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EXPERIMENTAL
ERRORS & STATISTICS
NURUL AUNI BINTI ZAINAL ABIDIN
FACULTY OF APPLIED SCIENCE
UITM NEGERI SEMBILAN
Scientific Notation
The number of atoms in 12 g of carbon:
602,200,000,000,000,000,000,000
6.022 x 1023
The mass of a single carbon atom in grams:
0.0000000000000000000000199
1.99 x 10-23
N x 10n
N is a number
between 1 and 10
n is a positive or
negative integer
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568.762
0.00000772
move decimal left
n>0
568.762 = 5.68762 x 102
move decimal right
n<0
0.00000772 = 7.72 x 10-6
Addition or Subtraction
1. Write each quantity with the same exponent n
2. Combine N1 and N2
3. The exponent, n, remains the same
4.31 x 104 + 3.9 x 103 = 4.31 x 104 + 0.39 x 104
= 4.70 x 104
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Multiplication
1. Multiply N1 and N2
2. Add exponents n1 and n2
(4.0 x 10-5) x (7.0 x 103) = ?
= (4.0 x 7.0) x (10-5+3)
= 28 x 10-2
(a x 10m) x (b x 10n) = (a x b) x 10m+n
= 2.8 x 10-1
Division
1. Divide N1 and N2
2. Subtract exponents n1 and n2
8.5 x 104 ÷ 5.0 x 109 = ?
= (8.5 ÷ 5.0) x 104 - 9
(a x 10m) ÷ (b x 10n) = (a ÷ b) x 10m-n
= 1.7 x 10-5
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Significant Figures
- The meaningful digits in a measured or calculated
quantity.
RULES:

Any digit that is not zero is significant
1.234 kg

4 significant figures
Zeros between nonzero digits are significant
606 m
3 significant figures
 Zeros to the left of the first nonzero digit are not
significant
0.08 L
1 significant figure
 If a number is greater than 1, then all zeros to the right of
the decimal point are significant
2.0 mg
2 significant figures
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 If a number is less than 1, then only the zeros that
are at the end and in the middle of the number are
significant
0.00420 g
3 significant figures
 Numbers that do not contain decimal points, zeros
after the last nonzero digit may or may not be
significant.
400 cm
1or 2 or 3 significant figures
4 x 102
1 significant figures
4.0 x 102
2 significant figures
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How many significant figures are in each of
the following measurements?
24 mL
2 significant figures
3001 g
4 significant figures
0.0320 m3
3 significant figures
6.4 x 104 molecules
2 significant figures
560 kg
2 significant figures
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Significant Figures
Addition or Subtraction
The answer cannot have more digits to the right of the
decimal point than any of the original numbers.
89.332
+1.1
90.432
3.70
-2.9133
0.7867
one significant figure after decimal point
round off to 90.4
two significant figures after decimal point
round off to 0.79
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Significant Figures
Multiplication or Division
The number of significant figures in the result is set by the
original number that has the smallest number of significant
figures
4.51 x 3.6666 = 16.536366 = 16.5
3 sig figs
round to
3 sig figs
6.8 ÷ 112.04 = 0.0606926 = 0.061
2 sig figs
round to
2 sig figs
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QUESTION
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LOGARITHMS AND ANTILOGARITHMS
log 957 = 2.981
Example
Characteristics
Mantissa
log 9.57 x 10-4 = -3.019
2
3
0.981
0.019
In converting a number to its logarithm, the number of
digits in mantissa of the log of the number (957)
should be equal to the number of SF in the number
(957).
For antilogarithm,
10 0.072 = 1.18
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Types of Errors in Chemical Analysis
1.
Absolute Error
Definition: The difference between the true value and
the measured value
E = xi – xt
Where xi = measured value
xt = true or accepted value
Example: If 2.62 g sample of material is analyzed to be
2.52 g, so the absolute error is − 0.10g.
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2.
Relative Error
Definition: The absolute or mean error expressed as a
percentage of the true value.
Er = xi – xt
xt
x
100%
The above analysis has a relative error of
− 0.10 g
2.62 g
x
100%
=
-3.8%
* We are usually dealing with relative errors of less than
1%. A 1% error is equivalent to 1 part in 100.
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2.1
Relative Accuracy
Definition: The measured value or mean expressed as
a percentage of the true value.
Er = xi
xt
x
100%
The above analysis has a relative accuracy of
2.52 g
2.62 g
x
100%
=
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96.2 %
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3.
Systematic Error or determinate error
Definition: A constant error that originates from a fixed
cause, such as flaw in the design of an equipment or
experiment.
It caused the mean of a set data to differ from the
accepted value.
This error tends to cause the results to either high
every time or low every time compared to the true
value.
There are 3 types of systematic error:
Oct
2008
Instrumental
Error
Personal
Error
Method
Error
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3.1
Instrumental Errors
All measuring devices contribute to systematic errors.
Glassware such as pipets, burets, and volumetric
flasks may hold volume slightly different from those
indicated by their graduations.
Occur due to significant difference in temperature
from the calibration temperatere.
Sources of uncertainties:
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3.2
Method Errors
Non-ideal analytical methods are often sources of
systematic errors.
These errors are difficult to detect.
The most serious of the 3 types of systematic errors.
Slow or incomplete
reaction
Interference
Instability of
reacting species
Occurrence of
side reaction
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3.3
Personal Errors
Involve measurements that require personal judgment.
For example :
i)
estimation of a pointer between tow scale
divisions.
ii)
color of solution.
iii)
level of liquids with respect to a graduation in
a burette.
iv)
prejudice.
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3.4
Effect of Systematic Errors
Constant
Error
Proportional
Error
• Does not change with size of the quantity
measured.
• Become more obvious as the size of the
quantity decreases.
• Approach to minimize the effect is use as
large a sample as possible.
• Increase and decrease in proportion to the
size of the sample for analysis.
• Source of error : Interference due to
contaminants in the sample.
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3.5
Detection and Control of Systematic Errors
Standard reference
materials (SRM)
Independent
analysis
Analysis of blank
sample
• There are certified samples containing a known
concentration or quantities of particular analytes.
• Can be purchased from a number of governmental
or industrial; sources such as U. S. National
Institute of Standards and Technology (NIST).
• If SRM are not available, an independent and largely
different analysis can be used in parallel with the
method evaluated.
• A statistical test must be used to determine
whether the difference is due to random errors in
the 2 methods.
• Blank contains the reagents and solvents used in
analysis but no analyte.
• Reveals errors due to interfering contaminants from
the reagents and vessels used in analysis.
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4
Random Error or Indeterminate error
Cause data to be scattered more or less symmetrically
around a mean value.
It reflects the precision of the measurement.
This error is caused by the many uncontrollable
variables in physical or chemical measurements.
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5
Gross Error
Differ from indeterminate and determinate errors.
They usually occur only occasionally, may cause a
result to be either high or low.
For example:
i)
part of precipitate is lost before weighing,
analytical results will be low.
ii)
touching a weighing bottle with your fingers
after empty mass will cause a high mass
reading for a solid weighed.
Lead to outliers, results in replicate measurements that
differs significantly from the rest of the results.
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