This linear Regression calculator helps to determine the linear regression equation by using the least square method for the data groups “X” and “Y”. Where the Data group “X” behaves as an independent variable and Data group “Y” behaves as a dependent variable.

The general form of linear regression equation can be represented by “Y = mX + b” and “m” represents the slope of the line or “b” shows the y-intercept (when the value X = 0) in graphical analysis. This calculator finds the value of the Y-intercept and slope of the line for the general form of a linear regression equation.

## What is linear regression?

In a simple Linear Regression model, the relationship between the dependent variable (Y) and the independent variable (X) is represented by a straight line:

**Where:**

- “Y” is the dependent variable (output/response)
- “X
_{i}” is the independent variable - “f” is a linear function
- “β” is an unknown parameter
- “e
_{i}” are the error terms

### What is a linear regression model?

Linear Regression model is an exclusive model technique to determine the relation between the dependent and independent variables. Additionally, this model gives the linear regression equation of the response variable based on the independent predictor variable.

The linear regression formula for this model is “Y = mX + b” where Y is the response variable that is dependent on the “X” predictor variable that is independent. While “m” is the slope of the model and “b” is the calculated intercept for y.

## How to calculate linear regression?

To calculate the linear regression manually, first find the value of the “m” slope and “b” y-intercept of the given data. Then put the values in the linear regression formula and get the final linear regression of the given data. For the value of “m” and “b” use the below formulas.

**m= [(n × ∑X**_{i} × Y_{i} ) − (∑X_{i }× ∑y_{i})] /[n × (∑X^{2}−(∑X)^{2})],

** **`b = (∑Y`_{i}) − (m × ∑X_{i})/n

To find the value of “m” and “b”, perform the correlation analysis by following the below steps and generating a linear regression equation.

First, find the mean of the given data groups.

Note the number of terms in the given data group.

Find the value of the sum of the data groups separately, its product, and the squares of the individual data groups.

Put the all above values in the “m” and “b” formulas.

After simplification, the final results of “m” and “b” are put into the general form of a linear regression equation and get the required linear regression model.

Alternatively, use our above regression line calculator to simplify the process of linear regression analysis of two data sets.

## Important Considerations:

While Linear Regression is a powerful and widely used statistical technique, it's essential to consider its assumptions and limitations:

**Linearity: **

The relationship between the independent and dependent variables must be linear. If the relationship is nonlinear, other methods may be more appropriate.

Independence: The observations should be independent of each other. In cases of time series or spatial data, other techniques may be more suitable.

**Homoscedasticity: **

The variance of the error terms should be constant across all levels of the independent variable.

**Normality: **

The error terms should be normally distributed.

When using Linear Regression, always validate the assumptions and evaluate the model's performance using appropriate metrics, such as the coefficient of determination (R-squared), residual analysis, and cross-validation.

## Linear regression examples:

In this section, we’ll describe the method of calculating linear regression between any two data sets.

**Example 1: **

Calculate the regression equation if:

X = 5, 22, 19, 8, 33, 10

Y = 9, 71, 31, 12, 44, 28

**Solution: **

**Step 1:** Calculate the mean of the data sets

**Mean of X** = μ_{x} = ΣX / n

μ_{x} = (5 + 22 + 19 + 8 + 33 + 10) / 6

μ_{x} = 97 / 6

μ_{x} = 16.67

Mean of Y = μ_{y} = ΣY / n

μ_{y} = (9 + 71 + 31 + 12 + 44 + 28) / 6

μ_{y} = 195 / 6

μ_{y} = 32.5

**Step 2:** Make a table:

X_{i} | Y_{i} | X_{i} * Y_{i} | X_{i}^{2} | Y_{i}^{2} |

5 | 9 | 45 | 25 | 81 |

22 | 71 | 1562 | 484 | 5041 |

19 | 31 | 589 | 361 | 961 |

8 | 12 | 96 | 64 | 144 |

33 | 44 | 1452 | 1089 | 1936 |

10 | 28 | 280 | 100 | 784 |

Σ X_{i} = 97 | Σ Y_{i} = 195 | Σ X_{i} * Y_{i} = 4024 | Σ X_{i}^{2} = 2123 | Σ Y_{i}^{2} = 8947 |

**Step 3:** Calculate slope “**m**”

m = {(n * ∑ X_{i} *Y_{i}) − (∑X_{i} * ∑Y_{i})} / {n * ∑X_{i}^{2 }− (∑X_{i})^{2}}

m = {(6 * 4024) – (97 * 195)} / {(6 *2123) - (9409)}

m = 5229 / 3329

**m = 1.5707**

**Step 4:** Calculate Y-intercept

b = {(∑Y_{i}) − (m * ∑X_{i})} / n

b = {(195) − (1.5707 * 97)} / 6

**b = 7.107**

**Step 5:** Make the linear equation

Y = (slope)x + (y-intercept)

## Frequently Asked Questions

**What are the types of Linear Regression?**

There are two types of linear regression depending on the dependent variable and independent variables such as

**What’s the linear regression formula represent?**

The linear regression formula is represented by the Y = mX + b, which represents the change in response variable “Y” by changing the values of the independent variable “X”. Also, the predicted value of “m” shows the change in y by changing the “X” or the predicted value of “b” shows where the regression line cuts the y-axis.

**How to find linear regression prediction?**

Use the simple linear regression formula which is “Y = mX + b” and find the value of “m” & “b” by using the correlation analysis and predict the results. Where “m” is the estimated slope and “b” is the estimated y-intercepts. To clever prediction, use the above regression equation calculator that estimates the values of slope and y-intercept.

**Which Assumptions Must Hold for Linear Regression Models?**

To interpret the proper results of the regression model some assumptions must hold:

The relationship between all variables is linear.

All predictor variables are independent of one another.

All variables are normally distributed.

The degree of all variables is one.

**What is the response and predictor variable in a linear regression?**

In linear regression analysis, the outcome variable is known as the response or dependent variable that is normally represented by “Y” and the variable that is used for the prediction is known as predictor or independent variable that normally represented by the “X”.