Download Lab 1: Analysis of Experimental Data: Statistics and Graphing Lab

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.