here is some description about errors present in any observation. like types of errors and much more....

1. THEORY OF ERRORS IN
OBSERVATIONS
Dr. Mahmood Arshad
Assistant Professor,
Dept. of Mining Engineering,
Faculty of Earth Sciences and
Engineering,
University of Engineering & Technology,
Lahore.
smarshad@uet.edu.pk
Min-E-240 Surveying
Lecture 3 – January 24, 2019

2. INTRODUCTION

3. DIRECT AND INDIRECT OBSERVATIONS

4. ERRORS IN MEASUREMENTS
 An error is the difference between an observed value for a
quantity and its true value, or
 It can be unconditionally stated that:
 No observation is exact,
 Every observation contains errors,
 The true value of an observation is never known, and,
 The exact error present is always unknown.

5. MISTAKES
 Mistakes are observer blunders and are usually caused by
misunderstanding the problem, carelessness, fatigue,
missed communication, or poor judgment.

6. SOURCES OF ERRORS IN MAKING OBSERVATIONS
 Natural errors
 Instrumental errors
 Personal errors

8. TYPES OF ERRORS
 Systematic errors, or cumulative errors
 Random errors, or accidental errors

9. PRECISION AND ACCURACY
 A discrepancy is the difference between two observed
values of the same quantity.
 Precision refers to the degree of refinement or consistency
of a group of observations and is evaluated on the basis of
discrepancy size.
 Accuracy denotes the absolute nearness of observed
quantities to their true values.

11. ELIMINATING MISTAKES AND SYSTEMATIC ERRORS
 Mistakes that do occur can be corrected only if discovered
 Comparing several observations
 Making a common sense estimate and analysis
 Repeat the observation
 Widely divergent result may be discarded
 It is seldom safe to change a recorded number

12. PROBABILITY
 Probability may be defined as the ratio of the number of
times a result should occur to its total number of
possibilities.
 Redundant observations are measurements in excess of
the minimum needed to determine a quantity.
 A residual is simply the difference between the most
probable value and any observed value of a quantity

13. GENERAL LAWS OF PROBABILITY
 Small residuals (errors) occur more often than large ones;
that is, they are more probable.
 Large errors happen infrequently and are therefore less
probable; for normally distributed errors, unusually large
ones may be mistakes rather than random errors.
 Positive and negative errors of the same size happen with
equal frequency; that is, they are equally probable.

15. MEASURES OF PRECISION
 Standard deviation and variance
σ = ±
𝑣2
𝑛 − 1
 Variance is σ2, equal to the square of the standard
deviation.

16. INTERPRETATION OF STANDARD DEVIATION
 Standard deviation establishes the limits within which
observations are expected to fall 68.3% of the time.
 Solve from book, Example 3.1

17. THE 50, 90, AND 95 PERCENT ERRORS
 The probability of an error of any percentage likelihood can
be determined. The general equation is
𝐸 𝑝 = 𝐶 𝑝σ
where 𝐸 𝑝 is a certain percentage error and 𝐶 𝑝 the corresponding
numerical factor taken from chart (Figure 3.5 from book) for
relation between error and percentage of area under normal
distribution curve (next slide)
 Refer to Example 3.1

19. Example 3.1
 Compute:
 the most probable value for the line length,
 standard deviation, and
 errors having 50%, 90%, and 95% probability.

20. Example 3.1 – Cont’d

21. Example 3.1 – Cont’d
 Conclusions:
 The most probable line length is 538.45 ft.
 The standard deviation of a single observation is ±0.08 ft.
Accordingly, the normal expectation is that 68% of the time a
recorded length will lie between 538.45 – 0.08 and 538.45 + 0.08
or between 538.37 and 538.53 ft; that is, about seven values
should lie within these limits. (Actually seven of them do.)
 The probable error (𝐸50) is ±0.05. Therefore, it can be anticipated
that half, or five of the observations, will fall in the interval 538.40
to 538.50 ft. (Four values do.)
 The 90% error is ±0.13 ft. and thus nine of the observed values
can be expected to be within the range of 538.32 and 538.58 ft.
 The 95% error is ±0.15 ft. so the length can be expected to lie
between 538.30 and 538.60, 95% of the time. (Note that all
observations indeed are within the limits of both the 90 and 95
percent errors.)

22. ERROR PROPAGATION
 The process of evaluating errors in quantities computed
from observed values that contain errors is called error
propagation.
𝐸𝑠𝑢𝑚 = ± 𝐸 𝑎
2
+ 𝐸 𝑏
2
+ 𝐸𝑐
2
+ ⋯
𝐸𝑆𝑒𝑟𝑖𝑒𝑠 = ± 𝐸2 + 𝐸2 + 𝐸2 + ⋯ = ± 𝑛𝐸2 = ±𝐸 𝑛
𝐸 𝑝𝑟𝑜𝑑 = ± 𝐴2 𝐸 𝑏
2
+ 𝐵2 𝐸 𝑏
2
𝐸 𝑚𝑒𝑎𝑛 =
𝐸𝑆𝑒𝑟𝑖𝑒𝑠
𝑛
=
𝐸 𝑛
𝑛
=
𝐸
𝑛

23. APPLICATIONS
 To analyze observations already made, for comparison with
other results or with specification requirements.
 To establish procedures and specifications in order that the
required results will be obtained.

24. CONDITIONAL ADJUSTMENT OF OBSERVATIONS
 In some types of problems, the sum of several observations
must equal a fixed value.
 Adjustment is made accordingly.

25. WEIGHTS OF OBSERVATIONS
 Concept of relative weights

26. LEAST-SQUARES ADJUSTMENT
 Misclosure
 To be discussed late in Lecture 16.

27. Homework 2
 Part – I: Solve the exercise questions in Chapter 3:
 Divide you roll number by 3 and get the remainder, i.e., 0, 1,
or 2.
 Solve all questions from the assigned chapters that give the
same remainder as in your roll number when question
number is divided by 3.
 Part – II: To be assigned in next lecture.