Vernier Graphical Analysis 3 - Manual (Page 23)

Graphical Analysis for Windows TEACHER’S MANUAL
25
General Notes
Statistics
The standard deviation used in statistical calculations is
std.dev. =
− ⋅ ∑
−
x n x
n
i m
2 2
1
where xm = mean
n = number of data points
This is sometimes referred to as the sample standard deviation.
The program calculates the “best fit” line using linear regression by the method of least squares. The equations used are
( ) ( ) ( )
( ) ( )
slope M = =
⋅ ∑ ⋅ − ∑ ⋅ ∑
⋅ ∑ − ∑
n
i
x
i
y
i
x
i
y
n x
i
x
i
2
2
( ) ( ) ( ) ( )
( ) ( )
intercept B = =
∑ ⋅ ∑ − ∑ ⋅ ∑
⋅ ∑ − ∑
x
i
y
i
x
i
i x y
n x
i
x
i
i
2
2
2
correlation coefficient =
( ) ( )
( ) ( )
x x y y
x x y y
m m
m m
1 1
1
2
1
2
− ⋅ −
− ⋅ −
∑
∑ ∑
( ) ( )
correlation coefficient =
⋅ ∑ − ⋅ ⋅
∑ − ⋅ ⋅ ∑ − ⋅
i
x
i
y n m x
m
y
x n m x y n y
i i m
2 2 2 2
( ) ( )
std. dev. of slope =
n
y
n x
i x
i
⋅
⋅ ∑ − ∑
2
2
2
σ
( )
( )
std. dev. of y - intercept =
y
x
n x
i
x
i
i
2
2
2
2
σ ⋅ ∑
⋅ ∑ − ∑
where n = number of data pairs
xm = mean of x-values
ym = mean of y-values
( )
y
n
i
y B M i x
2
1
2
2
σ =
−
⋅ − − ⋅ ∑
When Graphical Analysis performs automatic or manual curve fits, it calculates and reports the mean square error which
is
( )
mean square error =
− ′ ∑
2
i
y
i
y
n
where y
I
= the y-value of each data point
y
I
´ = the y-value of the curve fit
n = number of data points.