The document discusses measurement errors and how to account for them in scientific experiments and calculations. It defines two types of errors - random errors, which can be positive or negative, and systematic errors, which are due to limitations in instruments or techniques. Random errors can be reduced by taking multiple measurements, while systematic errors are minimized by using the best available techniques. The document also covers how to calculate the combined error when taking measurements, manipulating formulas, and quoting results with the associated uncertainty.

1. Khang Cao Nguyen

2. Outline
1. Why errors are important?
2. Standards of Measurement (SI system)
3. Two types of error: random and systematic
4. Treatment of errors
5. Quoting results and errors

3. "Science starts with the measurement."
Dmitri Mendeleev
FA=m.v.d
• Measurements are always somewhat
different from the “true value.”
These deviations from the true value
are called errors.

4. Bad news…
• No matter how good
you are… there will
always be errors.
• The question is…
How to deal with
them?
So, errors are important

5. System of measurement
• A system of measurement is a set of units of
measurement.
• A system of measurement of each country are
different.
• Some quantities are designated as
fundamental units, this quantities close to life.

6. “Sào”
• “Sào” is a family area average in Vietnam
• “Sào” is about 360 m2

7. “Thốn”
• “Thốn” as the width of an average man's thumb
at the base of the nail. “Thốn” unit is always
useful in oriental medicine until now.

8. Inch
One inch was equal to 3 barleycorn
One inch is distance from top of thumb to top of
index of King Henry

9. Copper pot with water inside Clock
Timer counter

11. Standards of Measurement
• SI units are those of the Système International
d’Unités adopted in 1960
• Used for general measurement in most
countries worldwide

12. Define the 7 Fundamentals
• Length meter m
• Mass kilogram kg
• Time second s
• Electric current ampere A
• Thermodynamic temp Kelvin K
• Luminous Intensity candela cd
• Amount of a substance mole mol

13. -Meter is one ten-millionth of the distance from the
Earth's equator to the North Pole (at sea level)
- Meter bar, made of an alloy of platinum and iridium, that
was the standard from 1889 to 1960
- The length of the path travelled by light in vacuum
during a time interval of 1/299,792,458 of a second
Define Meter

14. Standards
• The Meter :- the distance traveled by a
beam of light in a vacuum over a defined
time interval ( 1/299 792 458 seconds)
• The Kilogram :- a particular platinum-
iridium cylinder kept in Sevres, France
• The Second : the time interval between
the vibrations in the caesium atom (1 sec =
time for 9 192 631 770 vibrations)

15. After 1860

16. Experiment results
x x x  

17. Two types of error: random and systematic
• Errors can be divided into 2 main classes
• Random errors
• Systematic errors

18. Random errors
• An error that different between different measurements
• It may be positive or negative
• Always present in an experiment
• Sometimes referred to as reading errors
• Can be minimised by performing multiple measurements

19. Systematic Errors
• Object:
– Instrumental,
physical and human
limitations.
• Example: Device is
out-of calibration.
• How to minimize them?
– Best possible techniques.
• Are TYPICALLY
present.
• Measurements are given
as:
Measurement + Systematic Error
OR
Measurement - Systematic Error

20. Experiment results
x x x  
Errors = Random errors+Systematic Errors

21. The Gaussian Distribution

22. -3 -2 -1 0 +1 +2 +3
68%
x  x 
Gaussian PDF:
2
2
1 ( )
( ) exp
22
x x
P x
 
  
  
 
mean
Std. dev.
The Gaussian Distribution

23. -3 -2 -1 0 +1 +2 +3
96%
x  x 
Gaussian PDF:

p(x) 
1
 2
exp
(x  x)2
22






mean
Std. dev.
The Gaussian Distribution

24. -3 -2 -1 0 +1 +2 +3
99.7%
x  x 
Gaussian PDF:

p(x) 
1
 2
exp
(x  x)2
22






mean
Std. dev.
The Gaussian Distribution

25. Range centered on Mean
Measurements within
Range
68% 95% 99.7%
99.994
%
Measurements outside
Range
32%
1 in 3
5%
1 in
20
0.3%
1 in
400
0.006%
1 in
16,000
432
The Gaussian Distribution
x x x  

26. Treatment of errors
x1, x2, … xn.
1
1 n
i
i
x x
n 
 
| |i ix x x   1
| |
n
i
i
x
x
n


 

x x x  

27. If the errors are dependent A=f(x,y,z)
So
, , .x x x x y y z z z        
( , , )A f x y z
Treatment of errors

28. A A A
dA dx dy dz
x y z
  
  
  
Now
| | | | | |
A A A
A x y z
x y z
  
      
  
Treatment of errors

29.        
2 2 2 2
2 2 2 2
2 2 2 2
/
ln
exp
n m l
Z A B C Z A B C
Z A B C
Z ABC or AB C etc
Z A B C
Z A B C
Z A B C n m l
Z A B C
A
Z A Z
A
Z
Z A A
Z
         
          
          
       
          
          
       

  

  
Treatment of errors in formulae

30. Example of error manipulation 1
2
A r Where r=(50.5)m
A=78.5398m2
2 2
2
2 2
0.5
0.1
5
2 (2 0.1) 0.04
0.2 0.2 0.2 (78.5398 ) 16
r
r
A r
A r
A
hence A A A m m
A

 
    
      
   

        
Hence final result is A=(7916)m2

31. • P=2L+2W where L=(40.2)m and W=(50.2)m
• P=18m
•
• P=(18.00.3)m
 
2 2 2
(0.2) (0.2) 0.08
0.28
P
P m
   
  
Example of error manipulation 2

32. 2
l
g
  l=(2.50.1)m, g=(9.80.2)ms-2
=3.1735s
22 2
1 1
2 2
l g
l g


      
       
     
2 2
41 0.1 1 0.2
5.04 10
2 2.5 2 9.8
   
        
   
/=0.022 hence =0.022x3.1735=0.070
=(3.170.07)s
Example of error manipulation 3

33. • Value 44, error 5  445
• Value 128, error 32  13030
• Value 4.8x10-3, error 7x10-4  (4.80.7)x10-3
• Value 1092, error 56  109060
• Value 1092, error 14  109214
• Value 12.345, error 0.35  12.30.4
Don’t over quote results to a level inconsistent with
the error 36.6789353720.5
Quoting results and errors


34. Graphing with Errors
When graphing, plot error bars. A very sharp
pencil is good for getting the size just right.

35. Conclusions