error in chemical analysis

ANALYTICAL CHEMISTRY(Error in Chemical Analysis)

Dr.S.SURESH

Assistant Professor

Email:[email protected]

Mean value

A mean value is obtained by dividing the sum of a set ofreplicate measurements by the number of individualresults in the set. For example, if a titration is repeated fourtimes and the titre values are 10.1, 9.9, 10.0 and 10.2ml

Mean = 10.1 + 9.9 + 10 + 10.24

= 40.24

= 10.05This mean value is also called arithmetic mean or average.

The median

This is a value about which all other values in a setare equally distributed. Half of the values aregreater and the other half smaller numerically,compared to the median.

For example: If we have a set of values like 1.1, 1.2,1.3, 1.4 and 1.5 the median value is 1.3.

When a set of data has an even number of values,then the median is the average of the middle pair.

Accuracy

• Accuracy represents the nearness to ameasurement to its expected value. Anydifference between the measured value andthe expected value is expressed as error.

• For example: The dissociation constant foracetic acid is 1.75×10‒5 at 25 °C. In anexperiment, if a student arrives at exactly thisvalue, his value is said to be accurate.

Precision

• Precision is defined as the agreement between thenumerical value of two or more measurements ofthe same object that have been made in anidentical manner. Thus, a value is said to be precise,when there is agreement between a set of resultsfor the same quantity.

• However a precise value need not be accurate.

Methods of expressing precision

• Precision can be expressed in an absolute method.In the absolute way the deviation from the mean│xi ‒ │ expresses precision without considering sign

_

X

S.NoSample of an

organic compound% of carbon

Deviation from mean

│xi ‒ │

1 X1 38.42 0.20

2 X2 38.02 0.20

3 X3 38.22 0.00

= 38.22= 0.133

(Average deviation)

_

X

_

X

3

0.40

Absolute error

• The term accuracy is denoted in terms ofabsolute error E, E is the difference between theobserved value (Xi) and the expected value (Xt).

E = │ Xi – Xt│

• If a student obtains a value of 1.69×10‒5 for thedissociation constant of acetic acid at 25°C, theabsolute error in this determination is

E = │1.69 ×10‒5 – 1.75×10‒5 │

= │0.06 ×10‒5 │

Relative error

• Sometimes the term relative error is used toexpress the uncertainty in data. The relative errordenotes the percentage of error compared to theexpected value. For the dissociation constant valuereported.

Relative error = 0.06 ×10‒5 × 100

1.75×10‒5

= 3.4%

Problem:

• The actual length of a field is 500 feet. A measuringinstrument shows the length to be 508 feet. Find:a.) the absolute error in the measured length of the field.b.) the relative error in the measured length of the field.

Solution:

• (a)The absolute error in the length of the field is 8 feet.

E = │ Xi – Xt│ = 508-500 = 8 feet.

• b.) The relative error in the length of the field is

Relative error = 8 × 100

500

= 1.6%

ErrorsErrors are of two main types

• Determinate errors

• Indeterminate errorsDeterminate errors:

These errors are determinable and are avoided ifcare is taken. Determinate errors are classified intothree types

• Instrumental error

• Operative error

• Methodic errors

Instrumental error

• Instrumental errors are introduced due to the use

of defective instruments.

• For example an error in volumetric analysis will be

introduced, when a 20ml pipette, which actually

measures 20.1ml, is used.

• Sometimes an instrument error may arise from the

environmental factors on the instrument.

• For example a pipette calibrated at 20°C, if used at

30°C will introduce error in volume.

• Instrumental errors may largely be eliminated by

periodically calibrating the instruments.

Operative errors

• These errors are also called personal errors and areintroduced because of variation of personaljudgements.

• For example due to colour blindness a person mayarrive at wrong results in a volumetric orcolorimetric analysis.

• Using incorrect mathematical equations andcommitting arithmetic mistakes will also causeoperative errors.

Methodic errors

• These errors are caused by adopting defectiveexperimental methods.

• For example in volumetric analysis the use of animproper indicator leading to wrong results is anexample for methodic error.

• Proper understanding of the theoretical backgroundof the experiments is a necessity for avoidingmethodic errors.

Indeterminate errors

• These errors are also called accidental errors.

Indeterminate errors arise from uncertainties in

a measurement that are unknown and which

cannot be controlled by the experimentalist.

• For example: When pipetting out a liquid, the

speed of draining, the angle of holding the

pipette, the portion at which the pipette is held,

etc, would introduce indeterminate error in the

volume of the liquid pipette out.

Significant figure

• Data have to be reported with care keeping in mind

reliability about the number of figures used.

• For example, when reporting a value as many as six

decimal numbers can be obtained, when one uses a

calculator.

• However, reporting all these decimal numbers is

meaningless because, as is generally true, there may

be uncertainty about the first decimal itself.

• Therefore, experimental data should be rounded off.

Significant figure

• A zero is not a significant figure, when used tolocate a decimal. However, it is significant when itoccurs at the end.

• For example 0.00405 has three significant figures,the two zeros before the 4 being used to imply onlythe magnitude, but 0.04050 has four significantfigures, the zero beyond the 5 being significant.

Significant figure

• The number of significant figures in a given numberis found by counting the number figures from theleft to right in the number beginning with the firstnon-zero digit and continuing until reaching thedigit that contains the uncertainty. Each of thefollowing has three significant figures.

646 0.317 9.22 0.00149 20.2

Significant figure

• When multiplication and division are carried out, itis assumed that the number of significant figures ofthe result is equal to the number of significantfigures of the component quantity that contains theleast number of significant figures

Example = 0.1342

= 0.1310

0.122 11

Significant Figures

Rules for Counting Significant Figures

2. Zeros

a. Leading zeros - never count

0.0025 2 significant figures

b. Captive zeros - always count


c. Trailing zeros - count only if the number is written

with a decimal point

100 1 significant figure

100. 3 significant figures


Normal error Curve

• The normal error curve was first studied byCarl Friedrich Gauss as a curve for thedistribution of errors. He found that thedistribution of errors could be closelyapproximated by a curve called the normalcurve of errors.

Normal error Curve

• This normal distribution curve is a useful one tomeasure the extent of indeterminate error. It isgiven by

is the standard deviation

x = value of the continuous random variable.

µ = mean of the normal random variable

π = constant = 3.14

Normal error Curve

• In normal error curve, the frequency is plottedagainst mean deviation.

• When the frequency is maximum the error is nil.

• When the frequency decreases, the magnitude ofthe error increases

Normal error Curve

• When is very large, the curve obtained isbell shaped. When is very small, then asharp curve is obtained.

• When frequency increases, the willdecrease → sharp curve → nil error.

• When frequency decreases the willincrease → bell shaped curve → errorincreases

Normal error Curve

• The normal distributions are extremelyimportant in statistics and are often used inscience for real valued random variableswhose distributions are not known.

• How many significant figures are in:

1. 12.548

2. 0.00335

3. 504.70

4. 4000

5. 0.10200

(1) There are 5. All numbers are significant.

(2) There are 3. The zeros before the number is not significant.

(3) There are 5 significant figures.

(4) There is 1 significant figure.

(5) There are 5 significant figures.