Course Name: Statistical chemical measurements
Course Code: 4021103
Significant figures
Lecture 2
Types of errors
contents:
Detecting systematic errors
Significant figure
RULES FOR SIGNIFICANT FIGURES
• 1. All non-zero numbers are significant.
The number 33.2 has THREE significant figures because all of the digits present are non-
zero.
• 2. Zeros between two non-zero digits are significant.
2051 has four significant figures. The zero is between a 2 and a 5.
• 3. Leading zeros are not significant. They're nothing more than "place holders."
The number 0.54 has only two significant figures. 0.0032 also has two significant figures.
All of the zeros are leading.
3
Significant figure
• 4. Trailing zeros to the right of the decimal are significant.
There are four significant figures in 92.00.
• 92.00 is different from 92: a scientist who measures 92.00 milliliters knows his value
to the nearest 1/100th milliliter; meanwhile his colleague who measured 92 milliliters
only knows his value to the nearest 1 milliliter. It's important to understand that
"zero" does not mean "nothing." Zero denotes actual information, just like any other
number. You cannot tag on zeros that aren't certain to belong there.
• 5. Trailing zeros in a whole number with the decimal shown are significant.
Placing a decimal at the end of a number is usually not done. By convention, however,
this decimal indicates a significant zero. For example, "540." indicates that the trailing
zero is significant; there are three significant figures in this value.
4
Significant figure
• 6. Trailing zeros in a whole number with no decimal shown are not significant.
Writing just "540" indicates that the zero is not significant, and there are only two
significant figures in this value.
• 7. Exact numbers have an infinite number of significant figures. This rule applies to
numbers that are definitions.
For example, 1 meter = 1.00 meters = 1.0000 meters = 1.0000000000000000000 meters,
etc.
5
Significant figure
Rounding Data
In rounding a number ending in 5, always round so that the result ends with
an even number. Thus, 0.635 rounds to 0.64 and 0.625 rounds to 0.62
Examples
Find out the correct answer and write the answer in the correct round?
❖. 4.6 x 3.52
4.6 x 3.52 = 16.192 16
2 3
significant figure significant figure
7
Examples
Find out the correct answer and write the answer in the correct round?
❖. 5.64 x 12.458
5.64 x 12.458 = 70.26312 70.3
3 5
significant figure significant figure
8
Examples
Find out the correct answer and write the answer in the correct round?
❖. 96.752 / 3.541
96.752 / 3.541 = 27.32335498 27.32
5 4
significant figure significant figure
9
Exact Numbers
Numbers from definitions or numbers of objects are considered
to have an infinite number of significant figures
The average of three measured lengths; 6.64, 6.68 and 6.70 cm?
6.64 + 6.68 + 6.70
= 6.67333
3
= 6.67
Because 3 is an exact number
10
Examples
Multiplication and Division:
the answer contains the same number of significant digits as the
original number with the smallest number of significant digits.
For example,
(24 X 4.52)/ 100.0 = 1.08
= 1.1
(24 X 4.02)/100.0 = 0.965
= 0.96
Logarithms and Antilogarithms :
1. In a logarithm of a number, keep as many digits to the right of the decimal point as
there are significant figures in the original number.
2. In an antilogarithm of a number, keep as many digits as there are digits to the right
of the decimal point in the original number.
Example:
(a) log 4.000 X 10–5 = -4.3979400,
log 4.000 X 10–5 = - 4.3979
(b) antilog 12.5 = 3.162277 X 1012
antilog 12.5 = 3 X 1012
Errors in quantitative analysis
Suppose we perform a titration four times and obtain
values of 24.69, 24.73, 24.77 and 25.39 ml.
All four values are different, because of the errors
inherent in the measurements, and the fourth value
(25.39 ml) is substantially different from the other
three.
Errors in quantitative analysis
no quantitative results are of any value unless they are
accompanied by some estimate of the errors inherent
in them.
Analysts commonly perform several replicate
determinations in the course of a single experiment.
Types of Error
Gross error
Random
error
Systematic
error
Sources of Systematic Errors
1. Instrumental errors
Caused by unideal instrument behavior, by
faulty calibrations, or by use under
inappropriate conditions.
2. Method errors
Arise from unideal chemical or physical
behavior of analytical systems.
3. Personal errors
Results from the carelessness, inattention, or
personal limitations of the experimenter.
Detecting and overcome Systematic Errors
Systematic instrument errors
are usually corrected by calibration. Periodic calibration of equipment is always
desirable.
Personal errors
can be minimized by care and self-discipline. Errors that result from a known
physical disability can usually be avoided by carefully choosing the method.
Method errors or bias of an analytical method
is estimated by analyzing standard reference materials.
Accuracy:
• Is defined as the closeness of the test results obtained to the
true value of the standard used.
Note that We can never determine accuracy exactly because
the true value of a measured quantity can never be known
exactly.
accuracy is important:
To check how good a new instrument, method or
worker is.
Precision (repeatability):
• is defined as the closeness of the test results to each other.
• To determine the degree of agreement within the test results for a
particular sample.
• This is a measurement of the random errors of an analysis.
Example
4 student have analyzed the glucose concentration in a blood sample and the
results in the table below: Note that the true value is 125 mg/dL
Student Results of glucose concentration ( mg/dL) comment
A 126 125.3 125.9 125.4 125.2 Precise, biased
B 120 130 126 119 133 Imprecise, unbiased
C 115 124 120 118 122 Imprecise, biased
D 124.9 125 125.2 125.6 124.7 precise, unbiased
Note that:
If the range of the results is small, this mean precise.
But if the range of the results is large, this mean imprecise.
Perform the measurement
Why Method Validation?
To minimize analytical and instrumental errors.
To give reliable and reproducible results in accordance with the.
given specifications of the test method.
To ensure the quality of the test results.
To meet accreditation requirement.
Objective evidence for defense against challenges.
To be assured of the correctness of results.
21
