Uncertainty in Measurement: Accuracy, Significant Figure, Notation

Uncertainty in Measurement in Chemistry: It is possible to know exact numbers when measuring whole counting numbers of identifiable objects such as eggs, bananas, tables, chairs, and so on. Defined amounts, on the other hand, are precise. For example, one minute contains exactly \(60\) seconds. However, the accuracy of many scientific measurements using specific measuring devices is unknown. The accuracy of any such measurement is determined by \(\left( {\rm{i}} \right)\) the accuracy of the measuring device used, and \(\left( {\rm{ii}} \right)\) the skill of its operator.

The study of chemicals generally requires experimental data as well as theoretical calculations. In chemistry, there are many meaningful ways to use numbers without any complexity and to present data with as much certainty as possible. We will understand them in detail in this article.

What is Uncertanity in Measurement?

Let’s say we want to measure the length of a room with tape or by pacing it. We are likely to have different counts each time if we pace it off, or we will have a fraction of a pace left over. As a result, the measurement’s result isn’t entirely correct. The method of measurement has an impact on accuracy. The measure is more exact when using a tape than when pacing off a length. Repeating a measurement is one way to assess its quality. Take the average figure because each measurement is likely to yield a somewhat different result.

If the different measurements of the average value are close to the correct value, the measure is accurate (the individual measurements may not be comparable to each other).

If the different measurement values are near to one another and hence near to their mean value, the estimation is said to be precise. (The average value of different measurements may not be close to the correct value). The precision depends upon the measuring device as well as the skill of the operator.

Uncertainty in Measurement Formula

The measuring instrument in uncertainty is evaluated as \(+\) or \(- (±)\) half the smallest scale division. For a thermometer with a mark at every \({\rm{1}}.{\rm{0}}{\,^{\rm{o}}}{\rm{C}},\) the uncertainty is \(\pm {\rm{0}}.{\rm{5}}{\,^{\rm{o}}}{\rm{C}}.\) For example, if a scholar peruses a value from this thermometer as \({\rm{42}}.{\rm{0}}{\,^{\rm{o}}}{\rm{C}},\) they could give the result as \({\rm{42}}.{\rm{0}}{\,^{\rm{o}}}{\rm{C}}\, \pm \,{\rm{0}}.{\rm{5}}{\,^{\rm{o}}}{\rm{C}}.\)

Uncertainty in Measurement

All scientific measurements involve a certain degree of error or uncertainty. Precision and accuracy are two significant factors connected with these. Precision means how closely individual measurements agree with each other, and accuracy means how the experimental measurement agrees with the true or correct values. It may be noted that the errors which arise depend upon two factors.

1. Skill and Accuracy of the Worker: It is an important factor. Let us suppose that three different workers measure the length of a wire separately with the help of the same meter rod with the least count of \({\rm{0}}{\rm{.1}}\,{\rm{cm}}{\rm{.}}\) Their observations are as follows:

\({\rm{A}}\) reads the length of the wire as \({\rm{8}}{\rm{.1}}\,{\rm{cm}}{\rm{.}}\)

\({\rm{B}}\) reads the length of the wire as \({\rm{8}}{\rm{.2}}\,{\rm{cm}}{\rm{.}}\)

\({\rm{C}}\) reads the length of the wire as \({\rm{8}}{\rm{.3}}\,{\rm{cm}}{\rm{.}}\)

If the correct length of the wire is \({\rm{8}}{\rm{.2}}\,{\rm{cm}}{\rm{,}}\) person \({\rm{B}}\) has reported the result accurately, and person \({\rm{A}}\) and \({\rm{C}}\) have made certain errors.

2. Limitation of the Measuring Instrument: Now, let us suppose that the correct length of the wire is \({\rm{8}}{\rm{.24}}\,{\rm{cm}}\) and not \({\rm{8}}{\rm{.2}}\,{\rm{cm}}{\rm{,}}\) as reported above. Since the least count of the metering rod is only \({\rm{0}}{\rm{.1}}\,{\rm{cm}}{\rm{,}}\) it cannot give correct reading up to a second decimal place. Therefore, the measurement done by a meter rod will introduce an error. However, if we use a Vernier caliper with a least count of \({\rm{0}}{\rm{.01}}\,{\rm{cm}}{\rm{,}}\) then the length of the wire can be correctly reported to second place of decimal.

Thus, we conclude that the skill of the worker and the precision of the measuring scale are the two important factors upon which the accuracy of a particular measurement depends.

Significant Figures

We have noticed that every measurement done in the lab involves identical mistakes or uncertainty based upon the limitation of the measuring. To report scientific data, the term significant figures have been used. According to this, all digits written in a given data are certain to expect the last one, which is uncertain. For example, let us assume that the reading as reported by a measuring scale is \(11.64.\) It has four digits in all. Out of them, \(1, 1,\) and \(6\) are certain digits, while the last digit \(4\) is uncertain. Thus, the number possibly reported as follows:

The significant figures in some numbers are all certain digits plus one irresolute digit.

It may be prominent all digits describe in a number are remarkable. However, only the final digit is uncertain, while the rest are specific. Thus, the number \(11.64\) has all four digits as significant figures. Out of them, \(1,1\) and \(6\) are certain, while \(4\) have some uncertainty about it.

Rules for Reporting Significant Figures in a Number

The number of significant figures in any measured quantity is reported with the help of certain rules. These are discussed below:

Rule 1: All non-zero digits in a number are significant. For example,
\(54.3\) has three significant figures
\(5.232\) has four significant figures
\(11.164\) has \(5\) significant figures.

Rule 2: The zeros between two non-zero digits are always significant. For example,
\(7.01\) has three significant figures 
\(8.001\) has four significant figures.

Rule 3: The zeros written to the left of the first non-zero digit in a number are not significant. They indicate the position of the decimal point. For example,
\(0.523\) has three significant figures
\(0.014\) has two significant figures

Rule 4: All zeros placed to the right of a decimal point in a number are significant. They signify the accuracy of the measuring scale. For example, the length of a wire as measured with the help of meter rod, Vernier caliper, and screw gauge will have a different number of significant figures as given ahead.

1. The reading may be \({\rm{11}}{\rm{.0}}\,{\rm{cm}}\) on the meter scale with the least count of \({\rm{0}}{\rm{.1}}\,{\rm{cm}}{\rm{.}}\) The number of significant figures is three.
2. The reading maybe \({\rm{11}}{\rm{.00}}\,{\rm{cm}}\) on the Vernier caliper scale with the least count of \( {\rm{0}}{\rm{.01}}\,{\rm{cm}}{\rm{.}}\) The number of significant figures is \(4.\)
3. The reading maybe \({\rm{11}}{\rm{.000}}\,{\rm{cm}}\) on the screw gauge scale with the least count of \({\rm{0}}{\rm{.001}}\,{\rm{cm}}{\rm{.}}\) The number of significant figures is \(5.\)

Interestingly, when any number ends in zero, which is not to the right of the decimal point, then these zeros may or may not be significant. For example, a number \(18500\) may have three, four, or five significant figures. This ambiguity may be removed by expressing the value in terms of Exponential notation, also called scientific notations, which are being discussed.

Exponential Notations or Scientific Notations

In this notation, every number is written as \({\rm{N \times 1}}{{\rm{0}}^{\rm{n}}}{\rm{.}}\)

Here, \({\rm{N = a}}\) number with a single non-zero digit to the left of the decimal point

\({\rm{n = }}\) exponent of \(10.\) It may be a positive, negative integer, or zero.

To determine the value while writing a number as exponential notation, one should count the number of places; the decimal has to be moved. The exponent is positive if the decimal is moved to the left and negative when moved to the right. For example, 

\(40400.0 = 4.04 \times {10^4}\) (Decimal is moved four places to the left)

\(0.0000504 = 5.04 \times {10^{ – 5}}\) (Decimal is moved five places to the right)

The exponential notations are also quite useful in writing very small as well as huge numbers. For example,

1. A number \(0.000064\) is expressed as \(6.4 \times {10^{ – 5}}\) It has two significant figures.
2. The value of the Avogadro’s number according to the scientific notation is \(6.022 \times {10^{23}}.\) The number of significant figures is four.
3. The value of Planck’s constant is \(6.626 \times {10^{ – 34}}\) Joule second. The number of significant figures is \(4.\)

How Do You Calculate the Uncertainty of a Measurement?

We have studied that scientific measurements differ in their precision and accuracy depending upon the least count of the measuring instrument or scale. In most cases, these results have to be added, subtracted, multiplied, or divided to get the final result. It may be noted that the final computed result cannot be more precise or accurate than the least precise number involved in a particular calculation. The following rules obtain the number of significant figures in such mathematical calculations.

Rule 1: In addition, or subtraction of the numbers having different precisions, the final result should be reported to the same number of decimal places as having the least number of decimal places.

Addition of Numbers

Let us carry out the three numbers \(3.52, 2.3,\) and \(6.24\) having different precisions or different numbers of decimal places. The number having the least decimal places \(2.3.\) This means that the final result of addition should be reported only up to one place of decimal.

The final result has \(2\) decimal places, but the answer has to be reported only up to one decimal place. It means that digit \(6\) has to be deleted in the final result. The final result \(12.1\) has been calculated by applying the principle of rounding off the non-significant digits discussed.

Rounding off Non-significant Figures 

Rounding off a number means that the digits which are not significant have to be dropped. This exercise is done only to retain the significant figures in a number. Following rules are followed for rounding off a number.

1. If the digit to be dropped is more than five, add one to the preceding significant digit or figure and drop all other digits.
   For example, if the number \(12.06,\) if the digit to be dropped is \(6,\) the preceding digit is \(0\) is increased by one, and the final result is \(12.1.\)
2. If the digit to be dropped is less than \(5,\) it is deleted without bringing any change in the preceding significant digit or figure. For example, if a number \(24.123\) is to be reported up to four significant figures, then the last digit three is dropped, and the final answer is \(24.12.\)
3. If the digit to be dropped is five, then the preceding significant digit or figure in the number may be left unchanged if it is even and can be increased by one if it is odd.
   For example, \(1.6145,\) if reported up to \(4\) significant figures, will become \(1.614\) after rounding off. Similarly, \(1.6175\) will become \(1.618.\)
4. If more than one digit is to be dropped from a particular number during rounding off, they are dropped one at a time by following the above rules.
   For example, let us suppose that a number \(3.12456\) is to be reported up to three decimal places. It means that digits \(5\) and \(6\) are to be dropped. The correct answer, in this case, is obtained in two steps as follows:
   In \({1^{{\rm{st}}}}\) step: \(3.1246\) (the last digit to be dropped is greater than \(5\))
   Second step: \(3.125\) (the last digit to be dropped is greater than \(5\))

Note: If the problem involves more than one step, the rounding off must be done only in the final answer. The intermediate steps of calculations remained unchanged.

Some more problems relating to the addition of numbers may be solved as follows:

Subtraction of Numbers

The subtraction of numbers is done in the same way as the addition. For example,

The final result has four decimal places. But it has to be reported only up to two decimal places. Therefore, digits \(3\) and \(0\) are deleted, and the correct answer is \(11.36.\)

A few more problems relating to the subtraction of numbers as follows

Rule2: In the multiplication or division, the final result should be reported up to the same number of significant figures as present in the least precise number. Let us take up the examples of multiplication and division of numbers separately.

Multiplication of Numbers

If we multiply \(2.2120\) (having five significant figures) with \(0.011\)(have two significant figures), the value becomes \(0.0243322.\)

\(2.2120×0.011\)

\(=0.024332\) 

But according to the rule, the final answer has to be reported up to two significant figures. Therefore, the digits \(3, 3,\) and \(2\) have to be dropped by rounding off. The correct answer is \(=0.024.\)

Division of Numbers

If we divide \(4.2211\) (having \(5\) significant figures) by \(3.76\) (having three significant figures) the result comes out to be \(1.12263.\)

\(4.2211 \div 3.76\)

\(=1.12263\)

But according to the rule, the final answer has to be reported up to \(3\) significant figures only. Therefore, the digits \(2, 6, 3\) have to be dropped by rounding off. The correct answer is \(1.12.\)

Accuracy and Precision

Each experimental measurement is somewhat different from each other and the errors and uncertainties found in them depend on the efficiency of the measuring instrument and the person making the measurement. Accuracy denotes the closest value to the actual (true) value, that is, it shows the difference between the average experimental value and the actual value. Whereas precision refers to the closeness of the values ​​obtained by measurement.

Ideally, all measurements should be accurate and accurate. “A true value is ordinarily accurate, while it is not necessary that the exact value be accurate.”

Causes of Uncertainty in Measurements 

1. Lack of information (or knowledge) and data on the phenomena, systems, and events to be analyzed. 
2. “Abundance” of information. 
3. Conflicting nature of pieces of information/data. 
4. Measurement errors. 
5. Linguistic ambiguity. 
6. Subjectivity of opinions.

Summary

By quantifying how much uncertainty is related to results, the scientist can commune their findings more accurately. Scientific uncertainty normally means that there is a range of possible values within which the true value of the measurement lies. In this article, we learned about precision, accuracy, scientific notation, significant figures, rules for determining the number of significant figures in answers involving calculations.

FAQs

Q.1. Why do we calculate uncertainty in measurements?
Ans: If the uncertainty is too large, it is impossible to say whether the difference between the two numbers is real or just due to sloppy measurements. That’s why estimating uncertainty is so important! If the ranges of two measured values don’t overlap, the measurements are discrepant (the two numbers don’t agree). 

Q.2. How do you find the uncertainty of a single measurement?
Ans: The minor divisions on the scale are \(1-\)pound marks, so the least count of the instrument is \(1\) pound. In general, the uncertainty in a single measurement from a single device is half the least count of the instrument.

Q.3. What is the degree of uncertainty?
Ans: All measurements have a degree of uncertainty regardless of precision and accuracy. It is caused by two factors: the measurement instrument’s limitation (systematic error) and the experimenter’s skill in making the measurements (random error).  

Q.4. What does percentage uncertainty mean?
Ans: The per cent uncertainty is familiar. It is computed as the per cent uncertainty can be interpreted as describing the uncertainty that would result if the measured value had been \({\rm{100}}\,{\rm{units}}{\rm{.}}\) A similar quantity is a relative uncertainty (or fractional uncertainty).

Q.5. What is standard uncertainty?
Ans: The standard uncertainty \({\rm{u}}\left( {\rm{y}} \right)\) of a measurement result \({\rm{y}}\) is the estimated standard deviation of \({{\rm{y}}{\rm{.}}}\)