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Lab 1: Analysis of Experimental Data: Statistics and Graphing Lab Pre-Lab Show all calculations. 1. Given the numbers: 3, 4, 5, 3, 2, 7, 6, 4, 5, 2, 3 a. Calculate mean and standard deviation. b. Create a Histogram using 4 bins with proper graphing technique. 2. Suppose a rectangular book cover measures length, L, of 9 ± 0.02 inches and width, W of 5.5 ±0.02 inches. What is the area of the book and the uncertainty of the area? 3. When properly used, micrometer calipers may be used to measure dimensions within how many inches? (a) 0.001 (b) 0.0001 (c) 0.00001 (d) 0.0025 (e) 0.025 Analysis of Experimental Data: Statistics and Graphing Lab (read Experimental Methods for Engineers, Seventh Edition, Holman, McGraw-Hill, 2001,Chapter 3 pp.48-65, 71-77, and 233-235.) Objective: This experiment will introduce the student to various statistical concepts including mean, median, standard deviation, experimental uncertainty, normal distribution, and graphing technique. The use of vernier caliper and micrometer will also be introduced. Introduction: In an experiment, a set of data points are collected from the same instrument and the mean, median and sample standard deviation are calculated. The mean calculates the average value of the data, median is the middle value of the data set, and sample standard deviation calculates the deviation from the mean value collected from the sample population. The sample standard deviation shows how widespread the collected data is from the mean. What contributes to the deviation or error in the collected data? There are two types of errors: (1) fixed/systematic/biased error will cause repeat readings to be in error by the same amount and may be due to instrument error, calibration error, loading error, or environmental error, (2) random error is caused by uncontrolled variables in the measurement process such as random electronic fluctuation in the instruments. Uncertainties are estimates of experimental errors due to the above mentioned sources. Uncertainty Analysis is a method of estimating uncertainties. It combines the uncertainties of each measurement. The following describes the mathematics of calculating uncertainty: Given the result of an experiment, R, is based on a set of independent measurements, x1, x2, …, thus, R R( x1 , x2 ,..., xn ) let R be the uncertainty in the result and 1 , 2 ,…, n be the uncertainties in the independent variables, then 2 R 2 R 2 R R 1 2 ... n x1 x2 xn 1/ 2 Based on the uncertainty analysis, the overall accuracy of the result can be improved by reducing the error from the variable that has the largest R x x Another method to demonstrate the distribution of the errors in the collected data is by using a histogram. A histogram depicts the frequency distribution using columns so that the area of each column is proportional to the number of objects in the respective category. For example, Figure 1 shows the histogram of grade distribution for ME 325 (see guidelines for plotting graph). 6 4 frequency 2 0 50 60 70 80 90 100 grade (%) Figure 1: Histogram of grade. Often, a histogram is normalized and re-plotted so that it can be compared with the Gaussian distribution curve. When the sample size is large, the error for each data point should follow a Gaussian distribution and the probability that certain data values fall within a specified deviation from the mean value can then be determined. To calculate the Gaussian or normal error distribution, the probability density function, P(x) is used: P( x) 2 2 1 e ( x xm ) /(2 ) 2 where is the sample standard deviation and xm is the mean To calculate the Normalized Histogram, the frequency in each bin is recalculated based on the following equation: number of points in bin new frequency in each bin total data points bin width By plotting P(x) over the normalized histogram, one can see how close the experimental data follows the normal distribution curve. Materials: A packet of aluminum cylinders will be provided by the instructor. The only instruments needed will be a set of VERNIER CALIPERS and a MICROMETER. Procedure: 1. Each group member will be provided with ten aluminum cylinders. 2. Use the micrometer to measure the diameter and the vernier calipers to measure the length of the cylinders. 3. Record these data. Once all of the data have been recorded, the instructor will collect them. The combined data will then be distributed to everyone in the class. Lab Report: 1. Calculate the mean and standard deviation of the diameter and length data. 2. Plot a Normalized histogram for the diameter data and another for the length data. Use nine bins for each histogram. 3. On the appropriate histogram, plot the probability density function P(x) for each set of data using the calculated mean and standard deviation from part 1. 4. Use the calculated mean and standard deviation from part 1 to determine a nominal cylinder volume and calculate the uncertainty of the cylinder volume. 5. Describe the significance of the mean and standard deviation in the report. Attach the "raw" data sheets in the appendix. 6. Discuss the comparison between the experimental results and the Gaussian density function in the histogram. 7. Discuss possible reasons for any "questionable" data points.