So far,

• Central Tendency: A single value that reflects the nature of a distribution .
• Dispersion: Associating some number to the ‘spread’ of a distribution.

If you noticed, it’s about measuring various properties of ‘a single’ distribution.

But what if we have to compare two distribution?

Inspiration

Suppose one of your friends said, “Tall people tend to weigh more in my family”. It looks intuitively looks logical – but is it true scientifically?

To prove it, you and your friend collected information from your friend’s family: Dad, mom, your friend and his two siblings. You can find the average of height and weight separately, also find how ‘spread’ the data is (measure of dispersion). Now, how will you compare the “Heights” table and “Weights” table and make conclusions?

For that, we have a tool called Correlation Coefficient.

Correlation Coefficient

Correlation coefficient (developed by Karl Pearson) is a way to indicate how closely related two sets of data are. Correlation coefficient is given by:

Example 1
Calculate the correlation coefficient between marks in test 1 (X) and Test 2 (Y)

X	5	6	7	7	8
Y	7	8	5	2	2

Solution:

X	Y	XY	X^2	Y^2
5	7	35	25	49
6	8	48	36	64
7	5	35	49	25
7	2	14	49	4
8	2	16	64	4
∑X=33	∑Y=24	∑XY=148	∑X2 =223	∑Y2 =146

Calculating means,

Meaning of Correlation Coefficient

We got correlation coefficient value, and it is negative – what does it mean?

Correlation usually has one of two directions. These are positive or negative. If it is positive, then the two sets go up together. If it is negative, then one goes up while the other goes down.

In example 1, the scatter plot (plotting the points (x,y) on a graph) will give us a plot like this:

Clear that our values of Y are decreasing as value of X increases. Remember: we got our correlation coefficient as -0.8218!

But… it’s about checking if value of Y increases/decreases with value of X? Can’t we do it by observing the data or scatter plot? Not exactly. When your data set is bigger, or say ‘weakly’ correlated, it would be hard to catch such trends. Correlation coefficient helps us associate a degree of relation between the distributions as well. So, it’s a very useful tool!

Simplifying Calculations

Example 2
Find the correlation coefficient of the following marks scored by 5 students in two exams:

X	45	44	34	25
Y	50	49	35	20

Solution:

X	Y	XY	X^2	Y^2
45	50	2250	2025	2500
44	49	2156	1936	2401
34	35	1190	1156	1225
25	20	500	625	400
∑X=148	∑Y=154	∑XY=6096	∑X2 =5742	∑Y2 =6526

If you continue to calculate the correlation coefficient, you will get the answer as

But – big numbers, looks scary! Remember the trick we used for standard deviation? Turns out it will work for Correlation, too!

X	Y	U = X – 34	V = Y – 35	UV	U^2	V^2
45	50	11	15	165	121	225
44	49	10	14	140	100	196
34	35	0	0	0	0	0
25	20	-9	-15	135	81	225
		∑U=12	∑V=14	∑UV = 440	∑U2 = 302	∑V2 =646

Continue the calculation and find (Answer will be 0.9987)

Take a look at what happens to the scatter plot:

Look how the trend seen in scatter plot is the same, and how Y (and V) increases as X (and U) increases!

The Trick

• Choose the numbers
  • A (preferably the median of X),
  • B (preferably the median of Y) •
• Let U = X – A and V = Y – B
• Find r_UV
• r_UV = r_XY

Example 3:
Here’s the Height-Weight of 7 random entries in this Kaggle dataset: Find the correlation coefficient and make some observation about the correlation between height and weight.

Height (X)	174	189	185	195	189	195	192
Weight (Y)	96	87	110	104	104	81	101

(Try it yourself)

Some Properties of Correlation Coefficient

• r_XY = r_YX
• – 1 ≤ r ≤ 1
• r > 0 when X increases as Y increases
• r = 0 when X and Y have no connection
• r < 0 when X decreases as Y increases

Extra Reading

• Karl Pearson (Simple Wikipedia)
• Correlation (Simple Wikipedia)
• Introduction to Correlation (MathIsFun)