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three cases the term is actually measuring the width of the normal distribution curve

such that the narrower the curve the smaller the value of the term and the higher the

precision of the analytical data set.

Thestandard deviation(s) of a data set is a measure of the variability of the

population from which the data set was drawn. It is calculated by use of equation

1.10 or 1.11:

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxixÞ^2

n 1

s

ð 1 : 10 Þ

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x^2 iðxiÞ^2 =n

n 1

s

ð 1 : 11 Þ

(xix-) is the difference between an individual experimental value (xi) and the

calculated meanx-of the individual values. Since these differences may be positive

or negative, and since the distribution of experimental values about the mean is

symmetrical, if they were simply added together they would cancel out each other.

The differences are therefore squared to give consistent positive values. To compen-

sate for this, the square root of the resulting calculation has to be taken to obtain the

standard deviation.

Standard deviation has the same units as the actual measurements and this is one of

its attractions. The mathematical nature of a normal distribution curve is such that

68.2% of the area under the curve (and hence 68.2% of the individual values within

the data set) is within one standard deviation either side of the mean, 95.5% of the

area under the curve is within two standard deviations and 99.7% within three

standard deviations. Exactly 95% of the area under the curve falls between the mean

and 1.96 standard deviations. The precision (or imprecision) of a data set is commonly

expressed as1 SD of the mean.

The term (n1) is called thedegrees of freedomof the data set and is an important

variable. The initial number of degrees of freedom possessed by a data set is equal to

the number of results (n) in the set. However, when another quantity characterising

the data set, such as the mean or standard deviation, is calculated, the number of

degrees of freedom of the set is reduced by 1 and by 1 again for each new derivation

made. Many modern calculators and computers include programs for the calculation

of standard deviation. However, some use variants of equation 1.10 in that they usen

as the denominator rather thann1 as the basis for the calculation. Ifnis large,

greater than 30 for example, then the difference between the two calculations is

small, but ifnis small, and certainly if it is less than 10, the use ofnrather than

n1 will significantly underestimate the standard deviation. This may lead to false

conclusions being drawn about the precision of the data set. Thus for most analytical

biochemical studies it is imperative that the calculation of standard deviation is based

on the use ofn1.

Thecoefficient of variation(CV) (also known asrelative standard deviation)ofa

data set is the standard deviation expressed as a percentage of the mean as shown in

equation 1.12.

21 1.4 Quantitative biochemical measurements