Page 33 - 2017-v.2-Part 4(EN)

Annex A
(informative)

Explanation of the method of least squares for obtaining the line of best
fit and ±20 % limit lines for braking performance linearity

The readings taken in the test specified in 4.6.5.7 can be expected to lie near some straight line that can
be drawn through them. Although in practice one might draw a good straight line through the points by
eye, the method of least squares given here provides a criterion for minimizing the discrepancies and
permits a line to be selected that has a claim to be called the best fit.

The line of best fit is the line that minimizes the sum of the squares of the differences between the
observed results and the corresponding results predicted by the line.

The relationship between the variables is considered to be of the form

y = a + bx (A.1)

where

x is the independent variable, and is known precisely (in this case, the load applied to the pedal);

y is the dependent variable, and is observed but with a degree of uncertainty (in this case, the
braking force at the wheel).

a and b are unknown constants and have to be estimated.

For a series of n readings, this relationship can be resolved by taking a minimum of the sum of the
squares of the difference to give

x
n ∑ xy − ∑ ∑ y
b = (A.2)
2
x
n ∑ x − ∑ ∑ x

Taking ∑ ∑
y = and x = (A.3)
y x
n n

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