METHOD OF LEAST SQURE

3.  The method of least squares is a standard
approach to the approximate solution of over
determined systems, i.e., sets of equations in which
there are more equations than unknowns.
 "Least squares" means that the overall solution
minimizes the sum of the squares of the errors
made in the results of every single equation.
 The least-squares method is usually credited to Carl
Friedrich Gauss (1795), but it was first published
by Adrien-Marie Legendre.

4.  A linear regression is a statistical analysis assessing
the association between two variables. It is used to
find the relationship between two variables.
Regression Equation(y) = a + bx Slope(b) =
(NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N

5.  To fit the straight line y=a+bx:
 Substitute the observed set of n values in this
equation.
 Form the normal equations for each constant i.e.
Σy=na+bΣx, Σxy=aΣx+bΣx2 .
 Solve these normal equations as simultaneous
equations of a and b.
 Substitute the values of a and b in y=a+bx, which is
the required line of best fit.

6. Fit a straight line to the x and y values in the following
Table:
xi yi xiyi xi
2
1 0.5 0.5 1
2 2.5 5 4
3 2 6 9
4 4 16 16
5 3.5 17.5 25
6 6 36 36
7 5.5 38.5 49
28 24 119.5 140
28  i x   24.0 i y
140 2  i  x  i i 119.5 x y
3.428571
24
x   y  
7
4
28
7
3.428571
24
x   y  
7
4
28
7

7. 0.8392857
  
n x y 
x y
i i i i
2 2
 
n x x
( )

7  119.5  28 
24
2
7 140 28


a  y 
a x
 
0 1

3.428571 0.8392857 4 0.07142857
1
   
a
i i
*
* Y  a  a X
0 1
Y = 0.07142857 + 0.8392857x

8. Suppose if we want to know the approximate y value
for the variable x = 4. Then we can substitute the
value in the above equation.
Y *  a  a X
*
0 1
Y = 0.07142857 + 0.8392857x
Y = 0.07142857 + 0.8392857(4)
Y = 0.07142857 + 3.3571428
Y = 3.42857137 Ans

10. Fit the following Equation:
0 . 24  i y
to the data in the following table:
xi yi X*=log xi Y*=logyi
1 0.5 0 -0.301
2 1.7 0.301 0.266
3 3.4 0.477 0.534
4 5.7 0.602 0.753
5 8.4 0.699 0.922
15 19.7 2.079 2.141
b y  a x
log log( 2 )
2
log y log a b log x 2 2  
* *
Y y, X x,
 
let  log 
log
a log
a , a b
0 2 1 2
*
* Y  a  a X
0 1