Once the degree of relationship between variables has been established using co-relation analysis, it is natural to delve into the nature of relationship. Regression analysis helps in determining the cause and effect relationship between variables. It is possible to predict the value of other variables (called dependent variable) if the values of independent variables can be predicted using a graphical method or the algebraic method.

## Graphical Method

It involves drawing a scatter diagram with independent variable on X-axis and dependent variable on Y-axis. After that a line is drawn in such a manner that it passes through most of the distribution, with remaining points distributed almost evenly on either side of the line.

A regression line is known as the line of best fit that summarizes the general movement of data. It shows the best mean values of one variable corresponding to mean values of the other. The regression line is based on the criteria that it is a straight line that minimizes the sum of squared deviations between the predicted and observed values of the dependent variable.

## Algebraic Method

Algebraic method develops two regression equations of X on Y, and Y on X.

### Regression equation of Y on X

${Y = a+bX}$

Where −

- ${Y}$ = Dependent variable
- ${X}$ = Independent variable
- ${a}$ = Constant showing Y-intercept
- ${b}$ = Constant showing slope of line

Values of a and b is obtained by the following normal equations:

${\sum Y = Na + b\sum X \\[7pt]

\sum XY = a \sum X + b \sum X^2

}$

Where −

- ${N}$ = Number of observations

### Regression equation of X on Y

${X = a+bY}$

Where −

- ${X}$ = Dependent variable
- ${Y}$ = Independent variable
- ${a}$ = Constant showing Y-intercept
- ${b}$ = Constant showing slope of line

Values of a and b is obtained by the following normal equations:

${\sum X = Na + b\sum Y \\[7pt]

\sum XY = a \sum Y + b \sum Y^2

}$

Where −

- ${N}$ = Number of observations

### Example

**Problem Statement:**

A researcher has found that there is a co-relation between the weight tendencies of father and son. He is now interested in developing regression equation on two variables from the given data:

Weight of father (in Kg) | 69 | 63 | 66 | 64 | 67 | 64 | 70 | 66 | 68 | 67 | 65 | 71 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Weight of Son (in Kg) | 70 | 65 | 68 | 65 | 69 | 66 | 68 | 65 | 71 | 67 | 64 | 72 |

Develop

- Regression equation of Y on X.
- Regression equation of on Y.

**Solution:**

${X}$ | ${X^2}$ | ${Y}$ | ${Y^2}$ | ${XY}$ |
---|---|---|---|---|

69 | 4761 | 70 | 4900 | 4830 |

63 | 3969 | 65 | 4225 | 4095 |

66 | 4356 | 68 | 4624 | 4488 |

64 | 4096 | 65 | 4225 | 4160 |

67 | 4489 | 69 | 4761 | 4623 |

64 | 4096 | 66 | 4356 | 4224 |

70 | 4900 | 68 | 4624 | 4760 |

66 | 4356 | 65 | 4225 | 4290 |

68 | 4624 | 71 | 5041 | 4828 |

67 | 4489 | 67 | 4489 | 4489 |

65 | 4225 | 64 | 4096 | 4160 |

71 | 5041 | 72 | 5184 | 5112 |

${\sum X = 800}$ | ${\sum X^2 = 53,402}$ | ${\sum Y = 810}$ | ${\sum Y^2 = 54,750}$ | ${\sum XY = 54,059}$ |

## Regression equation of Y on X

Y = a+bX

Where , a and b are obtained by normal equations

\sum XY = a \sum X + b \sum X^2 \\[7pt]

Where\ \sum Y = 810, \sum X = 800, \sum X^2 = 53,402 \\[7pt]

, \sum XY = 54, 049, N = 12 }$

${\Rightarrow}$ 810 = 12a + 800b … (i)

${\Rightarrow}$ 54049 = 800a + 53402 b … (ii)

Multiplying equation (i) with 800 and equation (ii) with 12, we get:

96000 a + 640000 b = 648000 … (iii)

96000 a + 640824 b = 648588 … (iv)

Subtracting equation (iv) from (iii)

-824 b = -588

${\Rightarrow}$ b = -.0713

Substituting the value of b in eq. (i)

810 = 12a + 800 (-0.713)

810 = 12a + 570.4

12a = 239.6

${\Rightarrow}$ a = 19.96

Hence the equation Y on X can be written as

## Regression equation of Y on X

X = a+bY

Where , a and b are obtained by normal equations

\sum XY = a \sum Y + b \sum Y^2 \\[7pt]

Where\ \sum Y = 810, \sum Y^2 = 54,750 \\[7pt]

, \sum XY = 54, 049, N = 12 }$

${\Rightarrow}$ 800 = 12a + 810a + 810b … (V)

${\Rightarrow}$ 54,049 = 810a + 54, 750 … (vi)

Multiplying eq (v) by 810 and eq (vi) by 12, we get

9720 a + 656100 b = 648000 … (vii)

9720 a + 65700 b = 648588 … (viii)

Subtracting eq viii from eq vii

900b = -588

${\Rightarrow}$ b = 0.653

Substituting the value of b in equation (v)

800 = 12a + 810 (0.653)

12a = 271.07

${\Rightarrow}$ a = 22.58

Hence regression equation of X and Y is

Table of Contents

1.statistics adjusted rsquared

2.statistics analysis of variance

4.statistics arithmetic median

8.statistics best point estimation

9.statistics beta distribution

10.statistics binomial distribution

11.statistics blackscholes model

13.statistics central limit theorem

14.statistics chebyshevs theorem

15.statistics chisquared distribution

16.statistics chi squared table

17.statistics circular permutation

18.statistics cluster sampling

19.statistics cohens kappa coefficient

21.statistics combination with replacement

23.statistics continuous uniform distribution

24.statistics cumulative frequency

25.statistics coefficient of variation

26.statistics correlation coefficient

27.statistics cumulative plots

28.statistics cumulative poisson distribution

30.statistics data collection questionaire designing

31.statistics data collection observation

32.statistics data collection case study method

34.statistics deciles statistics

36.statistics exponential distribution

40.statistics frequency distribution

41.statistics gamma distribution

43.statistics geometric probability distribution

46.statistics gumbel distribution

49.statistics harmonic resonance frequency

51.statistics hypergeometric distribution

52.statistics hypothesis testing

53.statistics interval estimation

54.statistics inverse gamma distribution

55.statistics kolmogorov smirnov test

57.statistics laplace distribution

58.statistics linear regression

59.statistics log gamma distribution

60.statistics logistic regression

63.statistics means difference

64.statistics multinomial distribution

65.statistics negative binomial distribution

66.statistics normal distribution

67.statistics odd and even permutation

68.statistics one proportion z test

69.statistics outlier function

71.statistics permutation with replacement

73.statistics poisson distribution

74.statistics pooled variance r

75.statistics power calculator

77.statistics probability additive theorem

78.statistics probability multiplicative theorem

79.statistics probability bayes theorem

80.statistics probability density function

81.statistics process capability cp amp process performance pp

83.statistics quadratic regression equation

84.statistics qualitative data vs quantitative data

85.statistics quartile deviation

86.statistics range rule of thumb

87.statistics rayleigh distribution

88.statistics regression intercept confidence interval

89.statistics relative standard deviation

90.statistics reliability coefficient

91.statistics required sample size

92.statistics residual analysis

93.statistics residual sum of squares

94.statistics root mean square

96.statistics sampling methods

98.statistics shannon wiener diversity index

99.statistics signal to noise ratio

100.statistics simple random sampling

102.statistics standard deviation

103.statistics standard error se

104.statistics standard normal table

105.statistics statistical significance

108.statistics stem and leaf plot

109.statistics stratified sampling

112.statistics tdistribution table

113.statistics ti 83 exponential regression

114.statistics transformations

116.statistics type i amp ii errors

119.statistics weak law of large numbers