Scientific Measurements
Christopher G. Hamaker, Illinois State University, Normal IL
© 2005, Prentice Hall
Scientific Investigations
• Science is the methodical exploration of nature
followed by a logical explanation of the
observations.
• Scientific investigation entails:
–
–
–
–
planning an investigation
carefully recording observations
gathering data
analyzing the results
The Scientific Method
• The scientific method is a systematic
investigation of nature and requires proposing an
explanation for the results of an experiment in the
form of a general principle.
• The initial, tentative proposal of a scientific
principle is called a hypothesis.
• After further investigation, the original hypothesis
my rejected, revised, or elevated to the status of a
scientific principle.
The Scientific Method
•
•
•
•
It is a series of steps (not always in this order)
Making observations => question
Formulating hypotheses => answering question
inferring, predicting.
Testing hypotheses. => experimenting,
communicating, collecting data, and measuring.
Formulating theories => Confirming hypotheses
that are supported by data. It includes
constructing models, and predicting.
Uncertainty in Measurements
• A measurement is a number with a unit attached.
• It is not possible to make exact measurements,
and all measurements have uncertainty.
• We will generally use metric system units, these
include.
– the meter, m, for length measurements
– the gram, g, for mass measurements
– the liter, L, for volume measurements
Length Measurements
• Lets measure the length of a candy cane.
• Ruler A has 1 cm divisions, so we can estimate the
length to ±0.1 cm. The length is 4.2±0.1 cm.
• Ruler B has 0.1 cm divisions, so we can estimate
the length to ±0.05 cm. The length is 4.25±0.05
cm.
Uncertainty in Length
• Ruler A: 4.2 ±0.1 cm; Ruler B: 4.25 ±0.05 cm.
• Ruler A has more uncertainty than Ruler B.
• Ruler B gives a more precise measurement.
Mass Measurements
• The mass of an object
is a measure of the
amount of matter it
posses.
• Mass is measured with
a balance and is not
affected by gravity.
• Mass and weight are
not interchangeable.
Volume Measurements
• Volume is the amount of space occupied by
a solid, liquid, or gas.
• There are several instruments for measuring
volume, including:
– graduated cylinder
– syringe
– buret
– pipet
– volumetric flask
Significant Digits
• Each number in a properly recorded measurement
is a significant digit (or significant figure).
• The significant digits express the uncertainty in
the measurement.
• When you count significant digits, start counting
with the first non-zero number.
• Lets look at a reaction measured by three
stopwatches.
Significant Digits Cont.
• Stopwatch A is calibrated to seconds (±1 s), Stopwatch
B to tenths of a second (±0.1 s), and Stopwatch C to
hundredths of a second (±0.01 s).
Significant Digits and Placeholders
• If a number is less than one, a placeholder zero is
never significant.
• Therefore, 0.5 cm, 0.05 cm, and 0.005 cm all have
one significant digit.
• If a number is greater than one, a placeholder zero
is usually not significant.
• Therefore, 50 cm, 500 cm, and 5000 cm all have
one significant digit.
Exact Numbers
• When we count something, it is an
exact number.
• Significant digit rules do not apply to
exact numbers.
• An example of an exact number:
there are 3 coins on this slide.
Rounding Numbers
• All numbers from a measurement are significant.
However, we often generate nonsignificant digits
when performing calculations.
• We get rid of nonsignificant digits by rounding
off numbers.
• There are four rules for rounding off numbers.
Rules for Rounding Numbers
1. If the first nonsignificant digit is less than 5, drop all
nonsignificant digits.
Example:
A calculator displays 12.846239 and 3 significant
digits are justified.
The first nonsignificant digit is a 4, so we drop all
nonsignificant digits and get 12.8 as the answer.
Rules for Rounding Numbers
2. If the first nonsignificant digit is greater than or
equal to 5, increase the last significant digit by 1
and drop all nonsignificant digits.
A calculator display 12.856239 and 3
significant digits are justified.
The first nonsignificant digit is a 5, so the
last significant digit is increased by one to
9, all the nonsignificant digits are dropped,
and we get 12.9 as the answer.
Rounding Numbers
3. a) If the last digit is 5 and is preceded by an odd
number, then the last digit should be increased by .
Example: 4.635 is rounded to 4.64
b) If the last digit is 5 but is preceded by an even
number, then it stays the same or is rounded down
by 1.
Example: 4.625 is rounded to 4.62.
4. If a calculation has two or more operations, retain
all nonsignificant digits until the final operation and
then round off the answer.
Adding & Subtracting Measurements
• When adding or subtracting measurements, the
answer is limited by the value with the most
uncertainty.
• Lets add three mass measurements.
5
g
• The measurement 5 g has the
greatest uncertainty (±1 g).
• The correct answer is 15 g.
5.0
g
+ 5.00 g
15.00 g
Multiplying & Dividing Measurements
• When multiplying or dividing measurements, the
answer is limited by the value with the fewest
significant figures.
• Lets multiply two length measurements.
 5.15 cm × 2.3 cm = 11.845 cm2
• The measurement 2.3 cm has the fewest
significant digits, two.
• The correct answer is 12 cm2.
Exponential Numbers
• Exponents are used to indicate that a number has
been multiplied by itself.
• Exponents are written using a superscript; thus,
2×2×2×2 = 24.
• The number 4 is an exponent and indicates that
the number 2 is multiplied by itself 4 times. It is
read “2 to the fourth power”.
Powers of Ten
• A power of 10 is a number that results when 10 is
raised to an exponential power.
• The power can be positive (number greater than 1)
or negative (number less than 1).
Scientific Notation
• Numbers in science are often very large or very
small. To avoid confusion, we use scientific
notation.
• Scientific notation utilizes the significant digits in
a measurement followed by a power of ten. The
significant digits are expressed as a number
between 1 and 10.
Applying Scientific Notation
• To use scientific notation, first place a decimal
after the first nonzero digit in the number
followed by the remaining significant digits.
• Indicate how many places the decimal is moved
by the power of 10.
– A positive power of 10 indicates that the decimal
moves to the left.
– A negative power of 10 indicates that the decimal
moves to the right.
Scientific Notation Continued
• There are 26,800,000,000,000,000,000,000
helium atoms in 1.00 L of helium gas. Express
the number in scientific notation.
• Place the decimal after the 2, followed by the
other significant digits.
2.68 × 1022 atoms
• Count the number of places the decimal has
moved to the left (22). Add the power of 10 to
complete the scientific notation.
Another Example
• The typical length between two carbon atoms in a
molecule of benzene is 0.000000140 m. What is
the length expressed in scientific notation?
• Place the decimal after the 1, followed by the
other significant digits.
1.40 × 10-7 m
• Count the number of places the decimal has
moved to the right (7). Add the power of 10 to
complete the scientific notation.
Accuracy versus Precision
• Accuracy = proximity
of a measurement to
the true value of a
quantity.
• Precision = proximity
of several
measurements to each
other.
Summary
• A measurement is a number with an attached unit.
• All measurements have uncertainty.
• The uncertainty in a measurement is dictated by
the calibration of the instrument used to make the
measurement.
• Every number in a recorded measurement is a
significant digit.
Summary Continued
• Place holding zeros are not significant digits.
• If a number does not have a decimal point, all
nonzero numbers and all zeros between nonzero
numbers are significant
• If a number has a decimal place, significant digits
start with the first nonzero number and all digits
to the right are also significant.
Summary Continued
• When adding and subtracting numbers, the answer
is limited by the value with the most uncertainty.
• When multiplying and dividing numbers, the
answer is limited by the number with the fewest
significant figures.
• When rounding numbers, if the first
nonsignificant digit is less than 5, drop the
nonsignificant figures…If the number is 5 or
more, raise the first significant number by one and
drop all of the nonsignificant digits.
Summary Continued
• Exponents are used to indicate that a number is
multiplied by itself n times.
• Scientific notation is used to express very large or very
small numbers in a more convenient fashion.
• Scientific notation has the form D.DD × 10n, where
D.DD are the significant figures (and is between 1 and
10) and n is the power of ten.
• Accuracy refers to the proximity of a measurement to the
true value of a quantity.
• Precision refers to the proximity of several
measurements to each other.