Lecture 2

Affects measurement’s precision Can be minimized and controlled, but NEVER

ELIMINATED! Due to small and uncontrollable variables, which are

usually not even known Inevitably part of every analysis Accumulative effect causes replicate measurements to

fluctuate randomly around the mean of the set Generally follow a Gaussian Distribution (random or

normal distribution) Sometimes positive, sometimes negative Can be minimized by replicate analyses

Can be evaluated by the methods of statistics

Is done based on the assumption that random errors in an analysis follow a Gaussian or normal distribution

Significant departures from Gaussian behaviour can and do occur, but not often

Gaussian is based on normally distributed random errors

Defines the distribution of random error in events or measurements

equal probability of "high" value and "low" value Can also be used in conjunction with the standard deviation.

µ - population meanσ - population standard deviation

A sample of data is a finite number of experimental observation

The sample is treated as a tiny fraction of an infinite number of observations

The theoretical infinite number of data is a population or a universe of data

Statistical laws have been derived assuming a population data However, modifications have to be made when applied

to a small sample N.B. The difference between a statistical “sample” and

a chemical “sample”, chemical samples typically make up a statistical sample of the population of interest

This is a measure of precision of a population of data standard deviation, large data set

wherepopulation mean

N: number of replicate making up the population

N

xN

ii

1

2μ)(

•The mean occurs at the central point of maximum frequency•There is a symmetrical distribution of positive and negative deviation about the maximum•There is an exponential decrease in frequency as the magnitude of deviation increases•The standard deviation indicates precision(i.e. σB = 2σA, therefore precision associated withcurve B is half that of curve A.

•The curves are normalized wrt σ (x axis shows deviationfrom mean in units of σ). •Therefore, all Gaussiancurves plotted in this way are identical.

# of Standard Deviations from the Mean

% of Events Affected by Random Error that Occur

+/- 1 STD DEV 68.3 %

+/- 2 STD DEV 95.5 %

+/- 3 STD DEV 99.7 %

Standard Deviation defines a distribution of results around the mean.

Smaller standard deviation = greater precision Values are generally closer to the mean and

themselves

The standard deviation of each mean is known as the standard error of the mean and is given the symbol, Sm

Sm = S/√N

The Standard error is inversely proportional to the square root of the number of data, N

For analyses that are time consuming this is not practical

Samples accumulated over time can be pooled to provide an estimate of s that is superior to the value for any individual subset

We must assume the same random error in all measurements

Valid when samples have similar composition and analysed in the same way

Variance

Relative standard deviation

Coefficient of variation

100% x x

s = %CV

1000ppt x x

s = %RSD

1

)(1

2

2

N

xxs

N

ii

A sample of saliva is analyzed for NNN (n-nitrosonornicotine), and it is found to contain 1.0, 2.0, 3.0, 2.4, 2.5 ppm NNN.

USING your calculator, determine the mean, standard deviation and then %RSD for this analysis.

Calculate the following:

1. Mean

2. Range

3. Standard deviation

4. Relative standard deviation5. If the accepted value is 70.30 calculate, Relative error of the mean in ppt

Alternative equation for standard deviation

•Never round a standard deviation calculations until the very end•Never use this equation to calculate the standard deviation of numbers containing five or more digits

•Statistical Data Treatment and Evaluation•Confidence limits•Confidence interval•Null hypothesis•F-test•Q-test