Significant Figures and Experimental Errors
Name:
Partners’ names:
Date:
Course:
Objectives:
To understand scientific notation, significant figures, and different kinds of experimental errors
Theory:
The minimum number of digits required to report a value without loss of accuracy is the number of
significant figures of the value. It is important to use significant figures when recording a measurement so
that it does not appear to be more accurate than the equipment is capable of determining.
The most significant digit is the left-most non-zero digit, i.e. 2 in the number 0.002985. The least significant
digit is the right-most digit if the number includes the decimal, i.e. in the number 189.76970, 0 is the least
significant digit. The least significant digit is the right-most non-zero digit if the number has no decimal,
i.e., in 123000, 3 is the least significant digit.
There are different types of experimental errors we encounter during observation. Let’s learn about them in
some detail:
Random errors: are due to unknown and unpredictable causes such as fluctuations in temperature, voltages,
vibrations, and unbiased measurement errors.
Systematic errors: are due to instruments or techniques. Examples are improperly calibrated instruments or
observer bias. Reading a scale or meter stick from an improperly aligned eye position (parallax) can
introduce a systematic error.
Accuracy: how close a measurement is to the actual or accepted value.
Precision: agreement between several measurements. Precise measurements are very close to each other.
Least count: smallest sub-division on the measurement scale of an instrument (for example the least count of
a meter stick is a millimeter (mm)).
Significant figures: all the subdivisions that can be read from the scale of an instrument plus an additional
estimated subdivision (For example the length of a nail is measured to be 3.65 cm. In this case 3.6 cm were
read directly from a ruler; however the .05 cm was estimated by the measurer).
Most significant digit: the left-most non-zero digit
Example:
8 in 0.000812
2 in 291.76
Least significant digit:
a) No decimal point: the right-most non-zero digit is the least significant
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Example:
The 8 in 97780
b) With decimal points: the right-most digit, including zeros is the least significant
Example:
The 0 in 36.870
The 4 in 23.654
Note: all digits between the least and most significant are considered to be significant.
Rules for manipulating numbers:
i)
Multiplication or Division:
The number of significant figures in the final answer is equal to that of the term with the least significant
digits.
Example:
8.358 ---- has 4 significant figures
2.4
---- has 2 significant figures
8.345 x 2.4 = 20.0592 --- (by calculator)
However the answer can only have 2 significant figures (equal to that of the term with least significant
figures i.e. 2.4)
Answer: 20 --- (do not report the other numbers 20.0592)
ii) Addition and Subtraction:
The number of decimal places in the answer should be same as the term with the smallest number of decimal
places.
Example:
4.56968 + 2.5498 + 23 = 30.11948
This answer should be reported as: 30, (30.11948) because this is equal to the smallest number of decimal
places of the terms being added, namely that of. 23.
iii) Rounding off numbers:
Round off to the appropriate number of significant digits by increasing the last digit by one (if the number
after that is greater than 5) or leaving it as it is (if the number after that is less than 5).
Examples:
89.47976
to 4 significant figures is
89.48
0.025321
to 2 significant figures is
0.025 (The two zeroes are just place holders)
2.0048
to 4 significant figures is
2.005
iv) Scientific notation:
A scientific system for conveniently writing large or small numbers. It is written as:
a x 10b
a: coefficient or mantissa
b: exponent
Example:
2
936.87 to 2 significant figures is: 9.4 x 102
(Note: only ONE whole number followed by a
decimal point and then the correct power. The exponents are just place holders)
0.0201 to 1 significant figure is: 2 x 10-2
(The exponents are just place holders)
Equipment Required:
Calculator, A Digital Caliper, A meter tape.
Lab Procedures:
1. Significant Figures
a) Express the numbers in Table 1 to three significant figures. Write the numbers in normal notation in the
first column and in scientific notation in the second column.
Table 1:
Example
Normal Notation
(to 3 significant figures)
Scientific Notation
(to 3 significant figures)
1.064
224
10.08
0.1134
26.301
670
0.3141
29.36
0.0007964
98.00000001
b) Determine Pi
In this experiment we will try to measure the dimension of a soda can to determine the value of π.
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S. No.
Diameter (cm) ± Uncertainty
(Use caliper)
d ± ∆d
1
2
3
4
Average Value:
Circumference (cm )± uncertainty
(Use a meter tape)
C ± ∆C
±
±
Remember:
𝐶
Circumference = 2πr = πd. Hence 𝜋 = 𝑑
Using the above information, determine the value of π.
Determining uncertainty for π:
∆𝐶
The uncertainty in π is given by ∆𝜋 = 𝜋 ( 𝐶 +
∆𝑑
𝑑
).
Explanation: The value of ∆C/C yields a percentage of what ∆C is of C. Likewise for ∆d/d. Each of these
percentages contributes to the uncertainty of measurement of π. Adding these percentages and then
multiplying by π yields the uncertainty in the measurement of π, the ∆π.
Remember that the compound uncertainty is never smaller than the least value.
Hence,
Experimental value of π
±
The actual value of π is 3.1416.
Substitute the Experimental value of π you obtained above and the accepted value of π (3.1416) and calculate
the “Fractional error” and “Percent error”. You do not need to include Uncertainty in π for the calculation
below:
Vertical bars mean to take the “absolute value” or
𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙−𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑
Fractional Error = |
|
ignore any negative signs
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑
Fractional error
𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙−𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑
Percent Error = |
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑
| × 100%
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Percent error
2. A young physicist carried out an experiment to determine the acceleration due to gravity (g) and obtained
the following two values: 9.96 m/s2 and 9.72 m/s2. You know that the accepted value for g = 9.80 m/s2.
a) You wish to compare the two experimental results the physicist got, so you decide to find the percent
difference between these two values using the standard equation you learned in physics class:
Substitute the following values for E1 and E2 and calculate the percent difference using the formula given
below:
E1
9.96 m/s2
E2
9.72 m/s2
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
|𝐸2 − 𝐸1|
× 100%
𝐸2 + 𝐸1
(
2 )
Percent difference
b) You now wish to see how much each experimental value obtained above differs from the actual or
accepted value. You therefore decide to find the percent error using the equation you also learned:
Percent error =
|𝐸−𝐴|
𝐴
× 100%
Substitute the following values in the above equation and calculate the two percent error values.
E1
9.96 m/s2
E2
9.72 m/s2
A
9.80 m/s2
Record your percent error values in the table below:
Percent error for E1
Percent error for E2
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(c) Follow the rules underlined under the section “Rules for manipulating numbers” and complete the
following calculations. Make sure you have correct number of significant figures in your answer.
Perform the following Calculation. Write your answer to correct number of Significant Figures.
345.06 + 49.5 +26.0003 =
879.4 + 345.682 + 47.7721 =
7578393.34 – 9374.67839 =
8394.348973 – 4832.897234 =
23.45 X 34.4 =
4.567 X 0.0343 =
4837.343/454.567=
434.9867/2342.454 =
Answers the following questions:
1. Do experiments yield true (actual or accepted) values of physical quantities? Clarify.
2. What are the common causes of experimental errors? Give some examples.
3. How can you reduce experimental errors?
4. How do you determine the number of significant figures while reporting your answer? What are the
implications of reporting less/more than the significant digits?
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5. What is the difference between “Percent error” and “Percent difference?”
Conclusions (please write what you learned from this lab):
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