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Accuracy & Precision
in Measurement
Nagendra Bahadur Amatya
Associate Professor
nbamatya@ioe.edu.np
9851072341
Accuracy & Precision
• Accuracy:
• How close you are to the
actual value
• Depends on the person
measuring
• Calculated by the formula:
% Error = (YV – AV) x 100 ÷ AV
Where: YV is YOUR measured Value &
AV is the Accepted Value
• Precision:
• How finely tuned your
measurements are or
how close they can be
to each other
• Depends on the
measuring tool
• Determined by the
number of significant
digits
Accuracy & Precision
• Accuracy & Precision may be demonstrated
by shooting at a target.
• Accuracy is represented by hitting the bulls
eye (the accepted value)
• Precision is represented by a tight grouping
of shots (they are finely tuned)
Accuracy & Precision
Precision without
Accuracy
No Precision &
No Accuracy
Accuracy without
Precision
Accuracy - Calculating % Error
How Close Are You to the Accepted
Value (Bull’s Eye)
• Accuracy is the closeness of a measured
value to the true value.
• For example, the measured density of water has
become more accurate with improved
experimental design, technique, and equipment.
Density of H2O at 20° C
(g/cm3)
1
1.0
1.00
0.998
0.9982
0.99820
0.998203
Accuracy - Calculating % Error
• If a student measured the room width at
8.46 m and the accepted value was 9.45 m
what was their accuracy?
• Using the formula:
% error = (YV – AV) x 100 ÷ AV
• Where YV is the student’s measured value &
AV is the accepted value
Accuracy - Calculating % Error
• Since YV = 8.46 m, AV = 9.45 m
• % Error = (8.46 m – 9.45 m) x 100 ÷ 9.45
m
• = -0.99 m x 100 ÷ 9.45 m
• = -99 m ÷ 9.45 m
• = -10.5 %
• Note that the meter unit cancels during the division
& the unit is %. The (-) shows that YV was low
• The student was off by almost 11% & must
remeasure
•percent error is used to estimate the accuracy of a
surement.
•percent error will always Positive
What is the percent error if the measured density of
titanium (Ti) 45 g/cm3 and the accepted density of Ti
is 4.50 g/cm3?
remeasure -5%
5% remeasure
•Acceptable error is +/- 5%
•Values from –5% up to 5% are acceptable
•Values less than –5% or greater than 5% must be remeasured
Significant Digits
How to Check a Measurement for
Precision
Significant Digits & Precision
• The precision of a measurement is the smallest
possible unit that could be measured.
• Significant Digits (sd) are the numbers that
result from a measurement.
• When a measurement is converted we need to
make sure we know which digits are significant
and keep them in our conversion
• All digits that are measured are significant
Significant Digits & Precision
• How many digits are
there in the
measurement?
• All of these digits are
significant
• There are 3 sd
Length of Bar = 3.23 cm
cm
1 2 3 40
•What is the length of the bar?
Significant Digits & Precision
• If we converted to that measurement of 3.23 cm to
µm what would we get?
• Right! 32 300 µm
• How many digits in our converted number?
• Are they all significant digits (measured)?
• Which ones were measured and which ones were
added because we converted?
• If we know the significant digits we can know the
precision of our original measurement
Significant Digits & Precision
• What if we didn’t know the original
measurement – such as 0.005670 hm. How
would we know the precision of our
measurement.
• The rules showing how to determine the
number of significant digits is shown in
your lab manual on p. 19. Though you can
handle them, they are somewhat complex.
Types of Errors
Difference between measured result and true value.
•Illegitimate errors
•Blunders resulting from mistakes in procedure. You must be careful.
•Computational or calculational errors after the experiment.
•Bias or Systematic errors
•An offset error; one that remains with repeated measurements (i.e. a change of
indicated pressure with the difference in temperature from calibration to use).
•Systematic errors can be reduced through calibration
•Faulty equipment--such as an instrument which always reads 3% high
•Consistent or recurring human errors-- observer bias
•This type of error cannot be evaluated directly from the data but can be
determined by comparison to theory or other experiments.
Types of Errors (cont.)
• Random, Stochastic or Precision errors:
• An error that causes readings to take random-like values
about the mean value.
• Effects of uncontrolled variables
• Variations of procedure
• The concepts of probability and statistics are used to study
random errors. When we think of random errors we also
think of repeatability or precision.
Bias, Precision, and Total Error
•X True
•Total
error
•Bias Error
•Precision
Error
•X measured
Accuracy and Precision
•Accuracy is the closeness of a measurement (or set of
observations) to the true value. The higher the accuracy
the lower the error.
•Precision is the closeness of multiple observations to one
another, or the repeatability of a measurement.
Uncertainty Analysis
•The estimate of the error is called the uncertainty.
•It includes both bias and precision errors.
•We need to identify all the potential significant errors for the
instrument(s).
•All measurements should be given in three parts
•Mean value
•Uncertainty
•Confidence interval on which that uncertainty is based (typically 95%
C.I.)
•Uncertainty can be expressed in either
absolute terms (i.e., 5 Volts ±0.5 Volts)
or in
percentage terms (i.e., 5 Volts ±10%)
(relative uncertainty = ∆V / V x 100)
•We will use a 95 % confidence interval throughout this course (20:1 odds).
How to Estimate Bias Error
• Manufacturers’ Specifications
• If you can’t do better, you may take it from the
manufacturer’s specs.
• Accuracy - %FS, %reading, offset, or some combination
(e.g., 0.1% reading + 0.15 counts)
• Unless you can identify otherwise, assume that these
are at a 95% confidence interval
• Independent Calibration
• May be deduced from the calibration process
Use Statistics to Estimate Random Uncertnaty
• Mean: the sum of measurement values divided by the
number of measurements.
• Deviation: the difference between a single result and the
mean of many results.
• Standard Deviation: the smaller the standard deviation the
more precise the data
• Large sample size 
• Small sample size (n<30)
• Slightly larger value 
The Population
• Population: The collection of all items (measurements) of
the group. Represented by a large number of
measurements.
• Gaussian distribution*
• Sample: A portion of (or limited number of items in) a
population.
• *Data do not always abide by the Gaussian distribution. If
not, you must use another method!!
Student t-distribution (small sample sizes)
• The t-distribution was formulated by W.S. Gosset, a
scientist in the Guinness brewery in Ireland, who
published his formulation in 1908 under the pen name
(pseudonym) “Student.”
• The t-distribution looks very much like the Gaussian
distribution, bell shaped, symmetric and centered about
the mean. The primary difference is that it has stronger
tails, indicating a lower probability of being within an
interval. The variability depends on the sample size, n.
Student t-distribution
• With a confidence interval of c%
• Where α=1-c and ν=n-1 (Degrees of Freedom)
• Don’t apply blindly - you may have better
information about the population than you
think.
n
txX
n
tx ss σσ
νανα ,2/,2/ +<<−
Example: t-distribution
• Sample data
• n = 21
• Degrees of Freedom = ν = 20
• Desire 95% Confidence Interval
∀α = 1 - c = 0.05
∀α/2 = 0.025
• Student t-dist chart
• t=2.086
Reading Number Volts, mv
1 5.30
2 5.73
3 6.77
4 5.26
5 4.33
6 5.45
7 6.09
8 5.64
9 5.81
10 5.75
11 5.42
12 5.31
13 5.86
14 5.70
15 4.91
16 6.02
17 6.25
18 4.99
19 5.61
20 5.81
21 5.60
Mean 5.60
Standard dev. 0.51
Variance 0.26
Reading Number Volts, mv
1 5.30
2 5.73
3 6.77
4 5.26
5 4.33
6 5.45
7 6.09
8 5.64
9 5.81
10 5.75
11 5.42
12 5.31
13 5.86
14 5.70
15 4.91
16 6.02
17 6.25
18 4.99
19 5.61
20 5.81
21 5.60
Mean 5.60
Standard dev. 0.51
Variance 0.26
Estimate of Precision Error is Then:
23.060.5
21
51.0
086.260.5
,2
±
⋅±
⋅±
n
tx sσ
ν
α
• Precision error is
• ±0.23 Volts
How to combine bias and precision error?
• Rules for combining independent uncertainties for
measurements:
• Both uncertainties MUST be at the same Confidence
Interval (95%)
• Precision error obtained using Student’s-t method
• Bias error determined from calibration, manufacturers’
specifications, smallest division.
22
xxx PBU +=
Propagation of Error
• Used to determine uncertainty of a quantity
that requires measurement of several
independent variables.
• Volume of a cylinder = f(D,L)
• Volume of a block = f(L,W,H)
• Density of an ideal gas = f(P,T)
• Again, all variables must have the same
confidence interval.
RSS Method (Root Sum Squares)
• For a function y = f(x1,x2,...,xN), the RSS uncertainty is:
• First determine uncertainty of each variable in the
form ( xN ± ∆xN)
• Use previously established methods, including bias
and precision error.














∆++





∆+





∆=∆
22
2
2
2
1
1
... N
N
RSS x
x
f
x
x
f
x
x
f
u
∂
∂
∂
∂
∂
∂
Example Problem: Propagation of Error
• Ideal gas law:
• Temperature
• T±∆T
• Pressure
• P±∆P
• R=Constant
ρ =
P
RT
•How do we
estimate the error
in the density?
( )
2
2
222
1








+







=





+





= TΔ
TR
P
PΔ
TR
TΔ
T
ρ
pΔ
p
ρ
ρΔ RSS
∂
∂
∂
∂
•Apply RSS formula to density relationship:
22





 ∆
+




 ∆
=
∆
T
T
p
p
ρ
ρ
•Apply a little algebra: ρ =
P
RT

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Accuracy, Precision Measurement

  • 1. Accuracy & Precision in Measurement Nagendra Bahadur Amatya Associate Professor nbamatya@ioe.edu.np 9851072341
  • 2. Accuracy & Precision • Accuracy: • How close you are to the actual value • Depends on the person measuring • Calculated by the formula: % Error = (YV – AV) x 100 ÷ AV Where: YV is YOUR measured Value & AV is the Accepted Value • Precision: • How finely tuned your measurements are or how close they can be to each other • Depends on the measuring tool • Determined by the number of significant digits
  • 3. Accuracy & Precision • Accuracy & Precision may be demonstrated by shooting at a target. • Accuracy is represented by hitting the bulls eye (the accepted value) • Precision is represented by a tight grouping of shots (they are finely tuned)
  • 4. Accuracy & Precision Precision without Accuracy No Precision & No Accuracy Accuracy without Precision
  • 5. Accuracy - Calculating % Error How Close Are You to the Accepted Value (Bull’s Eye)
  • 6. • Accuracy is the closeness of a measured value to the true value. • For example, the measured density of water has become more accurate with improved experimental design, technique, and equipment. Density of H2O at 20° C (g/cm3) 1 1.0 1.00 0.998 0.9982 0.99820 0.998203
  • 7. Accuracy - Calculating % Error • If a student measured the room width at 8.46 m and the accepted value was 9.45 m what was their accuracy? • Using the formula: % error = (YV – AV) x 100 ÷ AV • Where YV is the student’s measured value & AV is the accepted value
  • 8. Accuracy - Calculating % Error • Since YV = 8.46 m, AV = 9.45 m • % Error = (8.46 m – 9.45 m) x 100 ÷ 9.45 m • = -0.99 m x 100 ÷ 9.45 m • = -99 m ÷ 9.45 m • = -10.5 % • Note that the meter unit cancels during the division & the unit is %. The (-) shows that YV was low • The student was off by almost 11% & must remeasure
  • 9. •percent error is used to estimate the accuracy of a surement. •percent error will always Positive What is the percent error if the measured density of titanium (Ti) 45 g/cm3 and the accepted density of Ti is 4.50 g/cm3?
  • 10. remeasure -5% 5% remeasure •Acceptable error is +/- 5% •Values from –5% up to 5% are acceptable •Values less than –5% or greater than 5% must be remeasured
  • 11.
  • 12. Significant Digits How to Check a Measurement for Precision
  • 13. Significant Digits & Precision • The precision of a measurement is the smallest possible unit that could be measured. • Significant Digits (sd) are the numbers that result from a measurement. • When a measurement is converted we need to make sure we know which digits are significant and keep them in our conversion • All digits that are measured are significant
  • 14. Significant Digits & Precision • How many digits are there in the measurement? • All of these digits are significant • There are 3 sd Length of Bar = 3.23 cm cm 1 2 3 40 •What is the length of the bar?
  • 15. Significant Digits & Precision • If we converted to that measurement of 3.23 cm to µm what would we get? • Right! 32 300 µm • How many digits in our converted number? • Are they all significant digits (measured)? • Which ones were measured and which ones were added because we converted? • If we know the significant digits we can know the precision of our original measurement
  • 16. Significant Digits & Precision • What if we didn’t know the original measurement – such as 0.005670 hm. How would we know the precision of our measurement. • The rules showing how to determine the number of significant digits is shown in your lab manual on p. 19. Though you can handle them, they are somewhat complex.
  • 17. Types of Errors Difference between measured result and true value. •Illegitimate errors •Blunders resulting from mistakes in procedure. You must be careful. •Computational or calculational errors after the experiment. •Bias or Systematic errors •An offset error; one that remains with repeated measurements (i.e. a change of indicated pressure with the difference in temperature from calibration to use). •Systematic errors can be reduced through calibration •Faulty equipment--such as an instrument which always reads 3% high •Consistent or recurring human errors-- observer bias •This type of error cannot be evaluated directly from the data but can be determined by comparison to theory or other experiments.
  • 18. Types of Errors (cont.) • Random, Stochastic or Precision errors: • An error that causes readings to take random-like values about the mean value. • Effects of uncontrolled variables • Variations of procedure • The concepts of probability and statistics are used to study random errors. When we think of random errors we also think of repeatability or precision.
  • 19. Bias, Precision, and Total Error •X True •Total error •Bias Error •Precision Error •X measured
  • 20. Accuracy and Precision •Accuracy is the closeness of a measurement (or set of observations) to the true value. The higher the accuracy the lower the error. •Precision is the closeness of multiple observations to one another, or the repeatability of a measurement.
  • 21. Uncertainty Analysis •The estimate of the error is called the uncertainty. •It includes both bias and precision errors. •We need to identify all the potential significant errors for the instrument(s). •All measurements should be given in three parts •Mean value •Uncertainty •Confidence interval on which that uncertainty is based (typically 95% C.I.) •Uncertainty can be expressed in either absolute terms (i.e., 5 Volts ±0.5 Volts) or in percentage terms (i.e., 5 Volts ±10%) (relative uncertainty = ∆V / V x 100) •We will use a 95 % confidence interval throughout this course (20:1 odds).
  • 22. How to Estimate Bias Error • Manufacturers’ Specifications • If you can’t do better, you may take it from the manufacturer’s specs. • Accuracy - %FS, %reading, offset, or some combination (e.g., 0.1% reading + 0.15 counts) • Unless you can identify otherwise, assume that these are at a 95% confidence interval • Independent Calibration • May be deduced from the calibration process
  • 23. Use Statistics to Estimate Random Uncertnaty • Mean: the sum of measurement values divided by the number of measurements. • Deviation: the difference between a single result and the mean of many results. • Standard Deviation: the smaller the standard deviation the more precise the data • Large sample size  • Small sample size (n<30) • Slightly larger value 
  • 24.
  • 25. The Population • Population: The collection of all items (measurements) of the group. Represented by a large number of measurements. • Gaussian distribution* • Sample: A portion of (or limited number of items in) a population. • *Data do not always abide by the Gaussian distribution. If not, you must use another method!!
  • 26. Student t-distribution (small sample sizes) • The t-distribution was formulated by W.S. Gosset, a scientist in the Guinness brewery in Ireland, who published his formulation in 1908 under the pen name (pseudonym) “Student.” • The t-distribution looks very much like the Gaussian distribution, bell shaped, symmetric and centered about the mean. The primary difference is that it has stronger tails, indicating a lower probability of being within an interval. The variability depends on the sample size, n.
  • 27. Student t-distribution • With a confidence interval of c% • Where α=1-c and ν=n-1 (Degrees of Freedom) • Don’t apply blindly - you may have better information about the population than you think. n txX n tx ss σσ νανα ,2/,2/ +<<−
  • 28. Example: t-distribution • Sample data • n = 21 • Degrees of Freedom = ν = 20 • Desire 95% Confidence Interval ∀α = 1 - c = 0.05 ∀α/2 = 0.025 • Student t-dist chart • t=2.086 Reading Number Volts, mv 1 5.30 2 5.73 3 6.77 4 5.26 5 4.33 6 5.45 7 6.09 8 5.64 9 5.81 10 5.75 11 5.42 12 5.31 13 5.86 14 5.70 15 4.91 16 6.02 17 6.25 18 4.99 19 5.61 20 5.81 21 5.60 Mean 5.60 Standard dev. 0.51 Variance 0.26 Reading Number Volts, mv 1 5.30 2 5.73 3 6.77 4 5.26 5 4.33 6 5.45 7 6.09 8 5.64 9 5.81 10 5.75 11 5.42 12 5.31 13 5.86 14 5.70 15 4.91 16 6.02 17 6.25 18 4.99 19 5.61 20 5.81 21 5.60 Mean 5.60 Standard dev. 0.51 Variance 0.26
  • 29. Estimate of Precision Error is Then: 23.060.5 21 51.0 086.260.5 ,2 ± ⋅± ⋅± n tx sσ ν α • Precision error is • ±0.23 Volts
  • 30. How to combine bias and precision error? • Rules for combining independent uncertainties for measurements: • Both uncertainties MUST be at the same Confidence Interval (95%) • Precision error obtained using Student’s-t method • Bias error determined from calibration, manufacturers’ specifications, smallest division. 22 xxx PBU +=
  • 31. Propagation of Error • Used to determine uncertainty of a quantity that requires measurement of several independent variables. • Volume of a cylinder = f(D,L) • Volume of a block = f(L,W,H) • Density of an ideal gas = f(P,T) • Again, all variables must have the same confidence interval.
  • 32. RSS Method (Root Sum Squares) • For a function y = f(x1,x2,...,xN), the RSS uncertainty is: • First determine uncertainty of each variable in the form ( xN ± ∆xN) • Use previously established methods, including bias and precision error.               ∆++      ∆+      ∆=∆ 22 2 2 2 1 1 ... N N RSS x x f x x f x x f u ∂ ∂ ∂ ∂ ∂ ∂
  • 33. Example Problem: Propagation of Error • Ideal gas law: • Temperature • T±∆T • Pressure • P±∆P • R=Constant ρ = P RT •How do we estimate the error in the density?
  • 34. ( ) 2 2 222 1         +        =      +      = TΔ TR P PΔ TR TΔ T ρ pΔ p ρ ρΔ RSS ∂ ∂ ∂ ∂ •Apply RSS formula to density relationship: 22       ∆ +      ∆ = ∆ T T p p ρ ρ •Apply a little algebra: ρ = P RT