Significant Figures

The record of any measurement is done by using a number that

includes all the digits those are known reliably plus one digit that is

uncertain. The reliable digits plus the one uncertain digit are known as

significant figures or significant digits of the measurement.

Significant figures indicate the precision of any measurement and thus

it depends on the least count of the measuring device.

For example, if a distance is measured with scale having least count

in millimeter (mm), a length of 20.6 centimeter (cm) has three

significant figures 2, 0 and 6. The digit 2, 0 are certain while the digit 6

is uncertain. Thus, 20.6 cm means the length is 20.60 cm ± 0.5 mm

(0.05 cm) i.e., it lies between 20.55 cm and 20.65 cm.

Determination of Significant Figures

Rules and conventions :

• All non-zero digits in a number are significant.

Example: Numbers 0.0000216, 0.0216, 21.6 and 216 have the same

number of significant figures namely three (2, 1, 6).

• All zeros between two non-zero digits are significant, no matter

where the decimal point is, if at all.

Example: In the numbers 0.0000206, 0.0206, 20.6 and 206, the zero

lying between the digits 2 and 6 is only significant.

• If the number is less than 1, the zeroes on the right of decimal point

but to the left of the first non-zero digit are not significant.

Example: In 0.0000206, the four zeros after decimal and before the

digit 2 has no significance. Similarly, in 0.0206, the zero after decimal

and before the digit 2 has no significance. So the number of

significant figures of these numbers are three (2, 0 and 6).

• The terminal or trailing zeros in a number without a decimal point

are significant depending on accuracy of measurement.

Example : In 2360 m, the terminal zero has no significance, if the

accuracy of measurement is 10 m then the number of significant

figures of this number is three (2, 3 and 6). If the accuracy of

measurement is 1 m, the terminal zero is significant figures of this

same number will be four (i.e. 2, 3, 6 & 0).

• The digit 0 conventionally put on the left of a decimal for a number

less than 1 is never significant. However, the zeros at the end of such

number are significant in a measurement.

Example: The number 0.120 has three significant numbers. The zero

before the decimal point is not significant.

• The terminal or trailing zeros in a number with a decimal point are

significant.

Example: In 23.60 m, the terminal zero has significance, so the

number of significant figures in this number is four (2, 3, 6 and 0).

NOTE

There can be some confusion regarding the trailing zeros. Suppose, a

length is recorded as 4.700 m. It is evident that the zeros here are

meant to convey the precision of measurement and are, therefore,

significant. Now, suppose the unit of the number is changed, i.e,

4.700 m = 470.0 cm = 4700 mm

Since the last number has trailing zeros in a number with no decimal,

it can be concluded erroneously that the number has two significant

figures, while in fact it has four significant figures.

To overcome such ambiguities in determining the number of

significant figures, measurements are to be reported in scientific

notation. In this notation, Every number is expressed as a x 10b

,

where a is a number between 1 and 10, and b is any positive or

negative exponent of 10. It is often customary to write the decimal

after the first digit. The significant number of the base represents the

significant number of the measurement. Thus,

4.700 m = 4.700 x 102

cm = 4.700 x 103

mm = 4.700 x 10-3

km

Each number in this case has four significant figures. Thus, “a choice

of change” of different units does not change the number of

significant digits or figures in a measurement.

Rounding off Numbers

The dropping of the excess digits in any number is being carried out

by rounding off numbers to the appropriate significant figures. The

rules for rounding off numbers are as follows:

• The preceding digit is raised by 1 if the last insignificant digit to be

dropped is more than 5 and left unchanged if the latter is less than 5.

For example (i) 4.796 becomes 4.80 (ii) 8.512 becomes 8.51.

• If the digit to be dropped is 5, the preceding digit should be nearest

even number.

For example

(i) 4.745 (after rounding off to three significant digits) becomes 4.74

(ii) 4.735 (after rounding off to three significant digits) becomes 4.74.

Rounding off and Significant Figures in Arithmetic Operations

The arithmetic calculation involving observed/measured quantities

should be such that the resulted quantity cannot be more precise

than the original observed/measured values. Thus, the final result

should not have more significant figures than the original data from

which it has been derived.

In order to achieve such result, following rules for arithmetic

operations are required to be followed:

• In addition or subtraction, the final result should be rounded off in

such a way as to retain as many decimal places as are there in the

original number with the least decimal places.

In multiplication or division, the final result should be rounded off in

such a way as to retain as many significant figures as there are in

the original number with the least significant figures.

For example

(i) 6.7153 x 4.67 = 31.360451 = 31.4 (Rounded off to three significant

figures)

(ii) (86.85 x 104

)2

= 7542.9225 x 108

= 754.3 x 109

(Rounded off to four

significant figures)

(iii) = 186.499 = 186 (Rounded off to three significant figures)

Error in Measurement

In case of repeated observation of any parameter, usually it has

been found to have variations, however small, in the resulting

measurement. Moreover, there is nothing definite in the amount of

variation i.e., variations are random in nature. Thus, a measurement

usually differs from its true value. The difference between a

measured and its true value is called the measurement error. Thus,

if x is a given measurement and x t is the true value, then the error e

is given by

e = x - x t

Error = measured value – true value.

If an estimated value of xt is usually known and is denoted by x1

.

Then, an estimate of error for a measurement value x of the

parameter is obtained as

e1

= x - x1

However, correction is the term more popularly being used to define

the magnitude of error but opposite in sign. Thus, rearranging the

error relation,

correction = (-e1

) = x1

- x

Or, correction = (estimated / designated) true value - measured

value.

Sources of Errors in Measurement

Depending on sources of origin, errors in measurements fall into

three classes. They are

Natural Errors

Instrumental Errors

Personal Errors

Natural Errors

These are caused due to variations in nature i.e., variations in wind,

temperature, humidity, refraction, gravity and magnetic field of the

earth.

Instrumental Errors

These result from imperfection in the construction or adjustment of

surveying instruments, and movement of their individual parts.

Personal Errors

These arise from limitations of the human senses of sight, touch and

hearing

Errors are traditionally been classified into three types.

Gross Error

Systematic Error

Random Error

Gross Error

Gross errors, also known as blunders or mistakes, are results from

Carelessness on the part of observer in taking or recording

reading;

Faults in equipments;

Adoption of wrong technique.

Misinterpretation.

The blunders or mistakes result into large errors and thus can easily

be detected by comparing with other types of errors (generally small

in value). The maximum permissible error in an observation is ± 3.29

s (where s is the standard deviation of sample distribution) and is

used to separate mistakes or blunders from the random errors. If any

error deviates from the mean by more than the maximum

permissible error, it is considered as a gross error and the

measurement is rejected.

After mistakes have been detected and eliminated from the

measurements, the remaining errors are usually classified either as

systematic or random error depending on the characteristics of

errors.

Systematic Error

Systematic errors occur according to a system. These errors follow

a definite pattern. Thus, if an experiment is repeated, under the

same conditions, same pattern of systematic errors reoccur. These

errors are dependent on the observer, the instrument used, and on

the physical environment of the experiment. Any change in one or

more of the elements of the system will cause a change in the

character of the systematic error. Depending on the value and sign

of errors in successive observation, systematic errors are divided

into two types.

Cumulative Error

Compensating Error

Systematic errors are dealt with mathematically using functional

relationships or models.

Cumulative Error

If the sign in error remains the same throughout the measuring

process, the error will go on accumulating all throughout the

process. This type of systematic error is termed as cumulative

error.

Compensating Error

If the sign of the systematic error changes, the resulting

systematic error is termed as compensating error.

Random Error

After mistakes are eliminated and systematic errors are corrected,

a survey measurement is associated with random error only. This

error is small and is equally liable to be plus or minus thus partly

compensating in nature. Random errors are unpredictable and they

cannot be evaluated or quantified exactly.

Random errors are determined through statistical analysis based

on following assumptions :

Small variations from the mean value occur more frequently

than large ones.

Positive and negative variations of the same size are about

equal in frequency, rendering their distribution symmetrical

about a mean value.

Very large variations seldom occur.

Thus, to eliminate random error in a measurement, observations

are repeated for number of times. The mean (average) of

observations is considered to be the true (or estimated) value of

the measurement. Normal or Gaussian distribution typifies the

distribution of samples of any measurement.

Propagation of Error

Measurements are used for calculation of different parameters.

As the measurements are fraught with errors, it is important to

know how these errors combine in various mathematical

operations.

Reliability of Measurement

The reliability of a measurement designates its worth as a

measurement value. Since, the true magnitude of a measured

quantity is never determinable; it’s worth is ascertained by making

use of reliability indicators. Standard deviation works as an indicator

of reliability of a set of observation and uncertainty associated with

an observation in the set provides an indicator of its reliability.

Uncertainty

An indicator to define the reliability, of any observation in a set of

repeated observations. Figure 7.1 represents the uncertainty involved

in an observation. For example, if an observation falls

within ( ± ) [where represents the mean and the standard

deviation of the set of the observations] then the observation lies

within 68.3% errors of the set of observation. Thus, 68.3% is

designated as the uncertainty of the sample.

To ascertain the reliability of an observation, it is required to find

within what percentage of error a particular observation lies. This

defines the uncertainty of the observation and consequently its

reliability. Lower the percentage of error within which a particular

observation lies, lower is its uncertainty and thus greater is its

reliability.

For example, an observation is 50% uncertain if it lies within -

0.6745 and + 0.6745 ; 90% uncertain if lies in -1.645 and +

1.645 ; 95% uncertain if it is within - 1.960 and + 1.960 and

99% uncertain for observation lying within - 2.576 and +

2.576 , (Figure 7.1).

Example : A distance of 10 m is measured having 95 percent error as

0.98 cm. Determine the uncertainly associated with an observation

10.05 m for this distance.

Solution : 95 percent equivalent to 1.96 . Hence, 0.98 = 1.96 or, =

5 cm. Given 10.05 = 10 m + 0.05 m = 10m (5cm (i.e. ) uncertainly

involved in 10.05 m is 50 %.

Quality of a measurement having a set of repeated observations is

being tested by using indicators like

Accuracy

Precision

Relative Precision

Accuracy

The accuracy of a set of repeated observations is being defined as

amount of closeness of their mean to the population or distribution

mean, i.e., closeness of the mean of observations to the true value.

Degree Of Accuracy

The degree of accuracy indicates the accuracy attained in the

measurements. It is usually expressed as the ratio of the error and

the associated measured value. For example, a degree of accuracy of

1 in 10,000 indicates that there is an error of 1 unit in 10,000 units of

measured / observed value.

Order Of Accuracy

The minimum degree of accuracy required for a particular survey and

the range of the allowed degree of accuracy is known as order of

accuracy. The most accurate work is designated as the work of the

first order accuracy. The work of the second order accuracy is less

accurate than that of the first order accuracy. Likewise, the work of

the third order accuracy and the fourth order accuracy are less

accurate than that of the second order and the third order accuracy,

respectively.

For example, the following standard of accuracy may be expected for

the horizontal distances for a particular survey.

First order accuracy 1/25,000

Second order accuracy 1/10,000

Third order accuracy 1/5,000

Precision

Precision pertains to the degree of closeness of observations among

each other in a set of repeated observations of a measurement. Thus,

if a set of observations for the same parameter are clustered together,

i.e.,observations have small deviations from their sample mean, then

the observations are said to have been obtained with high precision.

Relative Precision

Relative precision is defined as a ratio of the precision of a given

measurement and the value of the measurement itself. Thus, if d is a

measured distance, and sd is the standard deviation of the

measurement, then the relative precision is sd / d. It is expressed as

percentage or a fractional ratio such as 1/ 500 or by parts per million

(ppm)

Ex7-2 Observations for the distance between two points are found to

be as follows

Set I : 165.485 ± 0.005 m; Set II : 165.465 ± 0.010 m.

(i) State which of these sets of observation is more reliable and why?

(ii) State whether the sets of observation are significantly different or

not. Explain

(iii) Find the weighted mean of observation.

Solution :

(i) The standard deviation of Set I is less than that of Set II. Thus, Set I

observation is more reliable than Set II.

(ii) Difference between observations in Set I. & Set II is

165.485 – 165.465 = 0.020m

Thus the difference between the sets of observation in less

than 2 diff (i.e., 0.020 < 0.02236). Therefore sets I & II can be regarded

as value measurements of the same quantity and inspection indicates

that they can be combined to find the (weights) mean of observations.

(iii) The weighted mean of observations are :

Therefore, The weighted mean value of the distance is 165.481m