A simple linear regression is a method in statistics which is used to determine the relationship between two continuous variables. A simple linear regression fits a straight line through the set of n points. Learn here the definition, formula and calculation of simple linear regression. Check out this simple/linear regression tutorial and examples here to learn how to find regression equation and relationship between two variables. using the slope and y-intercept.

A regression is a statistical analysis assessing the association between two variables. In simple linear regression, a single independent variable is used to predict the value of a dependent variable.

x and y are the variables.
b = The slope of the regression line
a = The intercept point of the regression line and the y axis.
N = Number of values or elements
X = First Score
Y = Second Score
ΣXY = Sum of the product of first and Second Scores
ΣX = Sum of First Scores
ΣY = Sum of Second Scores
ΣX^{2} = Sum of square First Scores

To find the Simple/Linear Regression of

X Values | Y Values |
---|---|

60 | 3.1 |

61 | 3.6 |

62 | 3.8 |

63 | 4 |

65 | 4.1 |

To find regression equation, we will first find slope, intercept and use it to form regression equation.

Count the number of values. N = 5

Find XY, X^{2}
See the below table

X Value | Y Value | X*Y | X*X |
---|---|---|---|

60 | 3.1 | 60 * 3.1 =186 | 60 * 60 = 3600 |

61 | 3.6 | 61 * 3.6 = 219.6 | 61 * 61 = 3721 |

62 | 3.8 | 62 * 3.8 = 235.6 | 62 * 62 = 3844 |

63 | 4 | 63 * 4 = 252 | 63 * 63 = 3969 |

65 | 4.1 | 65 * 4.1 = 266.5 | 65 * 65 = 4225 |

Find ΣX, ΣY, ΣXY, ΣX^{2}.
ΣX = 311
ΣY = 18.6
ΣXY = 1159.7
ΣX^{2} = 19359

Substitute in the above slope formula given.
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX^{2} - (ΣX)^{2})
= ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)^{2})
= (5798.5 - 5784.6)/(96795 - 96721)
= 13.9/74
= 0.18784

Now, again substitute in the above intercept formula given. Intercept(a) = (ΣY - b(ΣX)) / N = (18.6 - 0.18784(311))/5 = (18.6 - 58.41824)/5 = -39.81824/5 = -7.964

Then substitute these values in regression equation formula
Regression Equation(y) = a + bx
= -7.964+0.188x.

Suppose if we want to know the approximate y value for the variable x = 64. Then we can substitute the value in the above equation.
Regression Equation(y) = a + bx
= -7.964+0.188(64).
= -7.964+12.032.
= 4.068
This example will guide you to find the relationship between two variables by calculating the Regression from the above steps.

This tutorial will help you dynamically to find the Simple/Linear Regression problems.