MEASUREMENT, PRECISION
AND ACCURACY
Dr. Abayomi A. A.
Measurement
• Measurement is the process observing and recording
the observations that are collected as a research effort.
• It is an act of assigning numbers to objects or events.
• Some of the measurements and calculations in
chemistry involve quantities such as pressure,
volume, mass and energy.
• Every quantity includes both a number and a unit.
For Example
If the mass of a sample is 20 grams, it means that the mass is
20 times the mass of 1 gram. Although 20 grams has the
dimension of mass, 20 is a pure dimensionless number, being
the ratio of two masses, that of the sample and that of the
reference, 1 gram.
International System of Units
• The international system of units (usually known as SI units, from
the French Sys-tème International) consists of several base units
from which all other units (such as those of volume or energy) are
derived.
• Because the base units are sometimes too large or too small for use,
SI prefixes are used to produce smaller or bigger units.
• the milligram (0.001 g, and symbolized mg) is used if small masses
are being recorded.
• The cubic metre (written m3) is too large for most purposes in
chemistry, and the cubic decimetre, dm3 (or litre) is commonly used
3
3
3
i.e 1m = 1000 dm = 1 000 000 cm
Base Unit
SI base unit
Base quantity
Length
Mass
Time
Electric current
Name
meter
kilogram
second
ampere
Symbol
m
kg
s
A
Thermodynamic
temperature
kelvin
K
Amount of
substance
Luminous intensity
mole
mol
candela
cd
Derived Unit
Derived quantity
Name
Symbol
area
square meter
m2
volume
cubic meter
m3
speed, velocity
meter per second
m/s
acceleration
meter per second squared
m/s2
wave number
reciprocal meter
m-1
mass density
kilogram per cubic meter
kg/m3
specific volume
cubic meter per kilogram
m3/kg
current density
ampere per square meter
A/m2
magnetic field strength
ampere per meter
A/m
amount-of-substance concentration mole per cubic meter
mol/m3
luminance
candela per square meter
cd/m2
mass fraction
kilogram per kilogram, which
may be represented by the
kg/kg = 1
number 1
Significant Figures
• The numerical value of every observed measurement is an
approximation, since no physical is exact.
• The accuracy of a measurement is always limited by the
reliability of the measuring instrument.
For Example
A recorded length of 15.7 to the nearest 0.1 cm means
that the actual length lies between 15.65 and 15.75 cm.
If the same measurement was made to the nearest 0.01
cm, it will recorded as 15.70 cm. While 15.7 has 3 sf.
15.70 has 4 sf.
Significant Figures (cont.)
• Measurement are often recorded to show the degree of
uncertainty in the measurement using the “±” sign.
• Thus the measured 15.7 cm could be written as 15.7 ± 0.1
• The recorded digits (both certain and uncertain) are called
significant figures.
• Generally, the ± could be dropped by assuming an
uncertainty of one unit in the rightmost.
• The greater the number of significant figures in a
maesurement, the greater is the uncertainty.
Significant Figures (cont.)
Determining Which Digits are Significant
i. Ensure the measured digit has a decimal point.
i.
Starting from the left, move to the right until the first non
zero digit is reached.
ii. This digit and every digit to its right is significant.
i.
v.
Zeroes that end a number and lie either after or before the
decimal point are significant.
Thus, 1.030, 5300 and 5.300 x 103 have 4 sf.
Sample Questions
i. For each of the following quantities, underline the zeros that
are significant figures and determine the number of
significant figures in each quantities:
a. 0.0030 L b. 0.1044 g
c. 53069 mL d. 0.00004715 m
e. 57,600 s f. 0.0000007160 cm3
Significant Figures and Calculations
 When doing arithmetic operations as common in most calculations
in chemistry, the urge to round off must be avoided.
 The following must be observed in such cases:
i. For multiplication and division, the answer should contain the
same number of sf as in the measurement with the lowest sf.
e.g What is the volume of a graphite composite with 9.2 cm length,
6.83 cm width and 0.3744 cm thickness?
V (cm3) = 9.2 cm x 6.83 cm x 0.3744 cm = 23 cm3
ii. For addition and subtraction, the answer will have the same no. of
decimal places as the measurement with the least no. of decimal
places
e.g What is the final volume of a mixture containing 38.5 mL
water and 23.28 mL protein solution
V (cm3) = 38.5 mL + 23.28 mL = 106.8 mL
Rules for Rounding Off
i.
If the digit removed is more than 5, the preceding number is
increased by 1. Thus 5.379 rounds off to 5.38
i.
If the digit removed is less than 5, the preceding number
remains unchanged. Thus 0.2413 becomes 0.241.
ii.
If the digit remove is 5, the preceding number is increased by
1 if it is odd and remains unchanged if it is even. Thus 17.75
rounds up to 17.8 while 17.65 become 17.6.
iii. Always carry one or two additional sf through multistep
calculation and round up.
Statistical Treatment of Data
• Significant Figures : number of digits
know with certainty + the first in doubt.
• Rounding off: use the same number of
significant figures.
•Addition and subtraction: 13.4+
1478.224 = 1491.624 ~ 1491.6
•Multiplication and division:
31x350.1=10,853.1~11,000
Statistical Treatment of Data
• Mean, arithmetic mean, and average
are synonyms.
N
x
 xi
i 1
N
• Median: is the middle result when
replicate data are arranged in order of
size.
Statistical Treatment of Data
• Accuracy: indicates the closeness of the
measurement to its true value or
accepted value. It is expressed by the
error.
• Precision: describes the reproducibility
of measurements. That is: the closeness
of results that have been obtained in
exactly the same way.
Precision and Accuracy
 Accuracy refers to the closeness of a measured value to a
standard or known value.
For example, if a weight measurement is 3.2 kg for a given
substance, but the actual or known weight is 10 kg, then the
measurement is not accurate.
 Precision refers to the closeness of two or more measurements
to each other.
For example: If the above measurement is made thrice to give
3.2, 3.1 and 3.2, then the measurement is precise.
Statistical Treatment of Data
Low accuracy, low precision
Low accuracy, high precision
High accuracy, low precision
High accuracy, high precision
Precision and Accuracy
 The concept of precision and accuracy are linked with two
types of errors:
i.
Systematic error: This produces values that are either all
higher or all lower than the actual or true value.
Such errors are common and may be due to a faulty or uncalibrated measuring device.
ii.
Random error: this produces some values that are higher or
lower than the true values. Random errors depend on the
measurer’s skill.
Precise measurements have low random error.
Accurate measurements have low systematic error and
generally low random error.


Statistical Treatment of Data
Distribution of Experimental Data
• Precision:
•Describes the reproducibility of
measurements.
•It can be represented by the deviation
from the mean. That is:
di  xi  x
Distribution of Experimental Data
Precision: Describes the reproducibility of
the measurements.
•It can be represented by:
•The deviation from the mean. di  xi  x
xi  x

•Average Deviation. d 
N
 x
N
•Standard Deviation s 
i 1
 x
2
i
N 1
Statistical Treatment of Data
• Kind of Errors:
•Systematic: instrument or the measuring
technique.
•Random: judgement of the observer,
fluctuations in conditions (temp., voltage,
pressure, etc.)
Statistical Treatment of Data
• Absolute Error: E  xi  xtrue
xi  xtrue
• Relative Error: E 
100%
xtrue
Standard Deviation
• Sample Standard deviation (for
use with small samples n< ~25)
• Population Standard deviation
(for use with samples n > 25)
– m = population mean
– IN the absence of systematic error,
the population mean approaches
the true value for the measured
quantity.
( xi  x )
s
n 1
2
( xi  m )

N
2
Example
• The following results were obtained in the
replicate analysis of a blood sample for its lead
content: 0.752, 0.756, 0.752, 0.760 ppm lead.
Calculate the mean and standard deviation for
the data set.
Standard deviation
• 0.752, 0.756, 0.752, 0.760 ppm lead.
x  0.755
( xi  x )
s
n 1
2
You’d report the amount of lead in this sample of blood as
Excel® Demo
Distributions of Experimental Data
• We find that the distribution of replicate data
from most quantitative analytical
measurements approaches a Gaussian curve.
• Example – Consider the calibration of a pipet.
Replicate data on the calibration of a
10-ml pipet.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Volume
9.988
9.973
9.986
9.980
9.975
9.982
9.986
9.982
9.981
9.990
9.980
9.989
9.978
9.971
9.982
9.983
9.988
Mean
9.982 ml
median
9.982 ml
spread
0.025 ml
Standard Deviation
Trial
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
0.0056 ml
Volume
9.975
9.980
9.994
9.992
9.984
9.981
9.987
9.978
9.983
9.982
9.991
9.981
9.969
9.985
9.977
9.976
9.983
Trial
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Volume
9.976
9.990
9.988
9.971
9.986
9.978
9.986
9.982
9.977
9.977
9.986
9.978
9.983
9.980
9.983
9.979
Frequency distribution
Volume
Range, mL
Number in Range
% in range
9.969 to 9.971
3
9.982 to 9.974
1
9.975 to 9.977
7
9.978 to 9.980
9
9.981 to 9.983
13
9.984 to 9.986
7
9.987 to 9.989
5
9.990 to 9.992
4
9.993 to 9.995
1
6
2
14
18
26
14
10
8
2
14
Average= 9.982
12
Number of measurements
Std. Dev = + 0.0056
10
8
6
4
2
0
9.965
9.970
9.975
9.980
9.985
9.990
Range of measured values
9.995
Confidence Intervals
For small data sets
ts
Confidence Interval for m  x 
n
small data sets
m is the true mean and the above equations express that
the “true mean” will be in the calculated range at a given
confidence.
Example
• The following results were obtained in the
replicate analysis of a blood sample for its lead
content: 0.752, 0.756, 0.752, 0.760 ppm lead.
Calculate the mean and standard deviation for
the data set.
0.755  0.003 ppm
Find (a) the 50% CL
and (b) the 90% CL
Confidence Intervals
?
ts
CL for m  x 
n
CL for m  0.755 
t  0.003
4
Confidence Intervals
?
ts
CL for m  x 
n
50% CL for m  0.755 
50% CL for m  0.755 
t  0.003
4
0.765  0.003
50% CL for m  0.755  0.001
4
Confidence Intervals
?
ts
CL for m  x 
n
t  0.003
90% CL for m  0.755 
4
2.353  0.003
90% CL for m  0.755 
4
90% CL for m  0.755  0.004
Confidence Intervals
90 % CI
50 % CI
0.750
0.755
0.760
There is a 50% chance that the true mean, m, lies in the range
0.755 + 0.001 ppm (of from 0.754 to 0.756 ppm)
Likewise, these calculations mean that there is a 90% chance
that the true mean, m, lies in the range 0.755 + 0.005 ppm (of
from 0.750 to 0.760 ppm)
Confidence limits and uncertainty
• Suppose we measure the volume of a vessel
five times and observe values:
6.372, 6.375, 6.374, 6.377, and 6.375 mL.
And find average = 6.374 mL
And s = 0.001 mL
Use a 90% CL to Estimate uncertainty!
Experimental Uncertainty
• Well, a 90% CI means that there is a 90%
chance that the true volume is within the
range.
And find average = 6.374 mL
And s = 0.001 mL
ts
CL for m  x 
n
CL for m  6.374 
t (0.001 )
5
Experimental Uncertainty
______(0.0018 )
90% CL for m  6.3746 
5