Measurements
Review of Basic Concepts
 Accuracy & Precision
 Types of Errors
 Significant Figures & Roundingoff values
 Precision of measuring devices
 Measurement of mass & volume
Measurements
Accuracy – how close a measurement is to the
true value
Precision – how close a set of measurements
are to each other
Measurements
Measurements
Estimating the accuracy and precision
of experimental data is extremely
important whenever we collect
laboratory results because data of
unknown quality are worthless.
Measure of Accuracy
1. Absolute Error
E = xi - xT
where
xi = experimental value
xT = true value or accepted value
(literature value)
Measure of Accuracy
2. Percent Relative Error
% RE =
xi - xT
xT
x 100
For our purpose, % RE < 5% is relatively
accurate
Measure of Precision
1. Range
R = xH - xL
where
xH = highest experimental value
xL = lowest experimental value
Measure of Precision
2. Standard Deviation
where
x = experimental value
x = mean
n = number of replicates / trials
Measure of Precision
2. Standard Deviation
• A low standard deviation means that
the data is very closely related to the
average, thus very reliable (usually
less than 5% of the mean)
• A high standard deviation means
that there is a large variance
between the data and the statistical
average, thus not as reliable.
Exercise
Show complete solution, observe correct units and
significant figures: ½ crosswise
A student determines the boiling point of ethanol in
three replicates, obtaining the following results:
78.450, 78.500, 78.390. If the literature value is
78.370, evaluate the accuracy and precision of the
data set by calculating the following:
a)
b)
c)
d)
Absolute error
Percent relative error
Range
Standard deviation
Exercise
Given: 78.450, 78.500, 78.390
Literature value = 78.370
Mean = (78.450 + 78.500 + 78.390) / 3 = 78.450
a) Absolute Error = 78.450 – 78.370 = 0.080
b) % Relative Error = 0.080 x 100 = 0.10%
78.370
% Relative Error less than 5% of the true
value, therefore good accuracy
Exercise
Given: 78.450, 78.500, 78.390
Literature value = 78.370
Mean = (78.450 + 78.500 + 78.390) / 3 = 78.450
c) Range = 78.500 – 78.390 = 0.110
d) SD =
√
(78.450–78.450)2 + (78.500–78.450)2
+ (78.390–78.450)2
3-1
SD = 0.060
SD less than 5% of the mean,
Therefore good precision
Review of Basic Concepts
 Accuracy & Precision
 Types of Errors
 Significant Figures & Roundingoff values
 Precision of measuring devices
 Measurement of mass & volume
Types of errors
Three general types of errors occur in
lab measurements:
1. Random error
2. Systematic error, and
3. Gross errors
Random or Indeterminate Errors
• Random (or indeterminate) errors are
caused by uncontrollable fluctuations in
variables that affect experimental
results
• This type of error causes data to be
scattered more or less symmetrically
around a mean value.
• Can be corrected or minimized by
doing replicates or multiple trials
• Random error in a measurement is
reflected by its precision.
Systematic or Determinate Errors
The errors that affect the accuracy of a
result. This type of error causes the mean
of a set of data to differ from the accepted
value. A systematic error caused the results
in a series of replicate measurements to be
all high (+ bias) or all low (- bias).
Bias has a definite value, an assignable
cause and are about the same magnitude for
replicate measurements. Bias affects all the
data in a set in the same way.
How do Systematic Errors Arise?
There are three types of systematic errors:
1. Instrumental errors are caused by the
imperfections in measuring devices and
instabilities in their components.
2. Method errors arise from non-ideal
chemical or physical behavior of analytical
systems.
3. Personal errors results from the
carelessness, inattention, or personal
limitations of the experimenter.
Instrumental Errors
All measuring devices are potential sources of
systematic errors.
• Calibrated glasswares
 using glasswares at
a temperature that
differs significantly
from the calibration
temperature
 distortions in container walls due to
heating
Instrumental Errors
• Electronic instruments are also subject to
systematic errors
 errors may emerge as the voltage of a
battery-operated power supply decreases
with use
 Errors can also occur if instruments are
not calibrated frequently or if they are
calibrated incorrectly
Method Errors
These are errors arising from the following
sources
• Chemical reagents – ex. wrong choice of
chemical indicator in an acid-base titration,
non-specific reagent
• Chemical reaction – reaction is too slow,
incomplete, or side reactions are happening
• Experimental procedure – correct reagents,
but wrong concentration of reagent specified
in the procedure
The titration curve of a strong acid with a
strong base.
Personal Errors
Many measurements require personal
judgements, judgements of this type are often
subject to systematic, unidirectional errors
• Estimating the position of a pointer between
two scale divisions
• Determining the color change at the end
point of a titration
• Reading the level of a liquid with respect to a
graduation in a pipet or buret
• Activating a timer
Personal Error
Parallax Error
– error from
incorrect reading
of volume
Effects of Systematic Errors
• Constant Errors: Constant Errors does
not depend on the size of the quantity
measured
• Proportional Errors: Proportional errors
decrease or increase in proportion to the
size of the sample taken for analysis. A
common cause of proportional errors is
the presence of interfering contaminants in
the sample.
Detecting 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.
Gross Errors
• Gross errors are caused by experimenter
carelessness or equipment failure. They
occur only occasionally.
• Gross error leads to outliers. This error
causes one result to differ significantly
from the rest of the results (either too
large or small)
• Often discarded when assessing data
Review of Basic Concepts
 Accuracy & Precision
 Types of Errors
 Significant Figures &
Rounding-off values
 Precision of measuring devices
 Measurement of mass & volume
Significant Figures in Measurements
- Measures the accuracy and degree
of precision of a measuring device
- Defined as all the certain digits plus
one uncertain digit
Significant Figures
Two different types of numbers
 Exact
Exact numbers are infinitely
important
 Measured
Measured number – measured with a
measuring device so these numbers
have ERRORS
Exact Numbers
An exact number is obtained when you count objects
or use a defined relationship.
• Counting objects are always exact
2 soccer balls
4 pizzas
• Exact relationships, predefined values, not
measured: 1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001
inches?
No, 1 foot is EXACTLY 12 inches.
Learning Check
1. Exact numbers are obtained by
A. using a measuring tool
B. counting
C. definition
2.
Measured numbers are obtained by
A. using a measuring tool
B. counting
C. definition
Measurement & Significant Figures
Every measurement
has a degree of
uncertainty.
10’s place is certain
1’s place is certain
The 1st decimal
place is uncertain
What is the Length?
1
2
3
We can’t see the markings between 1.6 - 1.7
We must guess between 1.6 & 1.7
We record 1.67 cm as our measurement
The last digit (7) was our guess...stop there
4 cm
Learning Check
What is
stick?
1)
2)
3)
the length of the wooden
4.5 cm
4.57 cm
4.574 cm
Measured Numbers
Do you see why Measured Numbers have
error…you have to make that Guess!
Significant figures are All certain digits plus
one uncertain digit.
The last significant figure is only the best
possible estimate.
Significant figures
Length of the wooden stick: 4.57 cm
Significant figures
Mass measurement: 373.35 g
Significant figures
36.5 mL
47 mL
20.38 mL
Rules in determining Significant Figures
1. All non-zero digits are significant
1.234 kg
4 significant figures
2. Rules for Zeros depend on position
a) Zeros between nonzero digits are
significant
606 m
3 significant figures
Rules in determining Significant Figures
b) Starting zeros are NOT significant
0.008 L
1 significant figure
c) Ending zeros
i. Significant if there is a decimal point
2.000 mg
4 significant figures
ii. Not significant if there is NO decimal
point
4002000 g 4 significant figures
How many significant figures are in
each of the following measurements?
24 mL
2 SF
3001 g
4 SF
0.0320 m3
3 SF
6.4 x 104 molecules
2 SF
560 kg
2 SF
Significant Figures
Addition or Subtraction
The answer must have the same number of
decimal places as the original number with the
least decimal place
89.332 + 1.1 = 90.432 = 90.4
3.70
- 2.9133
0.7867
round off to 0.79
Significant Figures
Multiplication or Division
The answer must have the same significant
figures as the original number with the
smallest number of significant figures
4.51 x 3.6666 = 16.536366 = 16.5
6.8 ÷ 112.04 = 0.0606926 = 0.061
Rounding Off Numbers
RULE 1. If the first digit you remove is 4 or
less, drop it and all following digits.
RULE 2. If the first digit removed is 5 or
greater, round up by adding 1 to the last
digit kept.
If a calculation has several steps, it is best to
round off at the END.
Rounding Off Numbers
Round off the following numbers into 3 SF:
1.5587
1.56
0.0037381
0.00374
13670
13700
128,522
129,000
1.6683 106
1.67 106
Examples of Rounding
Round off to 4 SF:
4965.03
4965
780,582
780,600
1999.5
2.000 x 103
Seatwork
Determine the
number of SF
Round-off to 3
SF:
1. 10470 m
6. 0.03458 m
2. 0.00054600 s
7. 10897 g
3. 540 min
8. 1.0539 s
4. 45.000 g
9. 17.000 g
5. 3.50 103 m
10.1.645 10-3 m
Review of Basic Concepts
 Accuracy & Precision
 Types of Errors
 Significant Figures & Roundingoff values
 Precision of measuring devices
 Measurement of mass & volume
Reading a measurement to the correct
precision
To what precision should a measurement be read?
The key is in the graduation or calibration of the
measuring device. If the graduation is by tenth
or hundredth or thousandth, then the precision is
10% of the graduation. If the graduation is by 2,
0.2, 0.02, then the precision is 25% of the
graduation.
Reading Temperature
Temperature
in °C
28.2 +
0.10C
Since Δ is 1, then the
reader can separate the
graduation in their mind
to 10 equal increments
and therefore the
precision is to the tenth of
the increment (+ 0.1).
Temperature
in °F
82.8 + 0.50F
Since Δ is 2, then the
reader can separate the
graduation in their mind to
four equal parts. Thus the
precision is 25% of the
increment (+ 0.5).
Reading Lengths
Since the smallest increment is 0.1 cm,
the the precision of the ruler is + 0.01
cm.
5
3
The length of the rod is 5.50 + 0.01 cm
Reading Mass
Mass = 126.4 + 0.1 lb
Mass = 6.50 + 0.05 g
When reading a digital scale, all shown digits are
significant. The uncertainty is the last digit on the
display. The bathroom scale above changes in increments
of 0.1 as the mass is increased. Therefore, the precision
of this scale is + 0.1 lb. The digital scale on the right,
however, changes in increment of 0.05 g. Therefore, the
precision of the digital pocket scale is + 0.05 g.
Reading Volumes
 = 1 mL
The precision of this
graduated cylinder is
0.1 mL.
Vol. = 52.5 + 0.1 mL
 = 0.2 mL
The precision of this
graduated cylinder is
0.05 mL.
Vol. = 6.60 + 0.05 mL
Determine the precision of the given
measuring tools?
+ 0.01 cm
+ 0.05 mL
Determine the precision of the given
measuring tools?
+ 0.01 g
Determining the density of the Glycerol sample
Initial mass (g)
Final mass (g)
Mass of added
Liquid (g)
Trial 1
21.50 + 0.01
25.40 + 0.01
3.90 + 0.02
Volume of liquid
(ml)
3.00 + 0.01
Density of liquid
(g/cc)
1.30 + 0.01
Review of Basic Concepts
 Accuracy & Precision
 Types of Errors
 Significant Figures & Roundingoff values
 Precision of measuring devices
 Measurement of mass &
volume
Mass
mass – measure of the quantity of matter
SI unit of mass is the kilogram (kg)
1 kg = 1000 g = 1 x 103 g
beam balance
electronic balance
Measurement of Volume
Volume – measure of the three dimensional
space occupied by matter
SI unit of mass is the cubic meter (m3)
Other Units :
cubic centimeters (cm3),
Cubic decimeters (dm3),
liters(l) and milliliters(ml)
1 m3 = 1000 dm3
1 dm3 = 1000 cm3 = 1 L
1 mL = 1 cm3
Measurement of Volume
The apparatus chosen depends on the
volume and accuracy needed.
Measurement of Volume
Beaker and Erlenmeyer Flask
- Approximate volume measurement
Measurement of Volume
Graduated Cylinder
- More accurate volume measurement
Measurement of Volume
Burette
- Used to accurately measure a volume of
reactant needed to react with another reactant
Measurement of Volume
Pipette
- Used to accurately transfer a fixed
volume of liquid