Measurement Uncertainty

Scientific Notation

Scientific notation is a more convenient way to write very large or very small numbers and follows the equation: a × 10b.

Learning Objectives

Recognize how to convert between general and scientific notation

Key Takeaways

Key Points

  • Scientific notation is expressed in the form [latex]a \times 10^b[/latex] (where “b” is an integer and “a” is any real number), such as [latex]6.02 \times 10^{23}[/latex].
  • Scientific notation allows orders of magnitude to be more easily compared.
  • E notation is another form of scientific notation, in which “E” replaces 10, such as 6.02 E 23. This number is the same as [latex]6.02 \times 10^{23}[/latex].
  • Basic operations are carried out in the same manner as with other exponential numbers.

Key Terms

  • Scientific notation: A way of writing numbers that are too big or too small to be conveniently written in standard form.
  • integer: An element of the infinite and numerable set {…,-3,-2,-1,0,1,2,3,…}.
  • Order of Magnitude: An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it.

Scientific Notation

Scientific notation is a more convenient way of writing very small or very large numbers. The general representation for scientific notation is [latex]a \times 10^b[/latex](where “b” is an integer and “a” is any real number). When writing in scientific notation, only include significant figures in the real number, “a.” Significant figures are covered in another section.

To express a number in scientific notation, you move the decimal place to the right if the number is less than zero or to the left if the number is greater than zero.For example, in 456000, the decimal is after the last zero, so to express this in scientific notation, you would need to move the decimal to in between the 4 and 5.

The decimal would move five places to the left to get 4.56 as our [latex]a[/latex] in [latex]a \times 10^b[/latex]. The number of times you move the decimal place becomes the integer “b.” In this case, the decimal moved five times. Therefore, our number in scientific notation would be: [latex]4.56 \times 10^5[/latex]. Keep in mind that zeroes are not included in “a” because they are not significant figures.

In order to go between scientific notation and decimals, the decimal point is moved the number of spaces indicated by the exponent. A negative exponent tells you to move the decimal point to the right, while a positive exponent tells you to move it to the left.

Scientific Notation: Introduction – YouTube: Learn to convert numbers into and out of scientific notation. Scientific notation is a way to express very big and very small numbers with exponents as a power of ten. It is also sometimes called exponential notation. This video includes an explanation and tutorial, as well as practice and example problems.

Examples of scientific notation:

  1. 0.0001 = 1 x 10-4
  2. .0256 = 2.56 x 10-2
  3. 4759000 = 4.759 x 106
  4. 5000 = 5 x 103

Another way of writing this expression, as seen on calculators and computer programs, is to use E to represent “times ten to the power of.” An example is shown here:

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Scientific notation for Avogadro’s number: Here is an example of scientific notation on a calculator. 6.02E23 means the same thing as 6.02 x 1023.

Scientific notation enables comparisons between orders of magnitude. As seen above, scientific notation uses base 10, and if a number is an order of magnitude greater than another, it is 10 times larger. For example, 4.759 x 106 is 3 orders of magnitude bigger than 5 x 103; it is 8 orders of magnitude bigger than 2.56 x 10-2.

Basic Math with Scientific Notation

Basic operations in scientific notation are carried out in the manner one would carry out exponential functions. Multiplication and division adds or subtracts exponents, respectively. Addition and subtraction require the exponents to be the same. A few examples are shown here:

Given two numbers in scientific notation,

[latex]x_{0}=a_{0}\times10^{b0}[/latex]

and

[latex]x_{1}=a_{1}\times10^{b1}[/latex]

Multiplication and division are performed using the rules for operation with exponential functions:

[latex]x_{0}x_{1}=a_{0}a_{1}\times10^{b0+b1}[/latex]

and

[latex]\frac{x_{0}}{x_{1}}=\frac{a_{0}}{a_{1}}\times10^{b0-b1}[/latex]

Some examples are:

[latex]5.67\times10^{-5}\times2.34\times10^{2}\approx13.3\times10^{-3}=1.33\times1-^{-2}[/latex]

and

[latex]\frac{2.34\times10^{2}}{5.67\times10^{-5}}\approx0.413\times10^{7}=4.13\times10^{6}[/latex]

Addition and subtraction require the numbers to be represented using the same exponential part, so that the significand can be simply added or subtracted: [latex]x_{1}=c\times10^{b0}[/latex]. Next, add or subtract the significands:

[latex]x_{0}\pm{x}_{1}=\left(a_{0}\pm{c}\right)\times10^{b0}[/latex]

An example:

[latex]2.34\times10^{-5}+5.67\times10^{-6}=2.34\times10^{-5}+0.567\times10^{-5}\approx2.91\times10^{-5}[/latex]

Basic operations: Basic operations, such as multiplication and addition, are done the same way as with exponentials.

Significant Figures

Significant figures are digits which contribute to the precision of a number.

Learning Objectives

Apply knowledge of significant figures to scientific calculations

Key Takeaways

Key Points

  • Significant figures are any non-zero digits or trapped zeros. They do not include leading or trailing zeros.
  • When going between decimal and scientific notation, maintain the same number of significant figures.
  • The final answer in a multiplication or division problem should contain the same number of significant figures as the original number with the fewest significant figures.
  • In addition and subtraction, the final answer should contain the same number of decimal places as the original number with the fewest number of decimal places.

Key Terms

  • Significant Figures: The digits that carry meaning in a number and contribute to its precision.
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Pi: This calculator shows eight significant figures of pi.

Significant figures of a number are digits which contribute to the precision of that number. Numbers that do not contribute any precision and should not be counted as a significant number are:

  • leading or trailing zeros (those are place holders)
  • digits that are introduced by calculations that give the number more precision than the original data allows.

Rules For Determining If a Number Is Significant or Not

  • All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4, and 5).
  • Zeros appearing between two non-zero digits (trapped zeros) are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1, and 2.
  • Leading zeros (zeros before non-zero numbers) are not significant. For example, 0.00052 has two significant figures: 5 and 2.
  • Trailing zeros (zeros after non-zero numbers) in a number without a decimal are generally not significant (see below for more details). For example, 400 has only one significant figure (4). The trailing zeros do not count as significant.
  • Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0, and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers. For example, if a measurement that is precise to four decimal places (0.0001) is given as 12.23, then the measurement might be understood as having only two decimal places of precision available. Stating the result as 12.2300 makes it clear that the measurement is precise to four decimal places (in this case, six significant figures).
  • The number 0 has one significant figure. Therefore, any zeros after the decimal point are also significant. Example: 0.00 has three significant figures.
  • Any numbers in scientific notation are considered significant. For example, 4.300 x 10-4 has 4 significant figures.

Conventions Addressing Significant Figures

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

  • A bar may be placed over the last significant figure, showing that any trailing zeros following this are insignificant. For example, 1300 with a bar placed over the first 0 would have three significant figures (with the bar indicating that the number is precise to the nearest ten).
  • The last significant figure of a number may be underlined; for example, “2000” has two significant figures.
  • A decimal point may be placed after the number. For example “100.” indicates specifically that three significant figures are meant.
  • In the combination of a number and a unit of measurement the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg, it is much clearer.

When converting from decimal form to scientific notation, always maintain the same number of significant figures. For example, 0.00012 has two significant figures, therefore the correct scientific notation for this number would be 1.2 x 10-4.

When multiplying and dividing numbers, the number of significant figures used is determined by the original number with the smallest amount of significant figures. When adding and subtracting, the final number should be rounded to the decimal point of the least precise number.

Examples:

1.423 x 4.2 = 6.0 since 1.423 has 4 significant figures and 4.2 only has two significant figures, the final answer must also have 2 significant figures.

234.67 – 43.5 = 191.2 since 43.5 has one decimal place and 234.67 has two decimal places, the final answer must have just one decimal place.

Another Way to Determine Sig Figs: The Pacific Rule & the Atlantic Rule

It can be challenging to remember all the rules about significant figures and whether each zero is significant or not significant. Here’s another way to determine significant figures (sig figs): the Pacific and Atlantic Rule.

If a number has a decimal Present, use the Pacific rule (note the double P’s). The Pacific Ocean is on the left side of the United States so start at the left side of the number. Start counting sig figs at the first non-zero number and continue to the end of the number. For example, since there is a decimal present in 0.000560 start from the left side of the number. Don’t start counting sig figs until the first non-zero number (5), then count all the way to the end of the number. Therefore, there are 3 sig figs in this number (5,6,0).

If a number has no decimal (the decimal is Absent) use the Atlantic rule (again, note the double A’s). Since the Atlantic Ocean is on the right side of the United States, start on the right side of the number and start counting sig figs at the first non-zero number. For example, since there is no decimal in 2900 start from the right side of the number and start counting sig figs at the first non zero number (9). So there are two sig figs in this number (2,9).

Significant Figures Made Easy! – YouTube: Don’t be confused by significant figures. It will just make sense with this video. The video introduces significant figures and discusses how to round for multiplication and division using significant figures.

Exact Numbers

Exact numbers are defined numbers or result from a count, unlike measured numbers.

Learning Objectives

Recognize exact and measured numbers

Key Takeaways

Key Points

  • An exact number has absolutely no uncertainty in it.
  • Exact numbers cannot be simplified and have an infinite number of significant figures.
  • Measured numbers have a limited number of significant figures.

Key Terms

  • Exact numbers: These numbers are either defined numbers or result of a count. They have an infinite number of significant figures.

Exact Numbers

Exact numbers are either defined numbers or the result of a count. For example, a dozen is defined as 12 objects, and a pound is defined as 16 ounces. An exact number can only be expressed in one way and cannot be simplified any further. Exact numbers have an infinite number of significant figures, but they often appear as integers.

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Exact numbers: There are exactly two chairs in this picture. The number of chairs is counted, not measured, so we are completely certain how many chairs there are.

Examples of exact numbers include:

  • Conversions within the American system (such as pounds to ounces, the number of feet in a mile, the number of inches in a foot, etc).
  • Conversions with the metric system (such as kilograms to grams, the number of meters in a kilometer, the number of centimeters in a meter).
  • The words per and percent mean exactly out of one or one hundred, respectively.
  • Counted numbers are exact: there are two chairs in the photograph. There are fifteen books on the shelf. Eighty-seven people attended the lecture.

Measured Numbers

In contrast, measured numbers always have a limited number of significant digits. A mass reported as 0.5 grams is implied to be known to the nearest tenth of a gram and not to the hundredth of a gram.

There is a degree of uncertainty any time you measure something. For example, the weight of a particular sample is 0.825 g, but it may actually be 0.828 g or 0.821 g because there is inherent uncertainty involved. On the other hand, because exact numbers are not measured, they have no uncertainty and an infinite numbers of significant figures.

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Measured numbers: Mass is an example of a measured number. When mass is reported as 0.5237 g, as shown on this scale, it is more precise than a mass reported as 0.5 g.

Examples of measured numbers:

  • The diameter of a coin, such as 10.2 mm.
  • The weight of an object, such as 8.887 grams.
  • The length of a pen, such as 12 cm.

Accuracy, Precision, and Error

Accuracy is how closely the measured value is to the true value, whereas precision expresses reproducibility.

Learning Objectives

Describe the difference between accuracy and precision, and identify sources of error in measurement

Key Takeaways

Key Points

  • Accuracy refers to how closely the measured value of a quantity corresponds to its “true” value.
  • Precision expresses the degree of reproducibility or agreement between repeated measurements.
  • The more measurements you make and the better the precision, the smaller the error will be.

Key Terms

  • systematic error: An inaccuracy caused by flaws in an instrument.
  • Accuracy: The degree of closeness between measurements of a quantity and that quantity’s actual (true) value.
  • Precision: Also called reproducibility or repeatability, it is the degree to which repeated measurements under unchanged conditions show the same results.

Accuracy and Precision

Accuracy is how close a measurement is to the correct value for that measurement. The precision of a measurement system is refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Measurements can be both accurate and precise, accurate but not precise, precise but not accurate, or neither.

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High accuracy, low precision: On this bullseye, the hits are all close to the center, but none are close to each other; this is an example of accuracy without precision.

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Low accuracy, high precision: On this bullseye, the hits are all close to each other, but not near the center of the bullseye; this is an example of precision without accuracy.

Precision is sometimes separated into:

  • Repeatability — The variation arising when all efforts are made to keep conditions constant by using the same instrument and operator, and repeating the measurements during a short time period.
  • Reproducibility — The variation arising using the same measurement process among different instruments and operators, and over longer time periods.

Accuracy and Precision – YouTube: This is an easy to understand introduction to accuracy and precision.

Error

All measurements are subject to error, which contributes to the uncertainty of the result. Errors can be classified as human error or technical error. Perhaps you are transferring a small volume from one tube to another and you don’t quite get the full amount into the second tube because you spilled it: this is human error.

Technical error can be broken down into two categories: random error and systematic error. Random error, as the name implies, occur periodically, with no recognizable pattern. Systematic error occurs when there is a problem with the instrument. For example, a scale could be improperly calibrated and read 0.5 g with nothing on it. All measurements would therefore be overestimated by 0.5 g. Unless you account for this in your measurement, your measurement will contain some error.

How do accuracy, precision, and error relate to each other?

The random error will be smaller with a more accurate instrument (measurements are made in finer increments) and with more repeatability or reproducibility (precision). Consider a common laboratory experiment in which you must determine the percentage of acid in a sample of vinegar by observing the volume of sodium hydroxide solution required to neutralize a given volume of the vinegar. You carry out the experiment and obtain a value. Just to be on the safe side, you repeat the procedure on another identical sample from the same bottle of vinegar. If you have actually done this in the laboratory, you will know it is highly unlikely that the second trial will yield the same result as the first. In fact, if you run a number of replicate (that is, identical in every way) trials, you will probably obtain scattered results.

As stated above, the more measurements that are taken, the closer we can get to knowing a quantity’s true value. With multiple measurements (replicates), we can judge the precision of the results, and then apply simple statistics to estimate how close the mean value would be to the true value if there was no systematic error in the system. The mean deviates from the “true value” less as the number of measurements increases.

Error and Percent Error – YouTube: How to calculate error and percent error.