Find the error associated with each data point and then square these errors and sum
them to obtain the quantity
∑N
i=1
Ei^2 called the sum of the errors squared. The “best
”linear least squares fit to all the data points is defined as the line which minimizes
the sum of the errors squared. This requires finding those values of αand βwhich
minimize the sum of the errors squared given by
E(α, β ) =
∑N
i=1
E^2 i=
∑N
i=1
(αx i+β−yi)^2 , to have a minimum value
(a) Show that the best linear least-squares fit requires that the coefficients αand β
be chosen to satisfy the equations
α=
N
∑N
i=1 xiyi−
(∑N
i=1 xi
)( ∑N
i=1 yi
)
∆
β=
(∑N
i=1 x^2 i
)( ∑N
i=1 yi
)
−
(∑N
i=1 xi
)( ∑N
i=1 xiyi
)
∆
,
where ∆ = N
∑N
i=1
x^2 i−
(N
∑
i=1
xi
) 2
.
(b) Given the data points
(1 ,10),(2 ,4),(2. 5 ,6),(3,12),(3. 5 ,5),(4,10)
Find the best linear least-squares fit. Hint: Construct a table of values of the
form
xi yi x^2 i xiyi
∑ 4
i=1 xi
∑ 4
i=1 yi
∑ 4
i=1 x^2 i
∑ 4
i=1 xiyi
(c) Plot the least squares straight line and the data points.
