Download Introduction Estimating and Uncertainty Accuracy and

Calculation Practice
01/05/2016
Cal-1
Math Calculations Practice Exercises
Introduction
The International System of Measurement (SI) is used worldwide and has been adopted as the official system of
measurement by most countries. It is commonly called the metric system. Our traditional American/English system of
measurement (e.g., miles, quarts, pounds) requires many conversion factors. Take length, for example – there are inches,
feet, yards, rods, chains, and miles! The metric system is much different. It is based on standard units that can be easily
converted by multiplying or dividing by factors of ten. Engineers and scientists most often use these standard metric
units: the meter, for length; the gram, for mass (or weight); the liter, for volume; and the degree Celsius (or less often
Kelvin) for temperature.
Estimating and Uncertainty
Whenever you take a scientific measurement, you are making a quantitative observation. When you use a device to make
a measurement, you obtain a measured number. When you report your data, you usually are estimating the last significant
figure. In other words, you will report the digits in the measurement that you are certain about plus one additional digit
that are you allowed to estimate. We will discuss significant figures later in this exercise. You can also look in your
lecture text or Appendix F of the lab manual if you need help with significant figures. Here are typical uncertainties of
common measuring devices:
Measuring Device
12 cm ruler
triple-beam balance
analytical balance
10 ml graduated cylinder
100 ml graduated cylinder
50 ml buret
25 ml volumetric flask
25 ml transfer pipet
Uncertainty
± 0.05 cm
± 0.05 g
± 0.0001 g
± 0.05 ml
± 0.5 ml
± 0.02 ml
± 0.02 ml
± 0.02 ml
The last digit in your measured number is the digit that is considered uncertain because it is estimated. For example, if
you are measuring the length of a piece of metal with a ruler that has marks down to units of 0.1 cm, you could estimate
the length down to the 0.05 cm, in other words, between tenths of a cm. The number of significant figures gives us an
idea of the accuracy of a measuring device. In this and future labs in this class, you will be expected to keep track of
significant figures for performing calculations and reporting results. Appendix F provides rules for rounding off the
results of calculations with significant figures.
When you use a conversion factor or a definition, those would be examples of an exact number – in other words, a
number for which you do not need to use a measuring device. Can you think of number relationships that would fit into
this category of number?
Accuracy and Precision
In common English, we often use the terms “accuracy” and “precision” interchangeably, to indicate how “correct” an
answer is. However, in science the two terms have different meanings. Accuracy is a measure of how closely an
observation is to the “true” or “accepted” value. Precision is a measure of how closely a group of observations are to one
another. If you think about a dartboard, “accurate” would be hitting the bulls-eye or center of the target; “precise” would
mean that all of your darts hit the target close to one another, without reference to whether or not you hit the bulls-eye.
So, it is possible to be precise (all the darts close together) but not accurate (missing the bulls-eye). Of course, we would
like to be both accurate and precise in our laboratory measurements. Appendix D (Useful Math Relationships) gives you
some mathematical ways of reporting accuracy and precision.
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Cal-2
Random and Systematic Errors
As we can see from the information above, each measurement has a certain amount of uncertainty associated with it,
which means that each measurement has a certain amount of error. Errors refer to the calculated difference between a
measured value and the “true” value. There are actually two kinds of error: random error and systematic error. Random
errors result from the uncertainty of your measurement device. They are not caused by a mistake in your technique, but
are caused by unpredictable or imperceptible factors that are beyond your control as an experimenter. An example would
be two different experimenters determining the mass of an object using two kinds of balances that have different
sensitivities for mass.
Errors that have definite causes are called systematic errors. In general, systematic errors are generally reproducible and
will result in values that are always higher than the true value or lower than the true value. An example would be a
thermometer that is not calibrated correctly and always gives a reading that is lower than the actual or true value. Random
errors are always present – but you want to eliminate or minimize the systematic errors in carrying out your experiments.
A. Measured and Exact Numbers
Example 1
Describe each of the following as a measured or exact number
a. 14 inches ______________________________________
b. Four chairs at the dining table _____________________
c. 60 minutes in 1 hour _____________________________
d. 7.5 kg _____________________________________
e. 2.54 cm in 1 inch ____________________________
f. 1000 mg in 1 g ______________________________
g. 12 inches in 1 ft ______________________________
h. 2.20 lb in 1 kg ________________________________
B. Scientific Notation
In scientific work, small numbers such as 0.000000025 m and large numbers such as 4,000,000 g are often expressed
using powers of 10, such as 2.5 x 10 -8 m and 4 x 10 6 g. The values of 2.5 and 4 are coefficients; the values 10 -8 and 10 6
are powers of ten. (See Table 1 and Example 2). The rules for converting standard numbers or scientific notation are
given below.
Calculation Practice
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Table 1
Standard number
10,000
1000
100
10
=
=
=
=
Cal-3
Some Powers of Ten
Power of ten
10 4
10 3
10 2
10 1
Standard number
0.1
0.01
0.001
0.0001
=
=
=
=
Power of ten
10 -1
10 -2
10 -3
10 -4
Rules for Writing Numbers in Scientific Notation
For numbers larger than 10:
a. Move the decimal point to the left until it follows the first non-zero digit in the number
b. Write a power of ten that is equal to the number of places the decimal was move to the left
For numbers smaller than 1:
a. Move the decimal point to the right until it is located after the first digit in the number
b. Write a negative power of ten that is equal to the number of places the decimal point was moved to the right
Example 2
Write the following standard numbers in scientific notation:
a. 35,000 km
_______________________________
b. 608 g _______________________________________
c. 0.0000815 m _________________________________
C. Significant Figures
In measured numbers, all the reported figures are called significant figures. The first significant figure is the first nonzero
digit. The last significant figure is always the estimated digit. Zeros between other digits or at the end of the decimal part
of a number (i.e., trailing zeros in a number with a decimal part) are counted as significant figures. However, leading
zeros are not significant; they are placeholders. Zeros are not significant in numbers equal to, or greater than 10, with no
decimal part (there may or may not be a decimal point); they are placeholders needed to express the magnitude of the
number.
In a number written in scientific notation, all the figures in the coefficient are significant. Examples of counting
significant figures in measured numbers are given in Table 2 and Example 3.
Table 2
Measurement
455.2 cm
0.800 m
50.2 L
0.0005 lb
25,000 ft
3.20 x 10 4 g
Examples of Counting Significant Figures
Number of Significant Figures
4
3
3
1
2
e
Reason
All nonzero digits are significant
The trailing zeros in the decimal part are significant
A zero between nonzero digits is significant
Leading zeros are not significant
Placeholder zeros are not significant
All the digits in a coefficient are significant
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Example 3
State the number of significant figures in each of the following measured numbers
a. 588 m
____________________
c. 1.5 x 10 3 mL
_________________
b. 0.0700 g
d. 0.00045 L ___________________
e. 45.0 kg ________________________
f.
g. 2.20 lbs
h. 1000 mm
_____________________
___________________
500 lb ______________________
___________________
D. Rounding Off
Often, you will use a measurement in a mathematical operation such as multiplication, division, addition or subtraction.
When the calculator display shows more numbers than the measurements support, it is necessary to round of the
calculated answer. If the numbers to be dropped begin with a number less than 5, they are simply dropped. However, if
the numbers dropped begin with a 5 or greater, the value of the last retained digit is increased by 1. For a calculator
display already in scientific notation, round off the coefficient to the correct number of significant figures.
On your calculator, an answer may appear in scientific notation, which means that a coefficient and a power of ten are
shown. In scientific notation, the correct number of significant figures is shown in the coefficient. Often the numbers
shown in a calculator display before the power of ten must be rounded. Be sure to keep the power of ten! See Table 3
and Example 4.
Table 3
Examples of Scientific Notation from Calculator Results
Calculator display
2.512
4.1585
8.775
05
12
-08
Number of significant figures to
be shown in the coefficient
2
3
2
Rounded and written in
scientific notation
2.5 x 10 3
4.16 x 10 12
8.8 x 10 -8
Example 4
Round off each of the following calculator displays to report answers with three significant figures and two significant
figures.
a. 75.6243 km
_______________
_____________
b. 0.528392 g
_______________
_____________
c. 387.600 miles
_______________
_____________
d. 0.007000 m
_______________
_____________
E. Multiplication and Division
If a calculated answer is obtained from multiplication and/or division, it is rounded off to the same number of significant
figures are the measured number with the fewest significant figures. See Examples 5 and 6.
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Example 5
Solve:
(0.025 𝑔)(4.62 𝑔)
3.44 𝑔
=
Solution: On the calculator, the steps are:
Enter keys
0.025
X
4.62
=
÷
3.22
=
Display reads
0.025
0.025
4.62
0.1155
0.1155
3.44
0.033575581
0.03
Two significant figures
Three significant figures
Three significant figures
Calculator display to 9 decimal places
Final answer rounded to two sig figs
Example 6
Solve
3.4 𝑥 10−4 𝑚
2.75 𝑥 10 8 𝑚
=
Solution: On the calculator, the steps are:
Enter keys
3.4
EXP ( or EE)
4
±
÷
2.75
EXP
8
=
Display reads
3.4
3.4
00
3.4
04
3.4
-04
3.4
-04
2.75
2.75
00
2.75
08
1.2363636
-12
1.2 x 10 -12
Two significant figures
Three significant figures
Calculator display
Coefficient rounded to two sig figs
F. Addition and subtraction
After you have added or subtracted measured numbers, you may need to round off the result. An answer from + or – has
the last significant figure in the column where all the numbers added or subtracted also have significant figures. See
Examples 7 and 8.
Example 7
Add 42.11 g + 4.056 g + 30.1 g
Solution:
+
+
42.11
4.056
30.1
__________
76.266
76.3
digits in this number end at the tenth’s place; all other numbers go further
calculator display
final answer rounded to give digit in tenth’s place
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Example 8
Subtract:
14.621 mL – 3.39 mL
Solution:
+
14.621
3.39
__________
11.231
11.23
digits end at hundredth’s place; all other numbers go further
calculator display
final rounded to give a digit in hundredth’s place
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Student Name: _____________________________________
Section: ______________
Pre-lab Activity Questions
1. Give one example of a measured number.
2. Give one example of exact numbers.
3. What is meant by “degree of uncertainty”?
4. Give 2 reasons why a zero would be considered NOT significant.
Exercises
A. Measured and Exact Numbers
Circle “M” or “E” to indicate whether each of the following numbers is measured or exact:
Counted 5 books
M
E
2.54 cm in 1 inch
M
E
5 lb
M
E
12 inches in 1 foot
M
E
9.25 g
M
E
361 miles
M
E
0.035 kg
M
E
100 cm in 1 m
M
E
B. Scientific Notation
Write the following numbers in scientific notation:
4,450,000
________________________
0.00032
____________________
38,000
________________________
25.2
____________________
0.0505
____________________
0.0000000021 ________________________
Write the following as standard numbers:
4 x 10 2
________________________
3 x 10 -4
____________________
5 x 10 3
________________________
8.2 x 10 -3
____________________
3.15 x 10 5
________________________
2.46 x 10 -6 ____________________
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C. Significant Figures
State the number of significant figures in each of the following measured quantities:
4.5 m
_____________________________
204.52 g
______________________
0.0004 L
_____________________________
625,000 mm
______________________
805 lb
_____________________________
34.80 km
______________________
D. Rounding Off
Round off each of the following to the number of significant figures indicated. Don’t forget placeholder zeros when
necessary!
0.4108
143.63212
532,800
5.448 x 10 2
8,950,000
Three significant figures
Two significant figures
E. Multiplication and Division
Do the following multiplication and division calculations. Give a final answer with the correct number of significant
figures.
4.51 ft x 0.28 ft
___________________________
0.184 cm x 8.00 cm x 0.034 cm
___________________________
2.51 x 10 4 ms -1 x 5.0 x 10 -7 m
___________________________
(42.4 𝑎𝑎𝑎)(1.45 𝐿) ÷ (4.8 𝐾)(0.0821 𝐿 𝑎𝑎𝑎 mol-1 K-1
___________________________
F. Addition and Subtraction
Do the following addition and subtraction calculations. Give a final answer with the correct number of significant
figures.
13.4 mL + 0.4552 mL
____________________________
145.5 m + 86.5 m + 1045 m
____________________________
245.625 g - 80.2 g
____________________________
(11.2 g – 2.11 g)/(10.2 cm x 2.3 cm x 1.0 cm)
____________________________