Correlation Coefficient Formula Problems

Problem 1: Calculate the correlation coefficient from the following table:

Solution:

Make a table from the given data and add three more columns of XY, X², and Y².

SUBJECT	AGE (X)	GLUCOSE LEVEL (Y)	XY	X²	Y²
1	42	98	4116	1764	9604
2	23	68	1564	529	4624
3	22	73	1606	484	5329
4	47	79	3713	2209	6241
5	50	88	4400	2500	7744
6	60	82	4980	3600	6724
∑	244	488	20379	11086	40266

∑xy = 20379

∑x = 244

∑y = 488

∑x² = 11086

∑y² = 40266

n = 6.

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}        [/Tex]                                                    

R = 6(20379) – (244)(488) / √[6(11086)-(244)²][6(40266)-(488)² ]                                                 

R = 3202 / √[6980][3452]        

R = 3202/4972.238

R = 0.6439

It shows that the relationship between the variables of the data is a strong positive relationship.

Problem 2: Calculate the correlation coefficient from the following table:

Solution:

Make a table from the given data and add three more columns of XY,  X², and Y².

SUBJECT	AGE (X)	Weight (Y)	XY	X²	Y²
1	40	99	3960	1600	9801
2	25	79	1975	625	6241
3	22	69	1518	484	4761
4	54	89	4806	2916	7921
∑	151	336	12259	5625	28724

∑xy = 12258

∑x = 151

∑y = 336

∑x² = 5625

∑y² 28724

n = 4

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}        [/Tex]     

R = 4(12258) – (151)(336) / √[4(5625)-(151)²][4(28724)-(336)²]        

R = -1704 / √[-301][-2000]        

R=-1704/775.886

R=-2.1961

It shows that the relationship between the variables of the data is a very strong negative relationship.

Problem 3:  Calculate the correlation coefficient for the following data:

X = 7,9,14 and Y = 17,19,21

Solution:

Given variables are,

X = 7,9,14

and,

Y = 17,19,21

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula.

X	Y	XY	X²	Y²
7	17	119	49	36
9	19	171	81	361
14	21	294	196	441
∑ 30	∑ 57	∑ 584	∑ 326	∑ 838

∑xy = 584

∑x = 30

∑y = 57

∑x² = 326

∑y² = 838

n = 3

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}        [/Tex]      

R = 3(584) – (30)(57) / √[3(326)-(30)²][3(838)-(57)²]        

R = 42 / √[78][-735]        

R = 42/-239.43

R = -0.1754

It shows that the relationship between the variables of the data is negligible relationship

Problem 4: Calculate the correlation coefficient for the following data:

X = 21, 31, 25, 40, 47, 38 and Y = 70,55,60,78,66,80

Solution:

Given variables are,

X = 21,31,25,40,47,38

And,

Y = 70,55,60,78,66,80

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula.

X	Y	XY	X²	Y²
21	70	1470	441	4900
31	55	1705	961	3025
25	60	1500	625	3600
40	78	3120	1600	6094
47	66	3102	2209	4356
38	80	3040	1444	6400
∑202	∑409	∑13937	∑7280	∑28265

 ∑xy = 13937

∑x = 202

∑y = 409

∑x² = 7280

∑y² = 28265

n = 6

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}        [/Tex]     

R = 6(13937) – (202)(409) / √[6(7280) – (202)²][6(28265) – (409)²]        

R = 1004 /√[2876][2909]        

R = 1004 / 2892.452938

R = 0.3471

It shows that the relationship between the variables of the data is a moderate positive relationship.

Problem 5: Calculate the correlation coefficient for the following data?

X = 5 ,9 ,14, 16 and Y = 6, 10, 16, 20 .

Solution:

Given variables are,

X = 5 ,9 ,14, 16

And

Y = 6, 10, 16, 20.

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula add all the values in the columns to get the values used in the formula

X	Y	XY	X²	Y²
5	6	30	25	36
9	10	90	81	100
14	16	224	196	256
16	20	320	256	400
∑44	∑52	∑664	∑558	∑792

∑xy = 664

∑x = 44

∑y = 52

∑x² = 558

∑y² = 792

n = 4

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= n(∑xy) – (∑x)(∑y) / √[n∑x²-(∑x)²][n∑y²-(∑y)²  [/Tex]

R = 4(664) – (44)(52) / √[4(558) – (44)²][4(792) – (52)²]

R = 368 / √[296][464]

R = 368/370.599

R = 0.9930

It shows that the relationship between the variables of the data is a very strong positive relationship.

Problem 6: Calculate the correlation coefficient for the following data:

X = 10, 13, 15 ,17 ,19 and Y = 5,10,15,20,25.

Solution:

Given variables are,

X = 10, 13, 15 ,17 ,19 and Y = 5, 10, 15, 20, 25.

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also  add all the values in the columns to get the values used in formula,

X	Y	XY	X²	Y²
10	5	50	100	25
13	10	130	169	100
15	15	225	225	225
17	20	340	340	400
19	25	475	475	625
∑74	∑75	∑1103	∑1144	∑1375

∑xy = 1103

∑x = 74

∑y = 75

∑x² = 1144

∑y² = 1375

n = 5

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}       [/Tex]

R = 5(1103) – (74)(75) / √ [5(1144) – (74)²][5(1375) – (75)²]

R = -35 / √[244][1250]  

R = -35/552.26

R = 0.0633

It shows that the relationship between the variables of the data is a negligible relationship.

Problems 7: Calculate the correlation coefficient for the following data:

X = 12, 10, 42, 27, 35, 56 and Y = 13, 15, 56, 34, 65, 26

Solution:

Given variables are,

X = 12, 10, 42, 27, 35, 56 and Y = 13, 15, 56, 34, 65, 26

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also add all the values in the columns to get the values used in the formula

X	Y	XY	X²	Y²
12	13	156	144	169
10	15	150	100	225
42	56	2352	1764	3136
27	34	918	729	1156
35	65	2275	1225	4225
56	26	1456	3136	676
∑182	∑209	∑7307	∑7098	∑9587

∑xy = 7307

∑x = 182

∑y = 209

∑x² = 7098

∑y² = 9587

n = 6

Put all the values in the Pearson’s correlation coefficient formula:

[Tex] R= \frac{n(∑xy) – (∑x)(∑y)}{\sqrt{[n∑x²-(∑x)²][n∑y²-(∑y)²}}       [/Tex]

R = 6(7307) – (182)(209) / √ {[6(7098) – (182)²][6(9587)-(209)²]}

R = 5804 / √[9464][13841] 

R = 5804/11445.139

R = 0.5071

It shows that the relationship between the variables of the data is a strong positive relationship.

Correlation Coefficient Formula: The correlation coefficient is a statistical measure used to quantify the relationship between predicted and observed values in a statistical analysis. It provides insight into the degree of precision between these predicted and actual values.

Correlation coefficients are used to calculate how vital a connection is between two variables. There are different types of correlation coefficients, one of the most popular is Pearson’s correlation (also known as Pearson’s R)which is commonly used in linear regression.

In this article, learn about the correlation coefficient formula, along with what is correlation, its types, examples, and problems.