**Fall 2018**

**ENGR 2411 Mechanics of Materials Lab**

**Section 001**

**Lab no. 1**

**Measurements, Statistics and Uncertainty**

**Submitted to:**

**Dr. Hyunju Jeong**

**College of Engineering**

**Submitted by:**

**Name: Omar Alhallak**

**Student ID: 50509697**

**TABLE OF CONTENTS**

TITLE |
PAGE. |

LIST OF TABLES |
3 |

LIST OF FIGURES |
4 |

ABSTRACT |
5 |

INTRODUCTION |
6 |

METHODOLOGY |
6 |

RESULTS AND DISCUSSION |
7 |

CONCLUSION |
10 |

WORKS CITED |
11 |

**LIST OF TABLES:**

TABLE NO. AND TITLE |
PAGE |

Table 1. Group 1 measurements in (mm) |
7 |

Table 2. Group 2 measurements in (mm) |
8 |

Table 3. Group 3 measurements in (mm) |
8 |

Table 4. Descriptive statistics of the object (Population). |
8 |

Table 5. Descriptive statistics of the object (Sample). |
9 |

Table 6. population and mean difference. |
9 |

**LIST OF FIGURES.**

Figure No. and Title |
PAGE |

Figure 1. Object dimensions |
6 |

Figure 2. Object Measurements |

**ABSTRACT.**

This Laboratory Study is about measuring an object with five different sides (a-e) using a Vernier Caliper. The main purpose was to determine the mean, median, deviation, standard deviation, and variance. It involves the sharing of each lab member’s measurements. The laboratory study discusses how much of a difference between a sample mean and a population mean. It also discusses the difference between a sample standard deviation and a population standard deviation.

** Keywords: **Vernier, Caliper, Measurement, Population, Sample,

I- **Introduction.**

No object’s measurements are precise. This is due to the various errors that present themselves in the form of systematic or statistical errors. Errors in measurements also subsidize to the undependable consistency of objects. Because of this error, tolerances and standard deviations are made to determine the stability of an object even with the error. In this laboratory study, the use of Vernier Calipers was used to measure dimensions of the object given. The object was to be measured by two groups of four people, and one group of five people. The purpose was to document the measurements and determine or compute various quantities. Quantities such as the mean, median, deviation, standard deviation and variance values for each component were to be determined.

II- **Methodology.**

**Equipment and procedure.**

A plastic object dimensions were measured by a Vernier Caliper. The object is shown in Figure 1.

Figure 1. Object Dimensions

The dimensions were measured in an (a-e) order by each member of each group. The Vernier Calipers was used because it renders a more accurate measurement that a standard ruler. The values were recorded in Appendix A. The object was passed to other members of the group to obtain Sample values. The component was also measured by other group in order to obtain population vales. Sample and Population values are shown in Appendix A. These values were also used to calculate the mean, median, deviation of each value, standard deviation and variance values for each component.

III- **Results And Discussion.**

The exact measurements of each member of the group. And the calculations of the mean, median, variance, deviation and standard deviation for the Sample and Population were calculated using Microsoft Excel shown in the tables below.

Column1 |
Person 1 |
Person 2 |
Person 3 |
Person 4 |

a | 44.40 | 44.70 | 48.80 | 48.80 |

b | 23.00 | 20.62 | 21.40 | 22.00 |

c | 33.25 | 33.26 | 33.40 | 32.30 |

d | 22.20 | 22.30 | 22.20 | 23.32 |

e | 11.30 | 10.40 | 11.68 | 11.54 |

**Table 1. Group 1 measurements in (mm)**

**Table 2. Group 2 Measurements in mm.**

Column1 |
Person 1 |
Person 2 |
Person 3 |
Person 4 |

a | 45.68 | 45.30 | 45.18 | – |

b | 21.68 | 22.16 | 22.04 | – |

c | 32.46 | 32.50 | 32.30 | – |

d | 22.96 | 23.00 | 23.04 | – |

e | 10.80 | 10.82 | 10.73 | – |

**Table3. Group 3 Measurements in mm.**

Column1 |
Person 1 |
Person 2 |
Person 3 |
Person 4 |

a | 45.30 | 45.34 | 44.08 | 44.80 |

b | 22.50 | 21.80 | 20.20 | 21.80 |

c | 32.80 | 32.60 | 34.00 | 33.00 |

d | 23.00 | 23.08 | 23.00 | 22.80 |

e | 11.58 | 11.66 | 21.76 | 11.60 |

After entering each member’s measured values, a plot was created to show the sample and population mean.

**Table 4. Descriptive statistics of the object (Population).**

Measurement |
A |
B |
C |
D |
E |

Mean (mm) | 45.31 | 21.73 | 32.88 | 22.83 | 12.05 |

Deviation (mm) | 7.829444 | 6.51 | 4.63 | 3.41 | 18.04 |

Deviation2 (mm2) | 61.3002003086423 | 42.3801 | 21.4343278549383 | 11.6281 | 325.4416 |

Median (mm) | 45.18 | 21.80 | 32.80 | 23.00 | 11.54 |

Standard Deviation (mm) | 1.189660692 | 0.7611214 | 0.49479773 | 0.371861522 | 3.0459134 |

Variance (mm2) | 1.415292562 | 0.579305785 | 0.244824793 | 0.138280992 | 9.27758843 |

**Table 5. Descriptive statistics of the object (Sample).**

Measurement |
A |
B |
C |
D |
E |

Mean (mm) | 45.68 | 21.76 | 33.05 | 22.51 | 11.23 |

Deviation (mm) | 6.250000 | 2.98 | 1.505 | 1.63 | 1.66 |

Deviation2 (mm2) | 39.0624999999999 | 8.8804 | 2.265025 | 2.6569 | 2.7556 |

Median (mm) | 44.75 | 21.70 | 33.26 | 22.25 | 11.42 |

Standard Deviation (mm) | 2.090255168 | 1.004042 | 0.506318 | 0.545374 | 0.575152 |

Variance (mm2) | 4.369166667 | 1.0081 | 0.256358 | 0.297433 | 0.3308 |

Table 6 shows the difference between the population mean and the sample mean which was also calculated by Microsoft Excel.

**Table 6. Population and Sample Mean Percent Difference.**

Column1 |
A |
B |
C |
D |
E |

Population mean | 45.31 | 21.73 | 32.88 | 22.83 | 12.05 |

Sample Mean | 45.68 | 21.76 | 33.05 | 22.51 | 11.23 |

% Difference | -0.008099825 | -0.001378676 | -0.005143722 | 0.0142159 | 0.0730187 |

**Discussion.**

The population mean and sample mean difference varied vaguely. This was due to human error. The average of all the deviations was calculated to be 3.0424 mm. The absolute average value of all of the deviations was calculated to be 2.075 mm. The sample standard deviation is more significant than the population standard deviation because the objective was to measure the dimensions of the object given. If the objective was to construct a consistent part, then it would be more significant to use the population standard because it shows the overall consistency of the dimensions.

Unfortunately, people think that uncertainty and error of measurements are the same, but are two different things. “Uncertainty of measurements is the doubt that exists about the result of any measurement. The uncertainty of measurements tells us something about its quality whereas error is the difference between the measured value and the ‘true value’ of the object being measured” (Bell, 2001). There are two ways to estimate uncertainties, by using Type A and Type B evaluations. Type A uses statistics to estimate evaluations of uncertainty. In Type B uses past experience of the measurement to estimate evaluations of uncertainty estimates. For example, by reading the calibration certificates, manufacturer’s specifications, from published information and common sense.

IV- **Conclusion.**

In this experiment, an object was measured by two groups of four and three groups of five using Vernier Calipers. The measurements were used to find the mean, median, deviation, standard deviation, variance and ratio of standard deviation and deviation of each measument. The mean was calculated for the population and sample and was used to find the percent difference. The percent difference for each segment (a-e) varied slightly due to the numerous human errors. If the object was measured by the same Vernier Caliper for each group, the percent difference would be reduced. However, the lab was very interesting and wonderful.

Works Cited Bell, Stephanie. “A Beginners’s Guide to Uncertainty of Measurement.” 2001. Document.