Regression refers to a statistical that attempts to determine the strength of the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables). Here the relation between selected values of x and observed values of y (from which the most probable value of y can be predicted for any value of x) are taken into consideration.

Formula

Regression Equation(y) = a + bx

Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX

Intercept(a) = (ΣY - b(ΣX)) / N

Where

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

ΣX2 = Sum of square First Scores

**Example:**

To find the Simple/Linear Regression of

X Values | Y Values |

2 | 2.1 |

5 | 2.6 |

7 | 2.8 |

4 | 4 |

8 | 4.1 |

**Step 1:** Count the number of values.

N = 5

**Step 2:** Find XY, X^{2}

See the below table

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

2 | 2.1 | 2 * 2.1 = 4.2 | 2*2 =4 |

5 | 2.6 | 5 * 2.6 = 13 | 5*5 = 25 |

7 | 2.8 | 7 * 2.8 = 19.6 | 7*7 =49 |

4 | 4 | 4 * 4 = 16 | 4*4 = 16 |

8 | 4.1 | 8 * 4.1 = 32.8 | 8*8 = 64 |

**Step 3:** Find ΣX, ΣY, ΣXY, ΣX^{2}.

ΣX = 26

ΣY = 15.6

ΣXY = 85.6

ΣX2 =158

**Step 4:** Substitute in the above slope formula given.

Slope (b) = (NΣXY - (ΣX)(ΣY)) / (NΣX^{2} - (ΣX)^{2})

= ((5)*(85.6)-(26)*(15.6))/((5)*(158)-(26)^{2})

= (428 – 405.6)/(790 - 676)

= 22.4/114

= 0.19649

**Step 5:** Now, again substitute in the above intercept formula given.

Intercept (a) = (ΣY - b(ΣX)) / N

= (15.6 - 0.196(26))/5

= (15.6 – 5.106)/5

= 10.494/5

= 2.0988

Regression Equation(y) = a + bx

= 2.0988 + 0.196x.

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