Understanding the Residual Errors - v



Simple Task

Given L=(l1,l2,l3,.....,ln). Assumption all measurements are free of gross errors and corrected for all systematic errors.

Processing using Microsoft Excel and Matlab

Observations of angles (sec)

L=[22.7 22.3 25.5 23.8 22.9 22.2 21.9 26.1 22.6 21.7 25.4 24.2 24.7 24.4 23.4 23.3 24.3 21.2 25.3 23.9 24.0 24.8 23.2 23.7 25.9 24.6 23.8 23.0 25.0 22.3 20.5 23.5 22.0 24.1 23.1 24.1 23.1 25.9 22.8 25.3 22.5 22.9 23.8 22.6 21.8 23.2 25.2 22.8 23.6 20.1]

After sorting – Matlab: sort=sort(L)then

Observation

u

1

20.1

2

20.5

3

21.2

4

21.7

5

21.8

6

21.9

7

22

8

22.2

9

22.3

10

22.3

11

22.5

12

22.6

13

22.6

14

22.7

15

22.8

16

22.8

17

22.9

18

22.9

19

23

20

23.1

21

23.1

22

23.2

23

23.2

24

23.3

25

23.4

26

23.5

27

23.6

28

23.7

29

23.8

30

23.8

31

23.8

32

23.9

33

24

34

24.1

35

24.1

36

24.2

37

24.3

38

24.4

39

24.6

40

24.7

41

24.8

42

25

43

25.2

44

25.3

45

25.3

46

25.4

47

25.5

48

25.9

49

25.9

50

26.1


Numerical Analysis

min, ū=

∑u / n

=

1175/50

=

23.5

Standard deviaton,σ=

√∑v² / ∑(n-1)

=

√[ (92.36 / (50-1) ]

=

± 1.373

PROBABLE ERROR AT 68.3%

t=

1.0009

E68.3=

t ( σ )

=

1.0009 ( ± 1.373 )

=

± 1.37

min range =

ū-E68.3

=

23.5-1.37

=

22.13

max range =

ū+E68.3

=

23.5+1.37

=

24.87

=

between 22.13 ‒– 24.87 , 34 observations ( 68 % )

PROBABLE ERROR AT 95%

t=

1.96

E95=

t ( σ )

=

1.96 ( ± 1.373 )

=

± 2.69

min range =

ū-E95

=

23.5-2.69

=

20.81

max range =

ū+E95

=

23.5+2.69

=

26.19

=

between 20.81 –‒ 26.19 , 47 observation ( 94 % )

PROBABLE ERROR AT 99%

t=

2.576

E99=

t ( σ )

=

2.576 ( ± 1.373 )

=

± 3.53

min range =

ū-E99

=

23.5-3.53

=

19.97

max range =

ū+E99

=

23.5+3.53

=

27.03

=

between 19.97 –‒ 27.03 , 50 observations ( 100 % )

PROBABLE ERROR AT 99.7%

t=

2.965

E99.7=

t ( σ )

=

2.965 ( ± 1.373 )

=

± 4.06

min range =

ū-E99.7

=

23.5-4.06

=

19.44

max range =

ū+E99.7

=

23.5+4.06

=

27.56

=

between 19.44 –‒ 27.56 , 50 observations ( 100 % )

PROBABLE ERROR AT 99.9%

t=

3.29

E99.7=

t ( σ )

=

3.29 ( ± 1.373 )

=

± 4.51

min range =

ū-E99.9

=

23.5-4.51

=

18.99

max range =

ū+E99.9

=

23.5+4.51

=

28.01

=

between 18.99 –‒ 28.01 , 50 observations ( 100 % )

Histogram




0 comments: