Correlation between {6.1, 6.0, 5.9, 5.8, . . . , 4.8} & {5.9, 5.7, 6.1, 5.2, . . . , 5.7}
Solved example problem, work with steps & calculation summary for correlation (r) between {6.1, 6.0, 5.9, 5.8, . . . , 4.8} & {5.9, 5.7, 6.1, 5.2, . . . , 5.7} to estimate the linear relationship or to find if the data linearly, non-linearly, positively or negatively correlated in statistical experiments.| Calculation Summary | |
|---|---|
| Data set x | {6.1, 6.0, 5.9, 5.8, 5.5, . . . . , 4.9 & 4.8} |
| Data set y | {5.9, 5.7, 6.1, 5.2, 5.7, . . . . , 4.1 & 5.7} |
| Correlation Coefficient (r) | 0.6328 |
Work with Steps for Correlation r = 0.6328
Question:
The following data collected from the survey taken at 11 persons. The purpose of this survey is to estimate the height correlation between each father & son in the group. All the measurements are given in feet & inches.
The following data collected from the survey taken at 11 persons. The purpose of this survey is to estimate the height correlation between each father & son in the group. All the measurements are given in feet & inches.
| All the measurement in feet & inches | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Father | 6.1 | 6.0 | 5.9 | 5.8 | 5.5 | 5.4 | 5.2 | 5.1 | 5.0 | 4.9 | 4.8 |
| Son | 5.9 | 5.7 | 6.1 | 5.2 | 5.7 | 4.8 | 5.5 | 4.3 | 4.3 | 4.1 | 5.7 |
Workout :
step 1 Address the formula, input parameters and values
data set x = {6.1, 6.0, 5.9, 5.8, 5.5, . . . . , 4.9 and 4.8}
data set y = {5.9, 5.7, 6.1, 5.2, 5.7, . . . . , 4.1 and 5.7}
Total number of elements (n) = 11
step 2 Find x̄ & ȳ
x̄ = 59.7/11
x̄ = 5
ȳ = 57.3/11
ȳ = 5
step 3 To find coefficient correlation follow below the table
step 4 Substitute ∑x, ∑y, ∑xy, ∑x2 & ∑y2 value in the below correlation coefficient formula
r = ∑XY√∑X2. ∑Y2
∑XY = 2.1073, ∑X2 = 2.1618, ∑Y2 = 5.1291
r = 2.1073√2.1618 x 5.1291
step 5 Simplify above expression
= 2.1073√11.0882
= 2.10733.3299
r = 0.6328
Thus 0.6328 is the correlation between x = {6.1, 6.0, 5.9, 5.8, 5.5, . . . . , 4.9 and 4.8} & y = {5.9, 5.7, 6.1, 5.2, 5.7, . . . . , 4.1 and 5.7}
step 1 Address the formula, input parameters and values
data set x = {6.1, 6.0, 5.9, 5.8, 5.5, . . . . , 4.9 and 4.8}
data set y = {5.9, 5.7, 6.1, 5.2, 5.7, . . . . , 4.1 and 5.7}
Total number of elements (n) = 11
step 2 Find x̄ & ȳ
x̄ = 59.7/11
x̄ = 5
ȳ = 57.3/11
ȳ = 5
step 3 To find coefficient correlation follow below the table
| x | y | X = x - x̄ | Y = y - ȳ | X2 | Y2 | XY |
| 6.1 | 5.9 | 0.6727 | 0.6909 | 0.4525 | 0.4773 | 0.4648 |
| 6.0 | 5.7 | 0.5727 | 0.4909 | 0.328 | 0.241 | 0.2811 |
| 5.9 | 6.1 | 0.4727 | 0.8909 | 0.2234 | 0.7937 | 0.4211 |
| 5.8 | 5.2 | 0.3727 | -0.0091000000000001 | 0.1389 | 0.0001 | -0.0034 |
| 5.5 | 5.7 | 0.0727 | 0.4909 | 0.0053 | 0.241 | 0.0357 |
| 5.4 | 4.8 | -0.027299999999999 | -0.4091 | 0.0007 | 0.1674 | 0.0112 |
| 5.2 | 5.5 | -0.2273 | 0.2909 | 0.0517 | 0.0846 | -0.0661 |
| 5.1 | 4.3 | -0.3273 | -0.9091 | 0.1071 | 0.8265 | 0.2975 |
| 5.0 | 4.3 | -0.4273 | -0.9091 | 0.1826 | 0.8265 | 0.3885 |
| 4.9 | 4.1 | -0.5273 | -1.1091 | 0.278 | 1.2301 | 0.5848 |
| 4.8 | 5.7 | -0.6273 | 0.4909 | 0.3935 | 0.241 | -0.3079 |
| ∑x = 59.7 | ∑y = 57.3 | ∑X = -0.0003 | ∑Y = -0.0001 | ∑X2 = 2.1618 | ∑Y2 = 5.1291 | ∑XY = 2.1073 |
step 4 Substitute ∑x, ∑y, ∑xy, ∑x2 & ∑y2 value in the below correlation coefficient formula
r = ∑XY√∑X2. ∑Y2
∑XY = 2.1073, ∑X2 = 2.1618, ∑Y2 = 5.1291
r = 2.1073√2.1618 x 5.1291
step 5 Simplify above expression
= 2.1073√11.0882
= 2.10733.3299
r = 0.6328
Thus 0.6328 is the correlation between x = {6.1, 6.0, 5.9, 5.8, 5.5, . . . . , 4.9 and 4.8} & y = {5.9, 5.7, 6.1, 5.2, 5.7, . . . . , 4.1 and 5.7}
