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
