the study of uncertainty in measurement
all measurements (no matter how carefully made) have an uncertainty to them, which is referred to as an error
uncertainty (error), is not due to mistakes and hence cannot be avoided. Therefore, it is important to minimize the uncertainties or errors in measurements
uncertainty is usually due to the instrument/tool which is used to make a measurement
Example 1: Making measurements with a ruler a) HB Notre Dame b)
0 1 2 3 4 5 6
HB Notre Dame
0 1 2 3 4 5 6
With the ruler used in a), the measurement of the length of the pencil would be 5.4 +/- 0.2 cm
The uncertainty (error) is 0.2 cm, hence this measurement indicates that the pencil can have a length from 5.2 cm to 5.6 cm
(5.2 cm

length

5.6 cm). The value of 0.2 cm is sometimes called the absolute uncertainty , or absolute error .
With the ruler used in b), the measurement of the length of the pencil would be
5.4
+/- 0.1 cm
The absolute error is 0.1 cm, hence this measurement indicates that the pencil can have a length from 5.3 cm to 5.5 cm (5.3 cm

length

5.5 cm). This measurement has less error (is more certain) than the previous measurement since the ruler has finer increments.
The relative error is a comparison between the absolute error of the measurement and the measurement. The relative error is often represented as a percent. For case b, the relative error would be relative error = absolute error/measurement = 0.1/5.4 = 0.019 or 1.9%
Example 2: Suppose you are measuring the time it takes for a ball to drop 2 meters. You repeat the experiment and come up with the following results. time (seconds): 0.64 0.69 0.54 0.44 0.75 0.90 0.43
Averaging the results would yield 0.63 seconds. The highest result was 0.90 seconds, while the lowest result yielded 0.43 seconds. So you could report your result as
0.63 +/- 0.30 seconds OR 0.63 +/- 0.20 seconds
For measurements that involve a stopwatch, the general rule of thumb is to use an error that is between 0.20 to 0.30 seconds. This error is due to human reaction time.
Justification of Uncertainty (error): For both of the examples, the uncertainty in the measurement needs to be justified.
In other words, you must indicate “why you chose the error you chose”.
Why is error analysis done?
for laboratory experiments, error analysis is used to compare your experimental results to theoretical results to show that your answer “agrees” with theory
if your experimental answer does not agree with the theory, then one must redesign the experiment or account for these differences (perhaps the current theory is incorrect and needs to be altered)
used in all branches of science to validate results

Error Problems/Calculations
Example 3: The dimensions of a rectangular shaped lawn are measured. The length of the lawn is measured to be 10.2 m
+/- 0.1 m. The width of the lawn is measured to be 5.6 +/- 0.1 m. a) Calculate the range of values (high and low values) for the length of the lawn. b) Calculate the range of values for the width of the lawn l
HIGH
= 10.2 + 0.1 = _______ m l
LOW
= 10.2 – 0.1 = ________ m w
HIGH
=
w
LOW
= c) Calculate the area of the lawn. e) Calculate the lowest possible value for the area of the lawn. d) Calculate the highest possible value for the area of the lawn (based on the errors in the measurement of the width and length). f) Calculate the absolute error on the area of the lawn.
Error Area = (A
HIGH
- A
LOW
)/2
Error Area = ___________
Error Area is sometimes called

A

A = ____________ h) Calculate the relative error in length, width and area. g) Why are the absolute errors in the length and width identical?
Example 4: The volume of water that is measured is 50 ml +/- 2 ml. The mass of the water is 50.2 +/- 0.1 grams a) Determine the range (possible values) for the measurement of the volume of water. b) Determine the range for the measurement of the mass of the water. c) Calculate the relative error for the volume of water. d) Calculate the relative error for the mass of water. e) Determine the range of values for density (highest and lowest possible value for density). f) Calculate the absolute error in density. g) Calculate the relative error in density. h) If you wanted to decrease the absolute/relative error in density, and had to either purchase a new scale to improve the precision on the measurement of mass or a new volumetric flask to improve the precision on the measurement of volume, which one would you choose and why (explain).
Example 5: A Hotwheels car was traveling at constant velocity. The distance it covered was 30.0 +/- 0.1 cm in 10.05 +/-
0.20 seconds. a) Determine the range (possible values) for the measurement of distance. b) Calculate the relative error for the measurement of distance. c) Determine the range for the measurement of time. d) Calculate the relative error for the measurement of time. e) Determine the range of values for the calculation of speed based on the high and low values for time and distance. f) Calculate the absolute error in speed. g) Calculate the relative error in speed. h) Explain how you could decrease the absolute error in speed?