Principle of Least Square, its Properties, Regression line and standard error of estimate

this is the presentation of Statistic in which the topics of presentations were being told in a precise way to help the student

1. Presented to: Mr. Hafiz Fiyaz Presented by: Group no. 9 Topic: (Principle of least square, Standard error of estimate, Properties of least square, Regression line )

2. Principle of least square “The principle of least square consists of determining the values of the values of the unknown parameters that will minimize the sum of squares of error” This concept was given by Karl F. Gauss (1777-1855)

3. Continue… The basic formula of Principle of least square by direct method is Y̑= a+bX+e a= X ²  Y -  xXY n X² - (X)² b= n XY – (X)(Y) nX² - (X)²

4. Properties of least square The least squares regression line always goes through the point ( X̅, Y ̅) the mean of the data. The sum of the deviations of the observed values of Y from the least square regression line is always equal to zero i.e. ( Y- Y ̑)=0 The sum of the squares of the derivations of the observed values fro the least-squares regression line is a minimum i.e. (Y- Y ̑) ² = minimum The least square regression line obtained from a random sample is the line of best fit because a and b are the unbiased estimates of the parameters of α & β

5. Standard error of Estimation the observed values of (X,Y) do not fall on the regression line but they scatter away from it. The degree of scatter or dispersion of the observed values about the regression line is measured by what is called standard error of estimate. S=

6. Regression line A first step in finding a relationship between two variables exists, is to plot each pair of independent-dependent observation on graph paper using X-axis for regression variable and Y-axis as dependent variable, this is called scatter diagram. If a relationship between them exist then the point show some sort of cluster around the line and these points are called regression line Y= a + bX α

7. Example