x*s_y) where n is the number of rows in the table (i.e. the number of x,y pairs.

The following table should help to illustrate the calculation:

	x	y	(x-x*)	(x-x*)2	(y-y*)	(y-y*)2	(x-x*)(y-y*)
	1	8.56	-4	16	1.865556	3.480297531	-7.46222222
	2	8.23	-3	9	1.535556	2.357930864	-4.60666667
	3	7.62	-2	4	0.925556	0.856653086	-1.85111111
	4	7.12	-1	1	0.425556	0.181097531	-0.42555556
	5	6.99	0	0	0.295556	0.087353086	0
	6	7.05	1	1	0.355556	0.126419753	0.355555556
	7	4.98	2	4	-1.71444	2.939319753	-3.42888889
	8	5.37	3	9	-1.32444	1.754153086	-3.97333333
	9	4.33	4	16	-2.36444	5.590597531	-9.45777778
mean x*	5		Total	60		17.37382222	-30.85
mean y*		6.694444	std dev	2.581989		1.38939724
		r=	-0.9555

In the above table, the x values represent the number of guests; and the y values represent the conversion rate given as a percentage. The columns headed by (x-x*)² and (y-y*)² are used in the calculation of the standard deviations for x and y – s_x and s_y. Once the last column is calculated, the values are totaled, giving the numerator (upper value of the fraction) in the equation for r.

For the above example, r is calculated as:

The use of the correlation co-efficient enables a determination as to whether or not there exists a relationship between the variables. A strong correlation does not indicate a causal relationship in the data; although causal relationships show strong correlation.

Note: the value for the correlation co-efficient r can range from -1 to 1. A value towards either end of the range indicates a strong correlation between the variables; values close to 0 indicate very little or no correlation.