Lab 1 Sources and Variation of Errors(1) PDF

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AEM 251: Mechanics of Materials Laboratory

Dr. M. E. Barkey Department of Aerospace Engineering and Mechanics The University of Alabama

Lab 1: Sources of Variation and Error

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A. Introduction and Objective Every test has at least two sources of variation that affect the results of the test. The first source of variation is due to the experimental procedure, such as using two different testing machines that have different calibrations, or different observers reading the same equipment differently. This type of variation is often called the experimental error. The second source of variation is inherent in the specimens (or sample population) themselves. In other words, no two specimens are exactly alike. The exact amount of this variation can be very difficult to determine, particularly if there is a large amount of experimental error. In any experiment, it is important to quantify both types of variation and to determine if the experimental error is more significant than the specimen variation. You are expected to address these issues in the conclusion of every lab report that you complete.

B. Theory 1. Distribution Plots A distribution plot shows the number of occurrences for some particular quantity. For example, a company that makes breakfast cereal may intend to fill each box with 500 grams of cereal. However, due to manufacturing variation, some boxes are filled with more or less than 500 grams of cereal. A certain number of cereal boxes can be tested, and then a distribution plot could be constructed based on the mass of the contents of each box of cereal. The plot has the mass on the x-axis and the number of occurrences for a range of mass on the y-axis (Figure 1). The type of distribution shown in Figure 1 is a discrete distribution of a finite sample population (size) because the x-axis (the mass) was broken down into 5gram intervals and the number of boxes tested was 6180—a finite number.

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Figure 1: Distribution Plot for a Discrete Sample Population

2. Measures of Variance

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The average, deviance, variance, and standard deviation are used to describe variation in a sample population. For a finite population, the average or mean is given by n

´x =

1 ∑x n i=1 i

where n is the number of the population and x i is the value of the ith sample. The deviation is the difference from the average for a certain sample: ith deviation=( x i−x´ )

If all of the deviations are small, then there is a relatively small amount of variation in the sample population. The deviation for a particular member of the sample population can be positive or negative, depending on which side of the mean it lies. A physical interpretation of the variation of a particular value is the signed distance from the mean of a particular member of the sample population. The sample variance of a finite population,

s

2

, is given by

n

s 2=

2 ( x i−´x ) ∑ i=1

n−1

The positive square root of the sample variance is the sample standard deviation. In the sample variance, the distance from the mean is squared to remove the negative sign. Extending the physical interpretation, the numerator in the expression is the summation of square areas whose sides are of the length given by the deviation for each particular member of the sample population. The denominator is one less than the total number of samples, because the sample population is finite. Two sample populations might have the same mean, but different values of the sample variance and standard deviation. This indicates that the dispersion of the distributions are different. 3. Continuous Distributions M.E.Barkey

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Often, these discrete distributions are represented by a continuous function, as in Figure 2. Continuous functions are useful because they are typically easier to work with than discrete distributions.

Figure 2: Discrete Distribution Represented by a Continuous Function

The continuous distribution shown in Figure 2 is a normal or Gaussian distribution. The tails of the continuous distributions typically extend to plus and minus infinity. Normal distributions have been demonstrated to be very useful in representing the distribution of a number of physical characteristics such as the variation of material yield strength, the distribution of measurement error, and even the distribution of exam grades in class. In the latter case, the distribution is often referred to as a bell- curve.

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Not all useful distributions can be represented by the normal distribution. In some cases, the data from the discrete population may be skewed to one direction or another. However, different types of continuous distributions can be used in these cases. Examples of different distributions are shown in Figure 3.

Figure 3: Normal and Skewed Distributions

It is important to note that you should not assume a particular distribution can be represented by the normal distribution unless there is evidence or prior experience to support that assumption. You will learn more about probability distributions if you take a course in probability and statistics.

C. Lab Procedure M.E.Barkey

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Obtain an aluminum can. Note any obvious imperfections. Place the aluminum can into the compression testing machine. The testing machine should be zeroed prior to testing. Record the crush load. Repeat the experiment as specified by the instructor. (Add/Remove Tables if necessary) Specimen Number Load (lbs.) Specimen Number Load (lbs.) Specimen Number Load (lbs.) Specimen Number Load (lbs.)

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D. Lab Report 1. Calculations Complete the columns in order in the table below to compute the sample mean, specimen variation, sample variance, and sample standard deviation. (You may fill in these columns or print out similar columns using a spreadsheet program.) Be certain to use the appropriate number of significant digits.

Number of samples,

x i (Crush Load)

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¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ n=¿ ¿ (x i− ´x ) (Deviation from Mean)

2 ( x i−´x ) (Squared

Deviation)

Page |9 ¿ ¿ ¿ x ¿ ¿ ¿ (¿ ¿ i− x´ )=¿ ¿ i=1 x ¿ ¿ ¿ (¿¿ i−´x )2 =¿ ¿ n 1 ´x = ∑ x i=¿ ¿ i=1 ¿ n ¿ n ¿ n i=1 ¿ ¿ ¿ ¿ ¿ ¿ Average value ( ¿ mean value ) =¿ ¿

¿ ¿ ¿ ¿ ¿ ¿ n

Summation of deviations , ∑ (x i−´x )=¿ ¿ i=1

¿ ¿ ¿ ¿ ¿ ¿ Sample variance , s 2=¿ ¿

¿ ¿ ¿ ¿ ¿ ¿ Sample standard deviation , s=¿ ¿

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2. Calculations Answer the following in complete sentences. 1. What are some sources of experimental error, and what are some sources of sample population variation for the aluminum cans?

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2. What do you estimate to be the maximum experimental error based on the sources of experimental error listed above?

3. Why is the sum of deviations zero or nearly zero? (Refer to column 2 of the table.)

4. Is the estimated experimental error greater than the sample standard deviation?

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5. What does it mean if the estimated experimental error is greater than the sample standard deviation?

6. How do your results compare with the historical data(historical data will be provided by the Instructor)?

3. Discussion of Results In every lab report that you complete, you must discuss the possible sources of variation and error in the experiment that was conducted. The discussion must include the estimated significance of the error (e.g. the load could only be determined to within ± 2.5 pounds). Reporting of results (including calculations) must be consistent with the proper number of significant digits attainable in the experiment.

Write the discussion using complete sentences. Use additional pages if necessary. M.E.Barkey

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E. References Devore, J. L. 1987. Probability and Statistics for Engineering and the Sciences. Second Edition, Brooks/Cole publishing company, Monterey, California.

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