Experiment 1: Errors, Uncertainties, and MeasurementsLaboratory Report

Kate Auditor, Bethanee Baes, Keana Balverde, Lina Lou Berdijo

Department of Occupational TherapyCollege of Rehabilitation Sciences, University of Santo Tomas

España Street, Manila PhilippinesAbstract

Though measurement has been standardized in order to reach universality and convenience, errors and uncertainties still presents itself, even in simple experiments. With the use of a foot rule, a vernier caliper and a micrometer caliper, the mean diameter of a metallic sphere were measured and derived from to get the average deviation of the mean diameter and the % error. The sphere measured a mean diameter and an average deviation of 1.51 cm and 0.03cm with the foot ruler, 1.582 cm and 0.0009cm with the vernier caliper and 1.5835 cm and 0.0002cm with the micrometer. By comparison, the micrometer caliper proves to be the most accurate with a 0.33% error, followed by the vernier caliper and the foot rule with a 0.61% error and 15.71% error respectively.

I. Introduction

Man has long put efforts to describe his environment in an objective and universal manner. As a result, man has created a special medium we generally know as measurement. Today, measurements continue to be a great part of man’s life; its use being utilize by all professions in every variety of settings. Healthcare practitioners have now been able to accurately give doses of medicine through the use of measurements. Similarly, engineers and architects have been building stronger infrastructures through the use of measurements. Even on a simpler field, mothers all around have been making dishes

and delicacies with the use of measurements. Society greatly relies on the application of measurements in order to progress. Because of this, measuring must be executed properly. To pursue this, man developed units of measurements.Tracing back to the ancient times, man used objects such as ropes, stones and sticks to measure objects. Units were also based using human body parts such as arms, hands and feet. However, body parts varied from person to person, making constancy a dilemma. To address the confusion, the French National Assembly appointed the committee that created the metric system in 1790. To this date, despite the convenient use of standardized measuring tools, errors and uncertainties still exist. Hence, this experiment aims to (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 set of average deviation of the mean, (4) to familiarize the students with the vernier calliper, micrometer calliper, and foot rule, (5) to compare the accuracy of these measuring devices, (6) and to determine the density of an object given its mass and dimensions.

II. Theory

In this experiment, uncertainty in measurement is shown by the result of

limitation in accuracy or precision. Accuracy

of measurement describes how well the results agree with an accepted value of the quantity being measured. Precision refers to the degree of exactness to which a measurement can be reproduced. For the experiment, the precision and accuracy is tested by using instruments with varying least count and, therefore, significant figures it can express. Least count of any precision instrument is defined as the least distance travelled by it. Generally, the number of significant figures is the number of reliably known digits. Below is the formula to determine the least count of an instrument:

one main scaleno . of divisions

=L .C .

The accuracy is determined by solving for average deviation and % error. Average deviation is referred as the average of the absolute values of the differences between individual numbers. Note that average deviation of each diameter is different from average deviation of the mean diameter. Percent (%) error is the measure of how faulty a measurement is, compared to the true/accepted value. Below are the formulas used:

∑∑ of deviations

no .of observations = average deviation

averagedeviation

√no . of observations = Average Deviation

of the mean diameter

% error=|experimental−accepted value|x100

accepted value

VI. Methodology

The foot rule, vernier caliper, and micrometer caliper were used to measure the diameter of the sphere, as well as its errors. First, the least count of the foot rule, vernier caliper, and micrometer caliper determined. Then the group took ten independent measurements to measure the diameter of the sphere using the foot rule. The ten independent measurements were added then divided by 10, to calculate for the mean diameter of the sphere. After that, the group computed for the deviation (d) of each measurement of diameter from the mean diameter. Then the average deviation (a.d.), which is the sum of the deviations (d) divided by the number (n) of observations, was calculated.

a .d .= Σ dn

Taking note of the average deviation (a.d.), the group then computed for the average deviation of the mean diameter (A.D.) with the use of the following equation:

A . D .=a . d .√ n

The volume of the sphere was then calculated using the following formula: (wherein r is the mean radius)

V= 43

π r 3

mean radius=mean diameter2

The weight of the sphere was determined by the use of the electronic gram balance. With that, the experimental density of

the sphere was calculated using the values that were obtained in computing the volume of the sphere and its mean radius.

The instructors gave the accepted value of density. Using that, the group computed for the % error for density.

% error=|experimental−accepted value|x100

accepted value

The second to the last steps were then repeated to gather the values of the vernier caliper and micrometer caliper.

Figure 1: Vernier Caliper

Figure 2: Micrometer Caliper

Image 1 and 2. The sphere was measured using the micrometer caliper

Image 3. The vernier caliper is used to measure the sphere

VII.Results and Discussion

The diameter of the sphere (in cm) measured using a foot rule, a vernier caliper and a micrometer caliper has been summarized in Table 1, Table 2, and Table 3 respectively. Ten trials were executed in order to get precise measurements using the instruments.

density= massvolume

Trial Foot Rule

1 1.60cm 1.60cm

2 1.50cm 1.50cm

3 1.60cm 1.60cm

4 1.60cm 1.60cm

5 1.50cm 1.50cm

6 1.20cm 1.20cm

7 1.50cm 1.50cm

8 1.70cm 1.70cm

9 1.50cm 1.50cm

10 1.40cm 1.40cm

Mean Diam. 1.51cm

a.d. 0.09cm

A.D. 0.03cm

Vol. 1.80cm3

Mass 16.27g

Exp. Val. 9.04(g/cm3)

Acc. Val. 7.8(g/cm3)

% Error 15.71%

Table 1. Diameter of Sphere using Foot Rule

Trial Vernier Caliper

1 1.580cm 1.580cm

2 1.585cm 1.585cm

3 1.585cm 1.585cm

4 1.575cm 1.575cm

5 1.580cm 1.580cm

6 1.585cm 1.585cm

7 1.585cm 1.585cm

8 1.580cm 1.580cm

9 1.580cm 1.580cm

10 1.580cm 1.580cm

Mean Diam. 1.582cm

a.d. 0.003cm

A.D. 0.0009cm

Vol. 2.073cm3

Mass 16.27g

Exp. Val. 7.848(g/cm3)

Acc. Val. 7.8(g/cm3)

% Error 0.62%

Table 2. Diameter of Sphere using Vernier Calliper

Table 3. Diameter of Sphere using Micrometer Calliper

Varying precision can be seen from the results. The least comes from measurements taken using the foot rule which show products ranging from 1.40 cm to 1.70 cm, making a hypothetical difference of 0.30 cm. The greatest precision is shown by measurements using the micrometer calliper, with results ranging from 1.583 cm to 1.584 cm, making its difference a hypothetical 0.001 cm. The vernier calliper follows next with its measurements ranging from 1.575 to 1.585, showing a hypothetical difference of 0.010. This confirms/exhibits that a relationship can be established with precision and least count: precision of an instrument is limited by the smallest division on the measurement scale. Aside from mentioned, observable is the differing accuracy of each instrument. However, the same instrument still yielded the same type of result. Least accurate came from measurements using the foot rule, with a 0.03 cm as average deviation of the mean and a 15.71% error. Most accurate amongst is the

Trial Micrometer Caliper

1 1.584cm 1.584cm

2 1.584cm 1.584cm

3 1.584cm 1.584cm

4 1.583cm 1.583cm

5 1.584cm 1.584cm

6 1.583cm 1.583cm

7 1.583cm 1.583cm

8 1.583cm 1.583cm

9 1.584cm 1.584cm

10 1.583cm 1.583cm

Mean Diam. 1.8535cm

a.d. 0.0005cm

A.D. 0.0002cm

Vol. 2.0789cm3

Mass 16.27g

Exp. Val. 7.8259(g/cm3)

Acc. Val. 7.8(g/cm3)

% Error 0.33%

micrometer calliper, which measurements gave a 0.0002 cm average deviation of the mean diameter and only 0.33% error. Next to this, the vernier calliper also exhibited good accuracy with a 0.0009 cm average deviation of the mean and a 0.62% error. This presents that there is an existing a relation between least count, significant figures and accuracy. Accuracy of an instrument depends on how well its performance is compared to a currently accepted value. More number of significant figures ensures definite value, which can be the basis of an instrument’s performance In order to have more significant figures present, the smaller the least count it, the better. The data gathered also show that precision and accuracy share a relationship with each other. Measurements can be accurate and, if so, will most often be precise to each other. However, because accuracy does depend of the instruments performance and not by smallest divisions, it is not necessary that measurements will be precise and also be accurate. Limitations to precision and accuracy may be attributed to errors encountered during the execution of the experiment. One of those encountered by the group is inexperience with the instruments. Another could be unconscious carelessness by the members. Worse of all is the false readings or measurements from a broken instrument because of a missing yet very vital part, such as a missing Rachet on a micrometer. Table 4 shows the width (in inches) of the thumb of the group members using a foot rule.

By contrast, it can be observed that most members have similar sizes of thumb. However a member does have a slightly smaller thumb. This illustrates that using body parts cannot be used as a standard as it breaks constancy. Constancy is essential to achieve standard measuring.

VIII. Conclusion

Error is always present in measurement. This may be because of (1) naturally unpredictable fluctuations in the readings of a measurement apparatus or in the experimenter's interpretation of the instrumental reading, known as random error and/or because of (2) imperfect calibration of measurement instruments or imperfect methods of observation, or interference of the environment with the measurement process, and always affect the results of an experiment in a predictable direction, known as systematic error. The former cannot be determined, though the latter can be predicted and easily be remediated. After 10 trials of measuring the metallic sphere with each instrument, the mean diameter gathered from the foot rule, vernier caliper and micrometer caliper are 1.51 cm, 1.582 cm and 1.5835 cm respectively. The densities derived are as follows: 9.04 g/cm3

using data from the foot rule, 7.848 g/cm3 from the vernier caliper and 7.8259 g/cm3 from the micrometer caliper. With the earlier data, the average deviation derived from all 10 trails using the foot rule, vernier caliper, and micrometer caliper are 0.09, 0.003, 0.0005 respectively, while average deviation of the mean diameter derived from the measurements from the foot rule, vernier caliper and micrometer caliper are 0.03, 0.0009, 0.0002 respectively. By comparison, the micrometer caliper is the most accurate with the availability of more number of significant figures expressed, yielding a smaller average deviation and % error. Measurements using the vernier caliper

Group Member

1 2 3 4

Thumbwidth

0.6 0.6 0.6 0.5

showed less % error than the foot rule, making it the least accurate amongst the three instruments.

VII. Application

Post Laboratory Questions:1. Which among the three measuring devices give you the least % error? Is the accuracy of a measurement affected by the least count of the measuring device? Superlative amongst the three is the Micrometer Caliper that gave only a 0.33% error. Taking this to account, the group affirms that the least count of the measuring device does affect the measurement. Following the principle of errors caused by limitation, the smaller the least count is (hence forth, smaller limitations), the smaller % error there would likely be.

2. What do you mean by error? What are the types of error? What are the errors you encountered in this experiment? Error is referred in Physics as a disparity or deviation between a measurement and the accepted or true value. Errors are classified into two (2) types: (1) systematic error, which are due to the limitations of the measuring instruments and the skill of carefulness of the experimenter, and (2) random error, which are caused by external factors beyond the control of the experimenter such as vibrations, noise, changes in atmospheric pressure and friction. The group did encounter a few errors such as parallax error, and the inexperience using the instruments. However, the group had a harder time resolving the dilemma with the measuring instrument since the first one the group was able to use was a broken micrometer with a missing rachet.

3. Sketch a.) a vernier caliper that reads 5.08cm b.) a micrometer caliper that reads 2.55 mm

(answers are attached at the back)

4. A student weighed himself using a bathroom scale calibrated in kilograms. He reported his weight in pounds. What is his percentage error in his reported weight if he used this conversion: 1 kg = 2.2 pounds? The standard kilogram is equal to 2.2046 pounds.

[ 2.2 pounds−2.2046 pounds2.2046 pounds ] x 100=0 . 21 %

error 5. In an experiment on determination of mass of a sample, your group consisting of 5 students obtained the following results: 14.43g, 14.32g, 14.33g, 14.30g, and 14.23g.

a. i. find the mean:

¿ 14.34 g+14.32 g+14.33 g+14.30 g+14.23 g5

= 71.52g

5

= 14.304 g

ii. find the a.d.:14.304 – 14.34 0.036

14.304 – 14.32 0.016

14.304 – 14.33 0.026

14.304 – 14.30 0.00414.304 – 14.23 /-0.074/

0.036+0.016+0.026+0.004+¿−0.074 /¿5

¿

0.1565

=0 . 0312 g

iii. find the A.D.:

0.0312

√5= 0.0139 g

b. Suppose that your group is required to make only four determinations for the mass of the sample. If you are the leader of the group, which data will you omit? Given the option, each of the members of the group would like to omit 14.23 g because it is not as precise as with the other data.

c. Recalculate the mean, a.d. and A.D. without this data.i. Mean:

¿ 14.34 g+14.32 g+14.33 g+14.30 g4

= 57.29 g

4

= 14.32 gii. a.d.:

14.304 – 14.34 0.036

14.304 – 14.32 0.016

14.304 – 14.33 0.026

14.304 – 14.30 0.004

0.036+0.016+0.026+0.0044

0.054

=0 . 0125 g

iii. A.D.0.0125

√4= 0.00625g

d.Which results will you prefer? The result without 14.23 g is preferred by the group because the results yield a smaller deviation, and perhaps a smaller % error.

VIII. Reference:

Edmonds, Dean S. Jr, Cioffari’s Experiments in College Physics, Massachusetts: D.C. Health and Company, 1988