**Statistics Calculators** ▶ Linear Regression Calculator

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**Table of Content**

An online linear regression calculator is programmed to determine the value of a dependent variable on the basis of an independent variable. Using this tool will assist you to determine the line of best fit for paired data.

Let us discuss the concept of linear regression in detail.

In statistics, a simple linear regression refers to the method to determine the value of a dependent variable on behalf of an independent variable. A linear regression always shows that there is a linear relationship between the variables. Linear-regression model is a way that is scientifically proven in order to predict the future. The online linear regression calculator is a free tool to determine the linear regression of any data of paired set.

Line of best fit, also known as “trend line” is a line that passes through a set of data points having scattered plot and shows the relationship between those points. By doing a simple regression analysis of one or two independent variables, we will always get a straight line. A free line of best fit calculator allows you to perform this type of analysis to generate a most suitable plot against all data points.

If multiple regression variables are involved in the process, then we may get a curved line.

A form of mathematical analysis that is adopted to determine the least squares regression line for a data set and provides proper graphical demonstration between the data points is known as least squares method. In practice, statisticians use this method to approach the line of best fit for any set of data given. Each and every point in data shows a proper relationship between a dependent variable that is unknown and an independent variable that is always known. A least squares regression line calculator uses the least squares method to determine the line of best fit by providing you with detailed calculations.

A mean is considered as the average of the values given.

**M = sum of the values given / No. of values**

You can evaluate the line representing the points by using the following linear regression formula for a given data:

**ŷ=bX+a**

where;

**ŷ** = dependent variable to be determined

**b**= slope of the line

X = independent variable

**a** = intercept ( the value of y when X = 0)

A regression equation calculator uses the same mathematical expression to predict the results.

You can determine the value of a and b by subjecting to the following equations:

**Value of a = MY − (b × MX)**

Where;

**Mx** = mean value for x

**My** = mean value for y

**Value of b = SP/SSx**

Where;

**SP (∑xy) = (X – Mx)*(Y – My)**

**SSx (∑x²) = (X – Mx)2**

The online linear regression calculator uses all these formulas to predict the results.

Let us solve a couple of examples to better understand the linear regression analysis:

**Example # 01:**

Determine the regression line equation for the set of data given below:

**{(1, 3), (5, 4), (4, 8), (5, 6), (2, 3)}**

**Solution:**

**Sum of X = 1+5+4+5+2**

** =17**

**Sum of Y = 3+4+8+6+3**

** = 24**

Now, the mean is calculated as follows:

**Mean of X = Mx = Sum of X / 5**

**= 3.4**

**Mean of Y = sum of Y /5**

**= 24 / 5**

**= 4.8**

Now , we have to calculator the following quantities as follows:

X – Mx |
Y – My |
(X – Mx)2 |
(X – Mx)*(Y – My) |

-2.4 |
-1.8 |
5.76 |
4.32 |

1.6 |
-0.8 |
2.56 |
-1.28 |

0.6 |
3.2 |
0.36 |
1.92 |

1.6 |
1.2 |
2.56 |
1.92 |

-1.4 |
-1.8 |
1.96 |
2.52 |

**SSx (∑x²) = (X – Mx)2**

**= 5.76+2.56+0.36+2.56+1.96**

**= 13.2**

**SP (∑xy) = (X – Mx)*(Y – My)**

**= 4.32-1.28+1.92+1.92+2.52**

**= 9.4**

The least squares regression line formula is given as follows:

**ŷ=bX+a**

First, we have to accumulate the value for a and b:

**b = SP/SSx**

**= 9.4 / 13.2**

**= 0.71212**

The values of a is determined as follows:

**a = MY−(b×MX)**

**= 4.8 – (0.71212 * 3.4)**

**= 2.378792**

By using line of best fit equation:

**ŷ=bX+a**

Putting the values of a and b :

**ŷ = 0.71212X + 2.378792**

The graphical plot of linear regression line is as follows:

Our free online linear regression calculator gives step by step calculations of any regression analysis.

**Example #02:**

Find the least squares regression line for the data set as follows:

**{(2, 9), (5, 7), (8, 8), (9, 2)}.**

Also work for the estimated value of y for the value of X to be 2 and 3.

**Solution:**

**Sum of X = 24**

**Sum of Y = 26**

The mean is evaluated as :

**Mean of X =Mx= 2+5+8+9 / 4**

**= 6**

**Mean of Y = My = 9+7+8+2 / 4**

**= 6.5**

Now, we have to calculate the following quantities:

X – Mx |
Y – My |
(X – Mx)2 |
(X – Mx)*(Y – My) |

-4 |
2.5 |
16 |
-10 |

-1 |
0.5 |
1 |
-0.5 |

2 |
1.5 |
4 |
3 |

3 |
-4.5 |
9 |
-13.5 |

**SSx (∑x²) = (X – Mx)2 **

**= 16+1+4+9**

**= 30**

**SP (∑xy) = (X – Mx)*(Y – My)**

**= -10-0.5+3-13.5**

**= -21**

Now , we have to determine the linear regression equation:

**ŷ= bX+a**

Determining the value of a and asb as follows:

**b = SP/SSx**

**= -21 / 30**

**= -07**

**a = MY−(b×MX)**

**= 6.5 – (-.07 * 6)**

**=10.7**

Now , putting all the values in linear regression formula::

**ŷ = -0.7x + 10.7**

For given values of X, the estimated values of Y are as follows:

Estimate |
Estimated Y |

2 |
9.3 |

3 |
8.6 |

The graphical plot of line of best fit is as follows:

Using free best fit line calculator assists you to generate estimated values for which you have to plot the line of best fit.

Whenever you are subjected to find the predicted value of Y and linear regression line for any set of data given, you can use our free online regression line calculator.

Let us see what to do:

**Input:**

- Enter the values of x
- After this, Enter the values of y
- If asked, then input the values of X to determine estimate values of Y
- Click ‘calculate’

**Output:**

Depending upon the inputs given, he calculator calculates:

- Sum of X and Y
- Mean of X and Y
- Regression line equation
- Step by step calculations
- Rectangular plot of the regression line

You can determine the linear regression in a variety of softwares including:

- R linear regression
- MATLAB linear regression
- Sklearn linear regression
- Linear regression Python

Excel linear regression

Linear regression has a vast use in the field of finance, biology, mathematics and statistics.

The difference between an observed value of the response variable and the value of the response variable predicted from the regression line is known as residual in the regression line.

A dependent variable is the one whose value is to be determined. Whereas, an independent variable is the one whose value is always given.

X is the independent variable and Y is the dependent variable.

A linear regression is considered as the linear approach that is used for modeling the relationship between a scalar response and one or more independent variables. Regression has a broad use in the field of engineering and technology as it is used to predict the future resulting values and considerable plots. A linear regression calculator is a free tool to solve any complicated problem regarding regression analysis.

From the source of wikipedia: Interpretation, Extensions, General linear models, Heteroscedastic models, Generalized linear models, Trend line, Machine learning

, Economics, Finance.

From the source of khan academy: Fitting a line to data, Equations of trend lines, Estimating the line of best fit .

From the source of lumen learning: Regression Analysis, Conditions for Regression Inference, A Graph of Averages, The Regression Fallacy.