Correlation and Regression - CMA

Regression Lines

A regression line is a line which is used to describe the behavior of a set of data. In other words, it gives the best trend of the given data. In this article, we will learn more about Regression lines and why it is important.

Why Regression lines are important?

Regression lines are useful in forecasting procedures. Its purpose is to describe the interrelation of the dependent variable(y variable) with one or many independent variables(x variable).

Using the equation obtained from the regression line acts as an analyst who can forecast future behaviors of the dependent variables by inputting different values for the independent ones.

Regression lines

Regression Line Formula: y = a + bx + u

Multiple Regression Line Formula: y= a + b1x1 +b2x2 + b3x3 +…+ btxt + u

Where linear regression is used?

Regression lines are used in the financial sector and in business. Various financial analyst employs linear regressions to forecast stock prices, commodity prices and to perform valuations for many different securities. Various companies employ linear regressions for the purpose of forecasting sales, inventories, and many other variables.

Browse more Topics under Correlation And Regression

The Least Square Regression

Generally, linear regression finds the straight line. It is also known as the least squares regression line. It represents in a bivariate dataset.

Let us suppose that y is a dependent variable. X is the independent variable. The population regression line is

Y = a0 + a1x

Where a0 is the constant and a1 is the regression coefficient and x is the value of the independent variable. If you are given a random sample of observation, the population regression line is estimated by

Y’ = a0 + a1x where a0 is the constant and b1 is the regression coefficient. Here you will, ‘x’ is the value of the independent variable and y’ is the predicted value of the dependent variable.

Some Important Properties of the Regression Lines

  • Regression coefficients values remain the same. Since shifting of origin takes place because of the change of scale. The property says
  • If the variables x and y are changed to u and v respectively u= (x-a)/p v=(y-c) /q, Here p and q are the constants.Byz =q/p*bvu Bxy=p/q*buv.
  • If there are two lines of regression. Both of these lines intersect at a specific point [x’, y’]. Variables x and y are taken into consideration. According to the property, the intersection of both the lines of regression i.e. y on x and y is [x’, y’]. This is the solution for both of the equations of variables x and y.
  • You will find the correlation coefficient between the two variables x and y is the geometric mean of both the coefficients. Also, the sign over the values of correlation coefficients will be the common sign of both the coefficients. So, if according to the property regression coefficients are byx= (b) and bxy= (b’) then the correlation coefficient is r=+-sqrt (byx + bxy) so, in some cases, both the coefficients give a negative value and r is also negative. If both the values of coefficients are positive the r will be positive.
  • The regression constant (a0) is equal to the y-intercept of the regression line. Where a0 and a1 are the regression parameters.

Solved Question on Regression Line

Q. How to calculate the Regression Line?

In statistics, you can calculate a regression line for two variables. you can use a scatterplot to get a graph.

Scatterplot of cricket chirps in relation to outdoor temperature

For the best fitting line (or regression line)

y=mx+b, Here m is the slope of the line and b is the y-intercept. it is the equation similar to algebra. But in statistics, the points do not lie perfectly on a line. it models around which the data lie if strong line pattern exists.

Here the slope of the line is the change over the change in X. So the slope is 10/3. it means the x value increases by 3 units, the value moves up by 10 units on average.

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