# Experiment 1 Errors, Uncertainties and Measurements

By asd0321
Jan 05, 2013
1872 Words

Experiment 1: Errors, Uncertainties and Measurements

Laboratory Report

Kendrick Don Reyes, Myrr Kea Rostrata, Josemarie Emmanuel Roxas, Lindley Susi, Jessica Tabuzo

Department of Biological Sciences

College of Science, University of Santo Tomas

España, Manila Philipines

Abstract

In this experiment, different measuring devices were used, namely the vernier calliper, micrometer calliper, foot rule, and the electronic gram balance. These devices were used to obtain the mean diameter, volume, mass, and the experimental value of density of the sphere of known composition. 1. Introduction

Measurement is the process or act of determining the size, length, quantity, etc. of something being observed or measured. The units of measurement evolved and changed greatly since the day it was made by humans. In different places, these measurements can vary and could well be different from each other. Thus, standards are used nowadays so that we can have a concrete basis and this also prevents fraud or the cheating of somebody especially in business matters. There are different systems of measurement used. We have this CGS system or known as centimeter-gram-second system which is a metric system derived from the meter-kilogram-second system or mks system. It uses centimeter (c) for

length, gram (g), second (s) for time, dyne for force, and erg for energy. The metric system has become a common system for weights and measures. Its simplicity is the reason why scientists use this system of measurement. You can easily change one unit of measure into another. The units of measurement in this system are all based on decimals. By simply moving the decimal point to the right or left, depending on whether the unit is being decreased or increased, you can change its unit. Greek decimal prefixes like deca, hecto, kilo is used to express units of ten multiples or greater. Despite setting all of these standards, one’s measurement can never be exact and can always have a certain amount of error. When a measurement is done, the outcome could depend on several factors like the measuring system, the procedure taken, the execution of techniques of the operator, and the condition of the environment (Bell, 1999). This dispersion of values that can be attributed to a measured quantity is what we call as measurement uncertainty. The flaws in measurement can come from the measuring instrument itself due to aging, wearing, poor readability or even noise. The item being measured, if not stable, can produce uncertainties.

There are two types of measurement error, systematic error and random error. Imperfect calibration of instrument, its age, wear, and tear, throughout the years which lead to errors can be classified as a systemic error. When you measure the weight of an object using a particular balance which is improperly tared and you get a certain amount of grams higher for all your mass measurements is an example of systematic error. Random errors, on the other hand, are caused by unknown and unpredictable changes in the experiments. Irregular changes in the environment can usually cause this and as well as the random noise on an electrical device (Exell).

The precision of a measurement is determines the exactness or accuracy of a number of measurements and how the same quantity agrees with each other. Accuracy tells the correctness, veracity or truthfulness of a measurement. The closer the measurement to the accepted value, the more accurate it is. In this experiment, the group aims to achieve the following objectives: (1) to study errors and how they propagate in simple experiment, (2) to determine the average deviation of a set of experimental values, (3) to determine the mean of a set of experimental values as well as a set of average deviation of the mean, (4) to familiarize the students with the vernier caliper, micrometer caliper, and the foot rule, (5) to compare the accuracy of these measuring devices, (6) to determine the density of an object given its mass and dimensions.

2. Theory

3. Methodology

The measuring devices were checked for error. The least count of the vernier caliper, micrometer caliper and the foot rule was determined. Ten independent measurements for the diameter of the sphere using the foot rule were made. This was done by taking measurements at different points along its circumference. The mean diameter of the sphere was calculated. The deviation (d) of each measurement of the mean diameter and the average deviation (a.d.) were also calculated. Then, the average deviation (A.D.) of the mean diameter was computed. The % error for the diameter was computed by considering A.D. as the error and the mean diameter as the standard value. The volume of the sphere was then computed. Significant figures were used. The sphere was weighed using the electronic gram balance. The density of the sphere was calculated using the values obtained from the volume and mass of the sphere. The instructor was asked for the accepted value of the density of the sphere. The % error was then computed. The same steps were used using the vernier caliper and micrometer caliper.

4. Results ad Discussion

Table 1 shows the diameters of the sphere using different instruments for measurements. Included also in Table 1 are the mean, average deviation (a.d.), average deviation of the mean (A.D.)volume, mass, density and % error for density.

Table 1: Dimensions, density, and percent error using different measuring instruments. | Diameter of Sphere (cm)|

Trial| Foot Rule| Vernier Caliper| Micrometer Caliper|

1| 1.20| 0.18| 1.385| 0.200| 1.5750| 0.0088|

2| 1.40| 0.02| 1.590| 0.005| 1.5845| 0.0007|

3| 1.48| 0.10| 1.580| 0.005| 1.5850| 0.0012|

4| 1.30| 0.08| 1.585| 0.000| 1.5855| 0.0017|

5| 1.40| 0.02| 1.590| 0.005| 1.5845| 0.0007|

6| 1.45| 0.07| 1.590| 0.005| 1.5850| 0.0012|

7| 1.50| 0.12| 1.585| 0.005| 1.5850| 0.0012|

8| 1.50| 0.12| 1.585| 0.005| 1.5850| 0.0012|

9| 1.30| 0.08| 1.680| 0.095| 1.5840| 0.0002|

10| 1.25| 0.13| 1.680| 0.095| 1.5850| 0.0012|

mean| 1.38| 1.585| 1.5838|

a.d.| 0.092| 0.042| 0.0018|

A.D.| 0.029| 0.013| 0.0057|

Volume(cm3)| 1.4cm3| 2.085 cm3| 2.0802 cm3|

Mass (g)| 16.29g|

Exp’l value of desnsity (g/cm3)| 12 (g/cm3)| 7.813 (g/cm3)| 7.831 (g/cm3)| Accepted value of density (g/cm3)| 7.8 (g/cm3)|

% error for density| 53.85 %| 0.17%| 0.40&|

Base from the data, measurements from the foot rule had the greatest % error for density (53.85 %) while measurements from the vernier caliper had the least % error (0.17%). Possible errors for the measurements are systematic error especially human error. For the foot rule, members of the group possibly commit an error because the foot rule has no handle so that the end of the sphere could easily see. It is also possible that members of the group wrongly read the values. Base from the the data, accuracy of the instruments can be infer; vernier caliper is more accurate that the foot rule. This is because the uncertain digit of the foot rule is certain in vernier caliper.

5. Conclusion

The calipers were checked for errors, and studied how they propagated in the experiment. The average deviation was computed for the foot rule, vernier caliper, and micrometer caliper , which is 0.092, 0.042, and 0.0018 respectively. The average deviation of the mean was also determined, namely, 0.029, 0.013, and 0.00057 respectively. The accuracy of the said measuring devices were compared and recorded in table form. The density of the sphere was determined, 12 g/cm3 , 7.813 g/cm3 , 7.831 g/cm3 . Based on the information written above, the objectives of this experiment were achieved.

6. Applications

Among the three measuring devices, the vernier caliper gave the least percent error. The accuracy of a measurement is affected by the least count of the measuring device. Its’ accuracy would always be uncertain because every measuring instrument has a distinctive amount of uncertainty in its measurement. Error is the deviation of a measured value from the actual value. It is the imprecision in measurements that cannot be avoided. There are two types of error, random and systematic. Repeated measurements obtained from a random error can still be reliably estimated. A systemic error occurs if there is a defect in the equipment or in the design of the experiment. The errors that we encountered during the experiment were more of systematic errors. Most errors are human errors; the some measurement in the vernier caliper are obtained from wrong used of the instrument. A student weighs himself using a bathroom scale calibrated in kilograms. He reported his weight in pounds. What is the percentage error in his reported weight if he uses this conversion: 1 kg=2.2 pounds? The standard kilogram is equal to 2.2046. % Error = |A-T|T×100

% Error = |2.2-2.2046|2.2046×100 = 0.21%

In an experiment on determination of mass of a sample, your group consisting of 5 students obtained the following results: 14.34 g, 14.32 g, 14.33 g, 14.30 g, and 14.32 g. Find the mean, a.d. and A.D. Suppose that your group is required to make only four determination for the mass of the sample. If you were the leader of the group, which date will you omit? Recalculate the mean, a.d. and A.D. without this data. Which results will you prefer?

Table 1. Mass and deviation of the sample in five trials

Trial| Mass (g)| Deviation (d)|

1| 14.34| 0.04|

2| 14.32| 0.02|

3| 14.33| 0.03|

4| 14.30| 0.00|

5| 14.23| 0.07|

Mean = 14.30

a.d. = 0.032

A.D. = 0.014

14.23 will be omitted because it is the farthest value from each other.

Table 2. Mass and deviation of the sample in the chosen four trials Trial| Mass (g)| Deviation (d)|

1| 14.34| 0.02|

2| 14.32| 0.00|

3| 14.33| 0.01|

4| 14.30| 0.02|

Mean = 14.32

a.d. = 0.0125

A.D. = 0.007

Base on the computed data above, the data from table 2 are preferred.

7. Reference

Bell, S. (1999). A beginner's guide to uncertainty measurement. United Kingdom: Crown. Error and Statistics. (2012, December 1). Retrieved from http:/www.lepla.org/en/modules/Activities/p04-error4.htm Exell. (2012, November 30). Error Analysis. Retrieved from http://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html "Metric System." Microsoft® Encarta® 2009 [DVD]. Redmond, WA: Microsoft Corporation, 2008.

Random Errors- Physics Laboratory Tutorial. (2012, December 1). Retrieved December 1, 2012, from http://phys.columbia.edu/~tutorial/rand_v_sys/tut_e_5_1.html Taylor, J. (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, 128-129. Undergraduate Physics- Error Analysis. (2012, December 1). Retrieved from http://felix.physics.sunysb.edu/~allen/252/PHY_error_analysis.html

Laboratory Report

Kendrick Don Reyes, Myrr Kea Rostrata, Josemarie Emmanuel Roxas, Lindley Susi, Jessica Tabuzo

Department of Biological Sciences

College of Science, University of Santo Tomas

España, Manila Philipines

Abstract

In this experiment, different measuring devices were used, namely the vernier calliper, micrometer calliper, foot rule, and the electronic gram balance. These devices were used to obtain the mean diameter, volume, mass, and the experimental value of density of the sphere of known composition. 1. Introduction

Measurement is the process or act of determining the size, length, quantity, etc. of something being observed or measured. The units of measurement evolved and changed greatly since the day it was made by humans. In different places, these measurements can vary and could well be different from each other. Thus, standards are used nowadays so that we can have a concrete basis and this also prevents fraud or the cheating of somebody especially in business matters. There are different systems of measurement used. We have this CGS system or known as centimeter-gram-second system which is a metric system derived from the meter-kilogram-second system or mks system. It uses centimeter (c) for

length, gram (g), second (s) for time, dyne for force, and erg for energy. The metric system has become a common system for weights and measures. Its simplicity is the reason why scientists use this system of measurement. You can easily change one unit of measure into another. The units of measurement in this system are all based on decimals. By simply moving the decimal point to the right or left, depending on whether the unit is being decreased or increased, you can change its unit. Greek decimal prefixes like deca, hecto, kilo is used to express units of ten multiples or greater. Despite setting all of these standards, one’s measurement can never be exact and can always have a certain amount of error. When a measurement is done, the outcome could depend on several factors like the measuring system, the procedure taken, the execution of techniques of the operator, and the condition of the environment (Bell, 1999). This dispersion of values that can be attributed to a measured quantity is what we call as measurement uncertainty. The flaws in measurement can come from the measuring instrument itself due to aging, wearing, poor readability or even noise. The item being measured, if not stable, can produce uncertainties.

There are two types of measurement error, systematic error and random error. Imperfect calibration of instrument, its age, wear, and tear, throughout the years which lead to errors can be classified as a systemic error. When you measure the weight of an object using a particular balance which is improperly tared and you get a certain amount of grams higher for all your mass measurements is an example of systematic error. Random errors, on the other hand, are caused by unknown and unpredictable changes in the experiments. Irregular changes in the environment can usually cause this and as well as the random noise on an electrical device (Exell).

The precision of a measurement is determines the exactness or accuracy of a number of measurements and how the same quantity agrees with each other. Accuracy tells the correctness, veracity or truthfulness of a measurement. The closer the measurement to the accepted value, the more accurate it is. In this experiment, the group aims to achieve the following objectives: (1) to study errors and how they propagate in simple experiment, (2) to determine the average deviation of a set of experimental values, (3) to determine the mean of a set of experimental values as well as a set of average deviation of the mean, (4) to familiarize the students with the vernier caliper, micrometer caliper, and the foot rule, (5) to compare the accuracy of these measuring devices, (6) to determine the density of an object given its mass and dimensions.

2. Theory

3. Methodology

The measuring devices were checked for error. The least count of the vernier caliper, micrometer caliper and the foot rule was determined. Ten independent measurements for the diameter of the sphere using the foot rule were made. This was done by taking measurements at different points along its circumference. The mean diameter of the sphere was calculated. The deviation (d) of each measurement of the mean diameter and the average deviation (a.d.) were also calculated. Then, the average deviation (A.D.) of the mean diameter was computed. The % error for the diameter was computed by considering A.D. as the error and the mean diameter as the standard value. The volume of the sphere was then computed. Significant figures were used. The sphere was weighed using the electronic gram balance. The density of the sphere was calculated using the values obtained from the volume and mass of the sphere. The instructor was asked for the accepted value of the density of the sphere. The % error was then computed. The same steps were used using the vernier caliper and micrometer caliper.

4. Results ad Discussion

Table 1 shows the diameters of the sphere using different instruments for measurements. Included also in Table 1 are the mean, average deviation (a.d.), average deviation of the mean (A.D.)volume, mass, density and % error for density.

Table 1: Dimensions, density, and percent error using different measuring instruments. | Diameter of Sphere (cm)|

Trial| Foot Rule| Vernier Caliper| Micrometer Caliper|

1| 1.20| 0.18| 1.385| 0.200| 1.5750| 0.0088|

2| 1.40| 0.02| 1.590| 0.005| 1.5845| 0.0007|

3| 1.48| 0.10| 1.580| 0.005| 1.5850| 0.0012|

4| 1.30| 0.08| 1.585| 0.000| 1.5855| 0.0017|

5| 1.40| 0.02| 1.590| 0.005| 1.5845| 0.0007|

6| 1.45| 0.07| 1.590| 0.005| 1.5850| 0.0012|

7| 1.50| 0.12| 1.585| 0.005| 1.5850| 0.0012|

8| 1.50| 0.12| 1.585| 0.005| 1.5850| 0.0012|

9| 1.30| 0.08| 1.680| 0.095| 1.5840| 0.0002|

10| 1.25| 0.13| 1.680| 0.095| 1.5850| 0.0012|

mean| 1.38| 1.585| 1.5838|

a.d.| 0.092| 0.042| 0.0018|

A.D.| 0.029| 0.013| 0.0057|

Volume(cm3)| 1.4cm3| 2.085 cm3| 2.0802 cm3|

Mass (g)| 16.29g|

Exp’l value of desnsity (g/cm3)| 12 (g/cm3)| 7.813 (g/cm3)| 7.831 (g/cm3)| Accepted value of density (g/cm3)| 7.8 (g/cm3)|

% error for density| 53.85 %| 0.17%| 0.40&|

Base from the data, measurements from the foot rule had the greatest % error for density (53.85 %) while measurements from the vernier caliper had the least % error (0.17%). Possible errors for the measurements are systematic error especially human error. For the foot rule, members of the group possibly commit an error because the foot rule has no handle so that the end of the sphere could easily see. It is also possible that members of the group wrongly read the values. Base from the the data, accuracy of the instruments can be infer; vernier caliper is more accurate that the foot rule. This is because the uncertain digit of the foot rule is certain in vernier caliper.

5. Conclusion

The calipers were checked for errors, and studied how they propagated in the experiment. The average deviation was computed for the foot rule, vernier caliper, and micrometer caliper , which is 0.092, 0.042, and 0.0018 respectively. The average deviation of the mean was also determined, namely, 0.029, 0.013, and 0.00057 respectively. The accuracy of the said measuring devices were compared and recorded in table form. The density of the sphere was determined, 12 g/cm3 , 7.813 g/cm3 , 7.831 g/cm3 . Based on the information written above, the objectives of this experiment were achieved.

6. Applications

Among the three measuring devices, the vernier caliper gave the least percent error. The accuracy of a measurement is affected by the least count of the measuring device. Its’ accuracy would always be uncertain because every measuring instrument has a distinctive amount of uncertainty in its measurement. Error is the deviation of a measured value from the actual value. It is the imprecision in measurements that cannot be avoided. There are two types of error, random and systematic. Repeated measurements obtained from a random error can still be reliably estimated. A systemic error occurs if there is a defect in the equipment or in the design of the experiment. The errors that we encountered during the experiment were more of systematic errors. Most errors are human errors; the some measurement in the vernier caliper are obtained from wrong used of the instrument. A student weighs himself using a bathroom scale calibrated in kilograms. He reported his weight in pounds. What is the percentage error in his reported weight if he uses this conversion: 1 kg=2.2 pounds? The standard kilogram is equal to 2.2046. % Error = |A-T|T×100

% Error = |2.2-2.2046|2.2046×100 = 0.21%

In an experiment on determination of mass of a sample, your group consisting of 5 students obtained the following results: 14.34 g, 14.32 g, 14.33 g, 14.30 g, and 14.32 g. Find the mean, a.d. and A.D. Suppose that your group is required to make only four determination for the mass of the sample. If you were the leader of the group, which date will you omit? Recalculate the mean, a.d. and A.D. without this data. Which results will you prefer?

Table 1. Mass and deviation of the sample in five trials

Trial| Mass (g)| Deviation (d)|

1| 14.34| 0.04|

2| 14.32| 0.02|

3| 14.33| 0.03|

4| 14.30| 0.00|

5| 14.23| 0.07|

Mean = 14.30

a.d. = 0.032

A.D. = 0.014

14.23 will be omitted because it is the farthest value from each other.

Table 2. Mass and deviation of the sample in the chosen four trials Trial| Mass (g)| Deviation (d)|

1| 14.34| 0.02|

2| 14.32| 0.00|

3| 14.33| 0.01|

4| 14.30| 0.02|

Mean = 14.32

a.d. = 0.0125

A.D. = 0.007

Base on the computed data above, the data from table 2 are preferred.

7. Reference

Bell, S. (1999). A beginner's guide to uncertainty measurement. United Kingdom: Crown. Error and Statistics. (2012, December 1). Retrieved from http:/www.lepla.org/en/modules/Activities/p04-error4.htm Exell. (2012, November 30). Error Analysis. Retrieved from http://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html "Metric System." Microsoft® Encarta® 2009 [DVD]. Redmond, WA: Microsoft Corporation, 2008.

Random Errors- Physics Laboratory Tutorial. (2012, December 1). Retrieved December 1, 2012, from http://phys.columbia.edu/~tutorial/rand_v_sys/tut_e_5_1.html Taylor, J. (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, 128-129. Undergraduate Physics- Error Analysis. (2012, December 1). Retrieved from http://felix.physics.sunysb.edu/~allen/252/PHY_error_analysis.html