Instant Notes: Analytical Chemistry

mass or concentration of the analyte, it is necessary to draw a line of best fit

through the plotted points before they can be used as a working curve. Although

this can be done by eye, a more accurate method is to employ linear regression.

It is invariably the case that, due to the effects of indeterminate errors on the data,

most of the points do not lie exactly on the line, as shown in Figure 1. Linear

regression enables a line of best fit through the points to be defined by calculating

values for the slope and y-axis intercept (band arespectively in equation (1)), and

the method of least squaresis commonly used for this purpose. An assumption

is made that only errors in the detector responses (y-values) are significant, any

errors in the values for the mass or concentration of the analyte being neglected.

The deviations in the y-direction of the individual plotted points from the

calculated regression line are known as y-residuals(Fig. 3) and the line repre-

sents the regression of yupon x. The method of least squares minimizes the sum

of the squares of the y-residuals by equating them to zero in defining equations

for the slope and intercept of the regression line.

For the slope, b

b= (3)

For the y-axis intercept, a

a=y

_

−b.x

_

(4)

N.B. As equation (4) is a re-arrangement of equation (1), it follows that the point

x

_

, y

_

, known as the centroid, must lie on the regression line.



i=N

i= 1

{(xi−x

_

)(yi−y

_

)}







i=N

i= 1

(xi−x

_

)^2

44 Section B – Assessment of data

1.2

1

0.8

0.6

0.4

0.2

0

0 20 40 60 80 100 120

Concentration (mg cm–3)

Absorbance at 325 nm

Slope = 0.00878

Intercept = 0.0686

y-residuals

Fig. 3. Calibration graph, regression line, slope and intercept values for the UV spectrophoto-

metric determination of an active ingredient in a sun cream. ——— =regression line; =

centroid, x

_

, y

_

; -----------=confidence limits lines at the 99 percent level; ——=confidence

limits for sample concentration of 30 mg cm-^3. Inset: Illustration of y-residuals.