a larger uncertainty associated with them. The uncertainty of the sample mean
deviating from the population mean decreases in the proportion of the reciprocal of
the square root of the number of values in the data set i.e. 1/√n.
Thus to decrease the uncertainty by a factor of two the number of experimental
values would have to be increased four-fold and for a factor of 10 the number of
measurements would need to be increased 100-fold. The nature of this relationship
again emphasises the importance of evaluating the acceptable degree of uncertainty of
the experimental result before the design of the experiment is completed and the
practical analysis begun. Modern automated analytical instruments recognise the
importance of multiple results by facilitating repeat analyses at maximum speed. It is
good practice to report the number of measurements on which the mean and standard
deviation are based as this gives a clear indication of the quality of the calculated data.Confidence intervals, confidence limits and the Student’stfactor
Accepting that the population mean is the best estimate of the ‘true’ value, the
question arises ‘How can I relate my experimental sample mean to the population
mean?’ The answer is by using the concept of confidence.Confidence levelexpresses
the level of confidence, expressed as a percentage, that can be attached to the data. Its
value has to be set by the experimenter to achieve the objectives of the study.
Confidence intervalis a mathematical statement relating the sample mean to the
population mean. A confidence interval gives a range of values about the sample
mean within which there is a given probability (determined by the confidence level)
that the population mean lies. The relationship between the two means is expressed in
terms of the standard deviation of the data set, the square root of the number of values
in the data set and a factor known asStudent’st (equation 1.13):¼xts
p
nð 1 : 13 Þwherexis the measured mean,mis the population mean,sis the measured standard
deviation,nis the number of measurements andtis the Student’stfactor. The term
s/√nis known as thestandard error of the meanand is a measure of the precision of
the sample mean. Unlike standard deviation, standard error depends on the sample
size and will fall as the sample size increases. The two measurements are sometimes
confused, but in essence, standard deviation should be used if we want to know how
widely scattered are the measurements and standard error should be used if we want
to indicate the uncertainty around a mean measurement.
Confidence level can be set at any value up to 100%. For example, it may be that a
confidence level of only 50% would be acceptable for a particular experiment. However,
a 50% level means that that there is a one in two chance that the sample mean is not
an acceptable estimate of the population mean. In contrast, the choice of a 95% or 99%
confidence level would mean that there was only a one in 20 or a one in 100 chance
respectively that the best estimate had not been achieved. In practice, most analytical
biochemists choose a confidence level in the range 90–99% and most commonly 95%.
Student’stis a way of linking probability with the size of the data set and is used
in a number of statistical tests. Student’stvalues for varying numbers in a data set24 Basic principles