Errors in Chemical Analysis

Impossible to eliminate errors.How reliable are our data?Data of unknown quality are useless!

•Carry out replicate measurements•Analyse accurately known standards•Perform statistical tests on data

Mean Defined as follows:

xx

N

i

N

= i = 1

Where xi = individual values of x and N = number of replicate measurements

Median

The middle result when data are arranged in order of size (for even numbers the mean of middle two). Median can be preferred whenthere is an “outlier” - one reading very different from rest. Median less affected by outlier than is mean.

Precision

Relates to reproducibility of results..How similar are values obtained in exactly the same way?

Useful for measuring this:Deviation from the mean:

d x xi i

Accuracy

Measurement of agreement between experimental mean andtrue value (which may not be known!).Measures of accuracy:

Absolute error: E = xi - xt (where xt = true or accepted value)

Relative error: Er

xi xtxt

100%

(latter is more useful in practice)

Illustrating the difference between “accuracy” and “precision”

Low accuracy, low precision Low accuracy, high precision

High accuracy, low precision High accuracy, high precision

Types of Error in Experimental Data

Three types:(1) Random (indeterminate) Error

Data scattered approx. symmetrically about a mean value.Affects precision - dealt with statistically (see later).

(2) Systematic (determinate) ErrorSeveral possible sources - later. Readings all too high or too low. Affects accuracy.

(3) Gross ErrorsUsually obvious - give “outlier” readings.Detectable by carrying out sufficient replicatemeasurements.

Sources of Systematic Error1. Instrument Error

Need frequent calibration - both for apparatus such asvolumetric flasks, burettes etc., but also for electronicdevices such as spectrometers.

2. Method ErrorDue to inadequacies in physical or chemical behaviourof reagents or reactions (e.g. slow or incomplete reactions)Example from earlier overhead - nicotinic acid does notreact completely under normal Kjeldahl conditions for nitrogen determination.

3. Personal Errore.g. insensitivity to colour changes; tendency to estimatescale readings to improve precision; preconceived idea of“true” value.

Systematic errors can be constant (e.g. error in burette reading -less important for larger values of reading) orproportional (e.g. presence of given proportion ofinterfering impurity in sample; equally significantfor all values of measurement)

Minimise instrument errors by careful recalibration and goodmaintenance of equipment.

Minimise personal errors by care and self-discipline

Method errors - most difficult. “True” value may not be known.Three approaches to minimise:•analysis of certified standards•use 2 or more independent methods•analysis of blanks

Sample Standard Deviation, s

The equation for must be modified for small samples of data, i.e. small N

sx x

N

ii

N

( )2

1

1

Two differences cf. to equation for :

1. Use sample mean instead of population mean.

2. Use degrees of freedom, N - 1, instead of N.Reason is that in working out the mean, the sum of the differences from the mean must be zero. If N - 1 values areknown, the last value is defined. Thus only N - 1 degreesof freedom. For large values of N, used in calculating, N and N - 1 are effectively equal.

Alternative Expression for s(suitable for calculators)

sx

x

NN

ii

N ii

N

( )( )

2

1

1

2

1

Note: NEVER round off figures before the end of the calculation

Two alternative methods for measuring the precision of a set of results:

VARIANCE: This is the square of the standard deviation:

sx x

N

ii

N

2

2 2

1

1

( )

COEFFICIENT OF VARIANCE (CV)(or RELATIVE STANDARD DEVIATION):Divide the standard deviation by the mean value and express as a percentage:

CVs

x ( ) 100%

Reproducibility of a method for determining the % of selenium in foods. 9 measurements were made on a single batch of brown rice.

Sample Selenium content (g/g) (xI) xi2

1 0.07 0.00492 0.07 0.00493 0.08 0.00644 0.07 0.00495 0.07 0.00496 0.08 0.00647 0.08 0.00648 0.09 0.00819 0.08 0.0064

xi = 0.69 xi2= 0.0533

Mean = xi/N= 0.077g/g (xi)2/N = 0.4761/9 = 0.0529

Standard Deviation of a Sample

s

0 0533 0 0529

9 10 00707106 0 007

. .. .

Coefficient of variance = 9.2% Concentration = 0.077 ± 0.007 g/g

Standard deviation:

Define some terms:

CONFIDENCE LIMITS interval around the mean that probably contains .

CONFIDENCE INTERVALthe magnitude of the confidence limits

CONFIDENCE LEVELfixes the level of probability that the mean is within the confidence limits

A set of results may contain an outlying result - out of line with the others. Should it be retained or rejected? There is no universal criterion for deciding this. One rule that can give guidance is the Q test.

Qexp xq xn /w

where xq = questionable result xn = nearest neighbour w = spread of entire set

Consider a set of results

The parameter Qexp is defined as follows:

Detection of Gross Errors

Qexp is then compared to a set of values Qcrit:

Rejection of outlier recommended if Qexp > Qcrit for the desired confidence level.

Note:1. The higher the confidence level, the less likely is rejection to be recommended.

2. Rejection of outliers can have a marked effect on mean and standard deviation, esp. when there are only a few data points. Always try to obtain more data.3. If outliers are to be retained, it is often better to report the median value rather than the mean.

Qcrit (reject if Qexpt > Qcrit)

No. of observations 90% 95% 99% confidencelevel

3 0.941 0.970 0.9944 0.765 0.829 0.9265 0.642 0.710 0.8216 0.560 0.625 0.7407 0.507 0.568 0.6808 0.468 0.526 0.6349 0.437 0.493 0.59810 0.412 0.466 0.568

The following values were obtained for the concentration of nitrite ions in a sample of river water: 0.403, 0.410, 0.401, 0.380 mg/l.Should the last reading be rejected?

Qexp . . ( . . ) . 0 380 0 401 0 410 0 380 0 7

But Qcrit = 0.829 (at 95% level) for 4 values

Therefore, Qexp < Qcrit, and we cannot reject the suspect value.

Suppose 3 further measurements taken, giving total values of:

0.403, 0.410, 0.401, 0.380, 0.400, 0.413, 0.411 mg/l. Should

0.380 still be retained?

Qexp . . ( . . ) . 0 380 0 400 0 413 0 380 0 606But Qcrit = 0.568 (at 95% level) for 7 values

Therefore, Qexp > Qcrit, and rejection of 0.380 is recommended.

Q Test for Rejection of Outliers