Lesson Progress:

Essential Skills for Physical Science

Lesson Content

Inquire: Recipe for Success in Science!

Overview

The information discussed in this lesson is to help you practice the essential skills that are necessary for success in physical science. One of them is producing higher quality records. These records rely on accurate calculations, scientific notation, descriptive statistics, significant figures, and so forth. These calculations go on to affect how others are able to view your research through graphs and models, to understand how you make measurements, and to interpret reported values properly. To achieve these goals, your related knowledge and underlying calculations must be sound.

Big Question

Can you use a real-life example to illustrate the difference between accuracy and precision in measurement?

Read: Essential Skills for Physical Science

Overview

This section explains the two major measurements used in science and the related key terms such as accuracy, precision, significant figures, scientific notations, mean, range, and so forth. Then, explore the roles of models and descriptive statistics in science.

Measurements in Science

Science is based on observation and experiment — that is, on measurement. Every measurement provides three kinds of information: the size or magnitude of the measurement (a number), a standard of comparison for the measurement (a unit), and an indication of the uncertainty of the measurement.

There are two major systems of measurement used in the world: the International System of Units or SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard, and SI units are also the standard agreed upon by scientists and mathematicians.

In science, therefore, we usually report results in SI units. This system is easier to use because all units are based on the number 10. Smaller units are multiplied or divided by 10 to get bigger or smaller units.

Accuracy and Precision

Scientists typically make repeated measurements to ensure the accuracy of their findings and to know the precision of their results. Accuracy is how close a measurement is to the correct data value for that measurement. Precision refers to how close the agreement is between repeated independent measurements (which are repeated under the same conditions). Accurate values agree with a true value, while precise values agree with each other.

One example for understanding accuracy and precision is shooting baskets. One situation is that if you shoot with no accuracy and precision, the ball will not hit the basket and will go anywhere at any given time. If you shoot with accuracy, your aim will always take the ball close to or into the basket. If you shoot with precision, your aim will always take the ball to the same location, which may or may not be close to the basket. If you are a good player, you will be both accurate and precise by shooting the ball the same way and making it into the basket each time.

Keeping Good Records

Science projects may take weeks, months, and sometimes even years, so keeping good records such as a special notebook or journal to make daily logs is important in order to document the process. These can serve as a map of where we have been and what we have learned.

Here are some tips for doing so:

  • Use a laboratory notebook, electronic device, or both. Use ink or a word processing program that backs up automatically so there is no lapse in records.
  • Record all of your materials and procedures.
  • Record every measurement and observation with precision.
  • Use drawings.
  • Date all entries, drawings, calculations, and conclusions.
  • Keep copies.
  • Document EVERYTHING that happens over the course of the research.

Significant Figures and Scientific Notation

When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a pen, you may measure it to be 8.1 cm. In this case, you could not express this value as 8.13 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. Be aware that the last digit has been estimated in some way by the person performing the measurement. All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits.

In order to determine the number of significant figures, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 8.1 cm has two significant figures. Note that the “zeros” may be a measured value when counting significant figures: e.g., there are three significant figures in 55.0 g.

Quantities in science may be very large or very small. Scientific notation is a way of writing very large or very small numbers using exponents. Numbers are written in an a×10b format where a is the decimal number and b is the exponent or power: e.g., 45000 is written as 4.5×104, and 0.00045 is written as 4.5 x 10-4.

Descriptive Statistics

Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (e.g., finding means and ranges). A mean is the average of a set of data found by adding all of the measurements together and then dividing by the total number of data points. For example, let’s say you take four exams in your math class and obtain scores of 88, 86, 94, and 90. You would calculate your mean by adding the four scores and dividing by four (your mean score would be 89.5). The range is how far the data is spread and is found by subtracting the smallest data point from the largest.

Graphs are visual representations of the data. Each type of graph has a specific use and purpose. Three common graphs are bar graphs, circle graphs, and line graphs. Bar graphs are used to compare things between different groups. Circle graphs are useful for showing percentages as a whole, and line graphs are used to show change over time.

Models in Science

A model is a representation of something that is often too difficult (or impossible) to display directly, which helps scientists analyze the data they have already collected. While a model is justified with experimental proof, it is only accurate under certain/limited situations.

Scientists use models for a variety of purposes. For example, models are developed by scientists to predict things (e.g., scientists use global climate models to predict the future of climate change).

Sometimes, things are too small to observe (e.g., atoms and molecules), therefore we have to create a model to visualize them mentally in order to understand their behaviors. Take the kinetic theory of gases for example; it’s a model in which a gas is viewed as being composed of atoms and molecules.

Models can also be used when field experiments are too expensive or dangerous, such as building fire simulation models used to predict how fire might develop in a building.

Overall, models have always been important in science and continue to be used to test hypotheses and predict information, and they are central to the process of knowledge building in science and to demonstrate how science knowledge is tentative.

Reflect: How to Improve Accuracy

Poll

After doing two weeks of experiments, you obtained some data to test your hypothesis. However, your adviser finds that the data has a lower accuracy than desired and needs to take steps to resolve this. Which method is best to improve your accuracy?

Expand: Scientific Notation and Significant Figures in Calculations

Discover

There are brief introductions about scientific notation and significant figures in the Read section. Here we will explore their calculations in depth, which are very important to ensure you make a measurement properly or to help you analyze a reported value to determine what is significant and what is not. We will work through these rules with a few examples in the section below.

Scientific Notation in Calculations

As mentioned before, scientific notation is used to express very large and small numbers as a product of two numbers. The first number is called a digit term, which is usually less than one and not greater than 10. The second number is called an exponential term, which is written as 10 with an exponent.

1. For addition:

Strategy: Convert all numbers to the same power of 10, add the digit terms of the numbers, and then convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

Example: Add 7.00 X 10-5 and 4.00 X 10-3

Solution:

4.00 X 10-3 = 400 X 10-5
(7.00 X 10-5.) + (400 X 10-5) = 407 X 10-5 = 4.07 X 10-3

OR

7.00 X 10-5 = 0.07 X 10-3
(0.07 X 10-3.) + ( 4.00 X 10-3) = 4.07 X 10-3

Both solutions have the same results.

2. For subtraction:

Strategy: Convert all numbers to the same power of 10, take the difference of the digit terms, and then convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

Example: Subtract 5.0 X 10-8 from 6.0 X 10-6

Solution:

5.0 X 10-8 = 0.05 X 10-6
(6.0 X 10-6) – (0.05 X 10-6) = 5.95 X 10-6

3. For multiplication:

Strategy: Multiply the digit terms in the usual way and add the exponents of the exponential terms.

Example: Multiply 3.5 X 103 by 4.0 X 10-6

Solution:

(3.5 X 103) X (4.0 X 10-6) = (3.5 X 4.0) X 10(3)+(-6) = 14.0 X 10-3

4. For division:

Strategy: Divide the digit term of the numerator by the digit term of the denominator and subtract the exponents of the exponential terms.

Example: Divide 4.9 X 107 by 7.0 X 10-5

Solution:

$frac{4.9 X 10^{7}}{7.0 X 10^{-5}}$ = $frac{4.9}{7.0}$X 107-(-5) = 0.70 X 1012 = 7.0 X 1011

5. For squaring:

Strategy: Square the digit term in the usual way and multiply the exponent of the exponential term by 2.

Example: Square the number 3.0 X 10-3

Solution:

(3.0 X 10-3)2 = (3.0)2 X 102 x (-3) = 9.0 X 10-6

6. For cubing:

Strategy: Cube the digit term in the usual way and multiply the exponent of the exponential term by 3.

Example: Cube the number 4 X 103
Solution:

(4 X 103)3 = 43 X 103×3 = 4 X 4 X 4 X 109 =6.4 X 1010

Significant Figures in Calculations

1. For addition and subtraction:

Strategy: The sum or difference should contain as many digits to the right of the decimal as that in the least certain of the numbers used in the computation (indicated by underscoring in the following example):

Example: Add 3.783 m and 1.0045 m.

Solution:

3.783 m ← Thousandths place
+ 1.0045 m ← Ten thousandths place: least precise
4.787 m ← Round to thousandths

Answer is 4.787 g

2. For multiplication and division:

Strategy: The product or quotient should contain no more digits than that in the factor containing the least number of significant figures.

Example: Multiply 0.532 by 7.2

Solution:

0.532 X 7.2 = 3.8 (round to two significant figures)
Three significant figures X Two significant figures → Two significant figures answer

Check Your Knowledge

Use the quiz below to check your understanding of this lesson’s content. You can take this quiz as many times as you like. Once you are finished taking the quiz, click on the “View questions” button to review the correct answers.

Lesson Resources

Lesson Toolbox

Additional Resources and Readings

SI Units

A comprehensive source of information on SI units at the National Institute of Standards and Technology (NIST) Reference

Using a Triple Beam Balance

A video showing the correct way to use a metric ruler to measure length and a beam balance to measure mass

Accuracy and Precision

A video explaining the difference between accuracy and precision

Rounding to Significant Figures

A video with more information on significant figures and rounding

Exponents: Level 1

A video containing a review of exponents useful in evaluating scientific notation

Scientific Notation

A resource allowing you to review the basics of scientific notation

The Very, Very Simple Climate Model

A simple but interesting interactive model explaining how the climate works

Lesson Glossary

Terms

AJAX progress indicator
  • accuracy
    how closely a measurement aligns with a correct value
  • bar graphs
    used to compare things between different groups
  • circle graphs
    useful for showing percentages as a whole
  • descriptive statistics
    organizing and summarizing data
  • graphs
    visual representations of the data
  • line graphs
    used to show change over time
  • mean
    the same as the arithmetic average found by adding all of the values and dividing by the number of values
  • model
    a representation of something that is often too difficult (or impossible) to display directly
  • precision
    refers to how close the agreement is between repeated independent measurements (which are repeated under the same conditions)
  • range
    the difference between the maximum and the minimum
  • scientific notation
    a way of writing very large or very small numbers that uses exponents
  • SI units
    (International System of Units); standards fixed by international agreement; the most widely used system of measurement

License and Citations

Content License

Lesson Content:

Authored and curated by Ja’Corie Maxwell, Jinxiu Yuan for The TEL Library. CC BY NC SA 4.0

Adapted Content:

Title: College Physics – 1.3 Accuracy, Precision, and Significant Figures. Rice University, OpenStax CNX. CC BY 4.0

Title: Chemistry – 1.4 Measurements. Rice University, OpenStax CNX. CC BY 4.0

Title: College Physics – 1.2 Physical Quantities and Units. Rice University, OpenStax CNX. CC BY 4.0

Title: Chemistry – 1.5 Measurement Uncertainty, Accuracy, and Precision. Rice University, OpenStax CNX. CC BY 4.0

Title: Introductory Business Statistics – 1.1 Definition of Statistics, Probability, and Key Terms. Rice University, OpenStax CNX. CC BY 4.0

Title: College Physics – 1.1 Physics: An Introduction. Rice University, OpenStax CNX. CC BY 4.0

Title: Chemistry – 1.4 Measurements. Rice University, OpenStax CNX. CC BY 4.0

Media Sources

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Mathematics Calculator FormulageraltPixabayCC 0
Mathematics Pay CountgeraltPixabayCC 0
Atmospheric Model SchematicNOAAWikimedia CommonsPublic Domain
Measuring Tape Inch+CMVariousWikimedia CommonsPublic Domain