EML3012C – Lab #1
Measurement, Instrumentation, Statistics and Error

Group 1A

Lab Performed: 9-5-2012

Report Submitted: 10-11-2012

Table of Contents:

I. Motive…………………………………………………………….…………….iii

II. Experimental…………………………………………………………………..iv

III. Results/Discussion ………………….………………….…………..…….v-viii
Part 1 Data…………………………………………..………………………….......v
Part 1 Histogram……………………………………...……………………………vi
Part 1 Calculations…………………………………...………………………….v-vi
Part 2 Data summary……………………………………..…………………….…vii
Part 2 Calculations……………………………………….………………………viii

IV. Conclusion…………………………………………………………………….ix

V. Appendix…………………….……………………………………………...x-xii

I. Motive:

The purpose of this lab is to analyze the error and deviation of manmade and manufactured objects. Measuring the marbles 100 times gives a population. The block’s dimensions (length, width, height) were measured 20 times each. This gives a sample for each dimension. From the population and samples, a histogram can be made of the data. Additionally, from these mean, mode, median, and standard deviation can be calculated. Lastly, the error and error propagation must be included because there is human and instrumental error.

II. Experimental:

This lab had two parts. For the first part, 100 glass spheres were measured. The A spheres were used. In the second part, the dimensions of block #15 were measured. A caliper was used to measure the spheres and the block. The caliper labeled C-MECH 019 with ILE of 0.001in. was used. There were many measurements taken, therefore, some data maybe inaccurate (due to human error). All units are in inches.

III. Results/discussion:

Part 1:

After measuring the marbles, the data was processed in Excel (see Table 3). Useful statistical information, Table 1, was found to use in analysis.

Table 1- Statistical data from 100 marble measurements
|Min |0.515 |
|Max |0.565 |
|Range |0.050 |
|Population(N) |100 |
|# of Bins |10 |
|Bin Size |0.005 |

This information is necessary to form a histogram (Figure A) of the raw data (Table 1). Information from Table 1 was used to formulate Table 2. This data (Table 2) is represented graphically in Figure A. This histogram shows the measurements of the marbles and should look more closely to a normal distribution (standard deviation curve). Looking at Figure A, it can be seen that the marbles are not exactly the same and there is error. The histogram is a graphical method to show there is deviation in the marble measurements.

Table 2- Data for Histogram (Figure A)
|Bin |Low |High |Frequency |
|1 |0.515 |0.520 |7 |
|2 |0.520 |0.525 |17 |
|3 |0.525 |0.530 |18 |
|4 |0.530 |0.535 |14 |
|5 |0.535 |0.540 |16 |
|6 |0.541 |0.546 |7 |
|7 |0.546 |0.551 |4 |
|8 |0.551 |0.556 |7 |
|9 |0.556 |0.561 |7 |
|10 |0.561 |0.566 |3 |

[pic]
Figure A: Histogram of the diameters of 100 spheres

The mean for this population can be calculated using equation (ii):
[pic]

This value is simply the average of the population and will be used in other statistical analysis. In this case, the mean was calculated using Excel and was found to be 0.5357. The mode is simply the value of a population that occurs most frequency. From the histogram, it can be seen that the mode is between 0.530 and 0.525. From excel, the mode is calculated as 0.535. Standard deviation for the population was found using the equation (iv):
[pic]
This value represents the difference from one measurement to the next. Using excel, the standard deviation was found to be 0.012. Limit error (LE) is found by adding Statistical Limit Error(SLE) and the Instrumental Limit Error (ILE). The ILE is 0.001.

The SLE is found using equation (vi):
[pic]
From this the SLE is calculated to be 0.003699 and the LE is 0.004699. All of the previous steps have led to a final diameter with the appropriate error. For this instance, the calculated mean of the population and calculated were used. The diameter (D) of the 100 spheres can be expressed as: D = 0.5357 ± 0.004699 in

Part 2:

The mean of each sample (length, width, and height) can be found using equation (i):
[pic]
This value represents the average value in each sample. From the above eqauation, standard deviation can be found for each of the samples. Equation (iii) was used to find all sample standard deviation: [pic]
Notice that equation (iii) is very similar to equation (iv). The above equation can only be used for a sample less than 30.. The mean of each sample (length, width, and height) can be found using equation (iii). When calculating the Limit error of each value, the Instrumental Limit Error was added to the Statistical Limit Error of each value (equation xiii).
[pic]
The ILE was 0.001. The results from these equations for each sample is summarized in Table 5 below Table 5- Summary of calculated data
|Mean (L): |3.0321 |
|Standard Deviation (L): |0.001774082 |
|SLE (L): |0.001190091 |
|LE (L): |0.002190091 |
|Mean (W): |0.86225 |
|Standard Deviation (W): |0.001681947 |
|SLE (W): |0.001128285 |
|LE (W): |0.002128285 |
|Mean (H): |0.62055 |
|Standard Deviation (H): |0.001468081 |
|SLE(H): |0.000984819 |
|LE (H): |0.001984819 |

The volume of the block is found by multiplying all of the dimensions. For this experiment, the mean values of the dimensions were used and Volume of block is 1.6228 (in3). The propagation of error in calculating the volume is found by using the Root Sum of Squares (V equation (ix):
[pic]
(where ( for each dimension is ILE and LE for each dimension respectively)
Error propagation is important to consider because there are several things being used together that have error. Error propagation represents the error building on error. This lab, for example, has error in calculating the block volume because error exists in the instruments. The error propagation in V (ILE) is 0.0041 and the error propagation in V (LE) is 0.002. Once the mean and the error propagation in the volume have been found, the final volume (using error propagation in volume due to LE)is: V = 1.6228 ± 0.002.

III. Conclusion:

It can be concluded from this experimental procedure, there is no such thing as perfectly the same man made objects. There will always be error. There was a high varience in the spheres and (from histogram and other data) a lot of error exists. Both parts have human error due to hand measuring. It can also be seen how the error in the dimensions of the block affect the error in the volume of the block. There is a high difference between the error calculated using ILE and the LE (sum of ILEand SLE).

IV. Appendix:

Table 3 – Measurments of 100 marbles
|Diameters of Spheres (in.) |
|0.559 |0.527 |0.535 |0.535 |0.559 |
|0.522 |0.54 |0.515 |0.537 |0.54 |
|0.55 |0.521 |0.54 |0.557 |0.526 |
|0.535 |0.546 |0.535 |0.527 |0.541 |
|0.517 |0.548 |0.536 |0.53 |0.54 |
|0.553 |0.538 |0.561 |0.558 |0.525 |
|0.545 |0.53 |0.555 |0.532 |0.552 |
|0.538 |0.533 |0.53 |0.534 |0.535 |
|0.525 |0.526 |0.534 |0.54 |0.565 |
|0.522 |0.545 |0.539 |0.528 |0.536 |
|0.532 |0.555 |0.541 |0.529 |0.526 |
|0.526 |0.518 |0.545 |0.52 |0.521 |
|0.518 |0.539 |0.525 |0.559 |0.525 |
|0.521 |0.547 |0.535 |0.551 |0.532 |
|0.525 |0.543 |0.562 |0.524 |0.536 |
|0.526 |0.527 |0.559 |0.523 |0.555 |
|0.532 |0.52 |0.539 |0.526 |0.53 |
|0.53 |0.525 |0.531 |0.521 |0.537 |
|0.525 |0.519 |0.557 |0.53 |0.527 |
|0.54 |0.525 |0.541 |0.552 |0.521 |

Table 4 – Measurments of Block #15
| |Block Dimensions (in) | |
|Length |Width |Height |
|3.033 |0.865 |0.62 |
|3.036 |0.864 |0.621 |
|3.031 |0.863 |0.62 |
|3.034 |0.861 |0.622 |
|3.032 |0.862 |0.621 |
|3.031 |0.861 |0.62 |
|3.036 |0.86 |0.618 |
|3.03 |0.863 |0.618 |
|3.032 |0.862 |0.622 |
|3.03 |0.861 |0.621 |
|3.031 |0.862 |0.622 |
|3.032 |0.86 |0.622 |
|3.031 |0.86 |0.622 |
|3.031 |0.863 |0.621 |
|3.032 |0.865 |0.618 |
|3.031 |0.861 |0.621 |
|3.032 |0.865 |0.622 |
|3.034 |0.864 |0.621 |
|3.033 |0.862 |0.621 |
|3.03 |0.861 |0.618 |

Equations:
(i)
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(ii)
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(iii)
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(iv)
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(v)
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(vi)
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(vii)
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(viii)

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(ix)
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